Estimation of common percentile of rainfall datasets in Thailand using delta-lognormal distributions
- Published
- Accepted
- Received
- Academic Editor
- Alban Kuriqi
- Subject Areas
- Statistics, Computational Science, Natural Resource Management, Environmental Impacts, Food, Water and Energy Nexus
- Keywords
- Confidence intervals, Common percentile, Fiducial generalized confidence interval, Adjusted method of variance estimates recovery, Bayesian approaches, Simulation, Coverage probability, Average lengths
- Copyright
- © 2022 Thangjai et al.
- Licence
- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ) and either DOI or URL of the article must be cited.
- Cite this article
- 2022. Estimation of common percentile of rainfall datasets in Thailand using delta-lognormal distributions. PeerJ 10:e14498 https://doi.org/10.7717/peerj.14498
Abstract
Weighted percentiles in many areas can be used to investigate the overall trend in a particular context. In this article, the confidence intervals for the common percentile are constructed to estimate rainfall in Thailand. The confidence interval for the common percentile help to indicate intensity of rainfall. Herein, four new approaches for estimating confidence intervals for the common percentile of several delta-lognormal distributions are presented: the fiducial generalized confidence interval, the adjusted method of variance estimates recovery, and two Bayesian approaches using fiducial quantity and approximate fiducial distribution. The Monte Carlo simulation was used to evaluate the coverage probabilities and average lengths via the R statistical program. The proposed confidence intervals are compared in terms of their coverage probabilities and average lengths, and the results of a comparative study based on these metrics indicate that one of the Bayesian confidence intervals is better than the others. The efficacies of the approaches are also illustrated by applying them to daily rainfall datasets from various regions in Thailand.
Introduction
Water is necessary for humans and the planet’s ecosystem. The amount of water that is available usually depends on the amount of rain that falls. A drought occurs when an abnormally long dry period uses up available water resources. Floods occur when rain swallows up land that is usually uncovered. These natural disasters are often made worse by human action. Thailand is divided into six geographical regions. Precipitation is not distributed equally around the country. Some regions barely see rain while others get more than their share. For example, heavy rains in central and northeastern Thailand caused flooding in 2021. The droughts and floods are a common feature and poses natural disaster. The droughts and floods cannot be eradicated but has to be managed. As a result, the common percentile is more appropriate than the common variance or the common coefficient of variation. Therefore, the common percentile is used to describe the rainfall dispersion in different regions.
Distributions of life science data are often skewed and contain a relatively large number of zero observations. In some situations, it is appropriate to model such data using a log-normal distribution for the positive values together with the probability of the point mass at zero. This distribution, commonly referred to as delta-lognormal (Aitchison, 1955), is often used to analyze environmental and medical data. For example, Zhou & Tu (2000) presented interval estimation for the mean of diagnostic test charge data for older adults with depression. Rainfall amount data series often contain zero values, which makes the delta-lognormal distribution appropriate for fitting them. Maneerat, Niwitpong & Niwitpong (2020) and Maneerat & Niwitpong (2021) proposed interval estimation for daily rainfall amount data in Thailand. Yosboonruang & Niwitpong (2021) studied confidence intervals for the ratio of two independent coefficients of variation of delta-lognormal distributions and demonstrated their efficacies with rainfall amount data from Thailand.
In some cases, the tails of several distributions can be different even though their means are the same. The percentile is a measure indicating the value below which a given percentage of observations in a group falls and can be used to describe key aspects of a distribution such as the central tendency and spread. It can play a significant role in the day-to-day statistical analyses of data and, when used in certain situations, can be more useful than the mean because it can be used to interpret a specific item, individual, or unit, which is not possible with the mean. The 25th and the 75th percentiles are the first and the third quartiles, respectively, and the interquartile range (the difference between the third and the first quartiles) is often used as a measure of spread in a distribution. Furthermore, the most well-known (50th) percentile is the median, which has been applied in statistic inference for diverse areas such as household income and survival time. The 95th percentile has been used to set premiums in the insurance industry by conducting interval estimation for normal distributions (Chakraborti & Li, 2007) and to compute the event rainfall depths (Shrestha, Fang & Zech, 2014). Krishnamoorthy, Mathew & Xu (2013) presented hypothesis testing for the upper percentile of a log-normal distribution based on samples with multiple detection limits and sample-size calculations.
Extreme rainfall events can lead to flooding. The extreme rainfall events play an important role in disaster prevention and preparedness. In practice, the extreme rainfall is practically quantified by estimating the extreme rainfall quantiles via quantiles estimation of various hydrological variables through selected statistical models using measured data. However, the measured data contain inherent randomness and uncertainty, the estimated quantiles are also uncertain. Therefore, the confidence intervals are used to estimate the rainfall information for reducing uncertainty and making hydrologic analysis effective (Chen et al., 2016; Serinaldi, 2009). Several researchers have studied the percentile or quantile of the rainfall data. For instance, Chen et al. (2016) constructed the confidence intervals of extreme rainfall quantiles using Bayesian, bootstrap, and profile likelihood approaches. Serinaldi (2009) constructed the confidence intervals for quantiles with emphasis on the extreme ones. Reis & Stedinger (2005) compared the confidence intervals for quantiles with historical information using the maximum likelihood estimator and Bayesian approaches. Lu et al. (2013) presented the confidence intervals for quantiles of extreme value distribution of hydrometeorology.
Let k be number of sample cases. For $k\phantom{\rule{thinmathspace}{0ex}}\u2a7e\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2$ independent samples of independent identically distributed random variables both with the same parameter (θ), one can pool the samples from different sources for inference purposes based on this common parameter. Herein, we are concerned with estimating common parameter θ. The problem of constructing confidence intervals for the common parameter from multiple distributions has been solved by several researchers. For example, Krishnamoorthy & Lu (2003) proposed the generalized variable approach for inference on the common mean of normal distributions. Lin & Lee (2005) developed a new generalized pivotal quantity based on the best linear unbiased estimator for constructing confidence intervals for the common mean of normal distributions. Tian (2005) presented procedures for inference on the common coefficient of variation of normal distributions. Tian & Wu (2007) provided the generalized variable approach for inference on the common mean of log-normal distributions. Ye, Ma & Wang (2010) presented procedures for hypothesis testing and interval estimation for the common mean of inverse Gaussian distributions. Ng (2014) constructed a confidence interval for the common coefficient of variation of several independent log-normal samples based on the generalized confidence interval (GCI) approach. Thangjai, Niwitpong & Niwitpong (2017a) investigated a new confidence interval for the common mean of several normal distributions by using an adjusted method of variance estimates recovery (Adjusted MOVER) approach. Thangjai & Niwitpong (2017) provided novel approaches to construct confidence intervals for the common coefficient of variation of weighted two-parameter exponential distributions. Thangjai, Niwitpong & Niwitpong (2017b) studied confidence intervals for the common mean of several one-parameter exponential populations. Thangjai & Niwitpong (2018) presented confidence intervals for common variance of several one-parameter exponential populations. Thangjai, Niwitpong & Niwitpong (2020a) proposed adjusted generalized confidence intervals for the common coefficient of variation of several normal populations. Thangjai & Ruengpeerakul (2020) constructed confidence intervals for the common inverse mean of several normal populations with unknown coefficients of variation. Thangjai & Niwitpong (2020a) established confidence intervals for the common signal-to-noise ratio of several log-normal distributions. Thangjai, Niwitpong & Niwitpong (2020b) presented confidence intervals for the common coefficient of variation of several rainfall data series in Thailand.
The problem of estimating parameters of interest from both classical and Bayesian perspectives has been extensively studied. For the classical methodology, the fiducial generalized confidence interval (FGCI) and adjusted MOVER approaches have been widely used to construct confidence intervals for the parameter of interest. For example, Hannig et al. (2006) introduced an FGCI approach to estimate simultaneous confidence intervals for the ratio of means of log-normal distributions. Kharrati-Kopaei, Malekzadeh & Sadooghi-Alvandi (2013) constructed simultaneous FGCIs for the successive differences of exponential location parameters under heteroscedasticity. Zhang & Falk (2014) proposed simultaneous confidence intervals constructed by using the FGCI approach for the ratio of the means of several log-normal distributions with heteroscedastic variances and unequal group sizes. Thangjai, Niwitpong & Niwitpong (2016) estimated simultaneous confidence intervals for the differences between the coefficients of variation of several log-normal distributions based on the FGCI approach. The FGCI approach can be used to estimate the confidence interval for complex parameters, but this approach is based on the simulation data for constructing the confidence interval. Thangjai & Niwitpong (2017) proposed the adjusted MOVER approach to estimate the confidence interval for the weighted coefficients of variation of two-parameter exponential distributions. Thangjai, Niwitpong & Niwitpong (2017a) proposed the confidence intervals for the common mean of several normal populations based on the adjusted MOVER approach. Thangjai, Niwitpong & Niwitpong (2017b) proposed the adjusted MOVER approach for constructing the confidence intervals for the common mean of several one-parameter exponential populations. The adjusted MOVER approach is easy to construct the confidence interval using the exact formula, but this approach is based on the initial confidence interval for the single parameter.
For the Bayesian methodology, Thangjai, Niwitpong & Niwitpong (2020b) proposed confidence intervals for the common coefficient of variation of log-normal distributions based on a Bayesian approach. Maneerat, Niwitpong & Niwitpong (2020) constructed Bayesian credible intervals for the difference between variances of delta-lognormal distributions. Yosboonruang, Niwitpong & Niwitpong (2020) proposed Bayesian credible intervals for measuring the difference between rainfall dispersions datasets for regions in Thailand. Thangjai, Niwitpong & Niwitpong (2020c) estimated Bayesian credible intervals for the means of normal distributions with unknown coefficients of variation. Thangjai, Niwitpong & Niwitpong (2021a) applied Bayesian credible intervals for the coefficients of variation of PM10 dispersion datasets. Thangjai, Niwitpong & Niwitpong (2021b) applied the Bayesian approach for estimating the coefficients of variation of normal distributions. The Bayesian approach can be used to construct the confidence interval for complex parameters, but it is based on the prior distribution of parameter.
To the best of our knowledge, approaches based on interval estimation for the common percentile of delta-lognormal distributions have not previously been reported. The objective of this study, the confidence intervals using the FGCI, adjusted MOVER, and two Bayesian approaches for the common percentile of delta-lognormal distributions are constructed and their efficacies are compared. The confidence intervals were used to environmental model such as rainfall data. The common percentile can help to describe the 5th percentage of the heavy rainfall data in different regions.
Materials and Methods
For one population, let $V=\left({V}_{1},{V}_{2},...,{V}_{n}\right)$ be a positive random variable from a log-normal distribution with parameter μ and σ^{2}. The probability density function of V_{i} is defined as
(1) $$f({v}_{i};\mu ,{\sigma}^{2})=\{\begin{array}{cc}{\displaystyle \frac{1}{{v}_{i}\sigma \sqrt{2\pi}}\mathrm{exp}\left(-{\displaystyle \frac{1}{2{\sigma}^{2}}{\left(\mathrm{ln}({v}_{i})-\mu \right)}^{2}}\right);}& {v}_{i}>0\\ 0;& otherwise\end{array}$$
Delta-lognormal population contains both zero and non-zero observed values. $Let{\delta}^{\mathrm{\prime}}=1-\delta $ be the probability of zero observations and let δ be the probability of non-zero observations. Suppose that $n={n}_{(0)}+{n}_{(1)}$ is the sample of size, where n_{(0)} is a number of zeros and n_{(1)} is a number of non-zeros. The zero observations follow a binomial distribution, n_{(0)} ∼ $B\left(n,{\delta}^{\mathrm{\prime}}\right)$. The non-zero observations follow a log-normal distribution. Let $X=\left({X}_{1},{X}_{2},...,{X}_{n}\right)$ be random sample drawn from a delta-lognormal distribution with parameters mean μ, variance σ^{2}, and probability of obtaining a zero observation δ^{′}. Moreover, let $x=\left({x}_{1},{x}_{2},...,{x}_{n}\right)$ be the observed value of $X=\left({X}_{1},{X}_{2},...,{X}_{n}\right)$.
According to Tian & Wu (2006), the probability density function of X_{i} is given by
(2) $$G({x}_{i};\mu ,{\sigma}^{2},{\delta}^{{}^{\prime}})=\{\begin{array}{cc}{\delta}^{{}^{\prime}};& x=0\\ {\delta}^{{}^{\prime}}+\delta F({x}_{i};\mu ,{\sigma}^{2});& x>0\end{array}\phantom{\rule{thinmathspace}{0ex}},$$where $F\left({x}_{i};\mu ,{\sigma}^{2}\right)$ is the log-normal cumulative distribution function.
Let $Y=\left({Y}_{1},{Y}_{2},...,{Y}_{n}\right)$ be independent log-transformed lognormal random variables. Let $\stackrel{\mathrm{\_}}{Y}$ and S^{2} be the mean and variance based on log-transformed sample from log-normal distribution. Also, $\stackrel{\mathrm{\_}}{y}$ and s^{2} be observed values of $\stackrel{\mathrm{\_}}{Y}$_{(1)} and S^{2}, respectively. Let $\stackrel{\mathrm{\_}}{Y}$_{(1)} and S^{2}_{(1)} be the mean and variance based on the log-transformed positive sample measurements. And let $\stackrel{\mathrm{\_}}{y}$_{(1)} and s^{2}_{(1)} be observed values of $\stackrel{\mathrm{\_}}{Y}$_{(1)} and S^{2}_{(1)}, respectively.
Following Hasan & Krishnamoorthy (2018), let q_{p} be the pth quantile of the delta-lognormal distribution. It is determined by the equation $G\left({q}_{p};\mu ,{\sigma}^{2},{\delta}^{\mathrm{\prime}}\right)=p$. It follows Eq. (2) that q_{p} = 0 if p < δ^{′}. For p > δ^{′}, using the relation between the normal and log-normal distributions, the q_{p} yields
(3) $${q}_{p}=\mathrm{exp}\left(\mu +{\mathrm{\Phi}}^{-1}\left({\displaystyle \frac{p-{\delta}^{{}^{\prime}}}{1-{\delta}^{{}^{\prime}}}}\right)\sigma \right),$$where ϕ is the standard normal distribution function.
Estimation of the pth quantile of the delta-lognormal distribution simplifies to estimation of
(4) $${\lambda}_{p}=\mu +{\mathrm{\Phi}}^{-1}\left({\displaystyle \frac{p-{\delta}^{{}^{\prime}}}{1-{\delta}^{{}^{\prime}}}}\right)\sigma .$$
For k populations, for i = 1,2,…,k, let X_{i} = (X_{i1}, X_{i2}, …, X_{ini}) be a sample with n_{i(0)} zeros from a delta-lognormal distribution. Let us assume that n_{i(0)} ∼ B(n_{i},δ^{′}_{i}) so that n_{i(1)} = n_{i} − n_{i(0)} sample measurements are greater than zero. Let Y_{i} = ln(X_{i}) ∼ N(μ_{i},σ^{2}_{i}), where X_{i} > 0. Let $\stackrel{\mathrm{\_}}{Y}$_{i(1)} and S^{2}_{i(1)} be the mean and variance based on the log-transformed positive sample measurements, respectively. Also, let $\stackrel{\mathrm{\_}}{y}$_{i(1)} and s^{2}_{i(1)} be the observed values of $\stackrel{\mathrm{\_}}{Y}$_{i(1)} and S^{2}_{i(1)}, respectively.
The estimator of the pth quantile of the delta-lognormal distribution is defined by
(5) $${\hat{q}}_{{p}_{i}}=\mathrm{exp}({\hat{\lambda}}_{{p}_{i}}),$$where ${\hat{\lambda}}_{{p}_{i}}={\overline{Y}}_{i(1)}+{\mathrm{\Phi}}^{-1}\left({\displaystyle \frac{{p}_{i}-{\delta}_{i}^{{}^{\prime}}}{1-{\delta}_{i}^{{}^{\prime}}}}\right){S}_{i(1)}$.
According to Yosboonruang, Niwitpong & Niwitpong (2022), the variances of $\stackrel{\mathrm{\_}}{Y}$_{i(1)} and S^{2}_{i(1)} are
$$Var\left({\overline{Y}}_{i(1)}\right)={\left(\mathrm{exp}\left({\mu}_{i}\right)\right)}^{2}\ast \left({\displaystyle \frac{{\sigma}_{i}^{2}}{{n}_{i(1)}}+1}\right)\ast {\left(\mathrm{exp}\left({\displaystyle \frac{\left({n}_{i(1)}-1\right){\sigma}_{i}^{2}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+3{\sigma}_{i}^{2}}}\right)\right)}^{2}$$
$$\ast \left({\displaystyle \frac{2\left({n}_{i(1)}-1\right)\left({\sigma}_{i}^{4}/{n}_{i(1)}\right)}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+\left(6{\sigma}_{i}^{4}/{n}_{i(1)}\right)}+1}\right)$$
(6) $$-{\left(\mathrm{exp}\left({\mu}_{i}\right)\ast \mathrm{exp}\left({\displaystyle \frac{\left({n}_{i(1)}-1\right){\sigma}_{i}^{2}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+3{\sigma}_{i}^{2}}}\right)\right)}^{2}$$and
(7) $$Var\left({S}_{i(1)}^{2}\right)=\left({\displaystyle \frac{2{\sigma}_{i}^{2}}{{n}_{i(1)}}\ast \mathrm{exp}\left(4{\mu}_{i}+2{\sigma}_{i}^{2}\right)}\right)\ast \left(2{\left(\mathrm{exp}\left({\sigma}_{i}^{2}\right)-1\right)}^{2}+{\sigma}_{i}^{2}{\left(2\mathrm{exp}\left({\sigma}_{i}^{2}\right)-1\right)}^{2}\right).$$
The variance of ${\hat{\lambda}}_{{p}_{i}}$ is
(8) $$Var({\hat{\lambda}}_{{p}_{i}})={\hat{\lambda}}_{{p}_{i}}^{2}(Var({\overline{Y}}_{i(1)})+{z}_{p}^{2}Var({S}_{i(1)}^{2})),$$where z_{p} is the 100p percentile of the standard normal distribution and Var( $\stackrel{\mathrm{\_}}{Y}$_{i(1)}) and Var(S^{2}_{i(1)}) are defined in Eqs. (6) and (7), respectively.
Therefore, the weighted average of ${\hat{\lambda}}_{{p}_{i}}$ based on k individual sample is defined by
(9) $${\hat{\lambda}}_{p}=\sum _{i=1}^{k}{\displaystyle \frac{{\hat{\lambda}}_{{p}_{i}}}{Var\left({\hat{\lambda}}_{{p}_{i}}\right)}/\sum _{i=1}^{k}{\displaystyle \frac{1}{Var\left({\hat{\lambda}}_{{p}_{i}}\right)},}}$$where $Var\left({\hat{\lambda}}_{{p}_{i}}\right)$ is defined in Eq. (8).
The common quantile of the delta-lognormal distributions is defined by
(10) $$\hat{\theta}=\mathrm{exp}({\hat{\lambda}}_{p}),$$where ${\hat{\lambda}}_{p}$ is defined in Eq. (9).
FGCI approach
The concept of FGCI uses a fiducial generalized pivotal quantity (FGPQ) which is a subclass of GPQ.
Definition: Let $Y=\left({Y}_{1},{Y}_{2},...,{Y}_{n}\right)$ be a random variable with the probability density function $f\left(y;\theta \right),$ where $\theta =\left(\mu ,{\sigma}^{2},{\delta}^{\mathrm{\prime}}\right)$ is a vector of unknown parameters. Let $y=\left({y}_{1},{y}_{2},...,{y}_{n}\right)$ be the observed value of $Y=\left({Y}_{1},{Y}_{2},...,{Y}_{n}\right)$ . Let $R\left(Y;y,\theta \right)$ be a function of Y, y and θ. The R(Y; y,θ) is called the FGPQ if the following two conditions are satisfied (Weerahandi, 1993; Hannig et al., 2006):
1. For a given Y = y, the conditional distribution of R(Y; y,θ) is free of the nuisance parameter.
2. For every y, the observed value of $R\left(Y;y,\theta \right)$ is the parameter of interest.
The FGPQ for $\eta =\left(p-{\delta}^{\mathrm{\prime}}\right)/\left(1-{\delta}^{\mathrm{\prime}}\right)$ can be found by replacing δ^{′} by its FGPQ.
Let ${R}_{\mu},{R}_{\sigma},and\phantom{\rule{thinmathspace}{0ex}}{R}_{\eta}$ be FGPQs for μ, σ, and η, respectively. According to Hasan & Krishnamoorthy (2018), the FGPQ for λ_{p} can be expressed as follows:
$${R}_{{\lambda}_{p}}={R}_{\mu}+{\mathrm{\Phi}}^{-1}({R}_{\eta}){R}_{\sigma}$$
$$={\overline{y}}_{(1)}+{\displaystyle \frac{Z}{{U}_{(1)}}{\displaystyle \frac{{s}_{(1)}}{\sqrt{{n}_{(1)}}}+{\mathrm{\Phi}}^{-1}({R}_{\eta}){\displaystyle \frac{{s}_{(1)}}{{U}_{(1)}}}}}$$
(11) $$={\overline{y}}_{(1)}+{\displaystyle \frac{{s}_{(1)}}{\sqrt{{n}_{(1)}}}\left({\displaystyle \frac{Z+{\mathrm{\Phi}}^{-1}({R}_{\eta})\sqrt{{n}_{(1)}}}{{U}_{(1)}}}\right),}$$where $Z\sim N\left(0,1\right)$ and ${U}_{(1)}\sim \sqrt{{\chi}_{{n}_{(1)}-1}^{2}/({n}_{(1)}-1)}$.
Let R_{δ′} be the FGPQ for δ^{′}. It can be generated by generating from uniform (0, 1). The fiducial quantity R_{η} must be in the interval (0, 1), or equivalently,
(12) $$0<{\displaystyle \frac{p-{R}_{{\delta}^{{}^{\prime}}}}{1-{R}_{{\delta}^{{}^{\prime}}}}<1.}$$
To estimate the percentiles of R_{λp} , it is enough to estimate the percentiles of the term within parentheses in Eq. (11). Therefore, the 100pth of the delta-lognormal distribution is (1 − α) upper confidence limit for exp(λ_{p}).
Let q_{pi} be the pth quantile of the delta-lognormal distribution, where i = 1,2,…,k. It is determined by the equation G(q_{pi} ; μ_{i},σ^{2}_{i},δ^{′}_{i}) = p_{i}. Thus, q_{pi} = 0 if p_{i} < δ^{′}_{i}. For p_{i} > δ^{′}_{i}, the q_{pi} is given by
(13) $${q}_{{p}_{i}}=\mathrm{exp}\left({\mu}_{i}+{\mathrm{\Phi}}^{-1}\left({\displaystyle \frac{{p}_{i}-{\delta}_{i}^{{}^{\prime}}}{1-{\delta}_{i}^{{}^{\prime}}}}\right){\sigma}_{i}\right)=\mathrm{exp}\left({\lambda}_{{p}_{i}}\right),$$where i = 1,2,…,k.
The FGPQ for λ_{pi} can be expressed as follows:
(14) $${R}_{{\lambda}_{{p}_{i}}}={\overline{y}}_{i(1)}+{\displaystyle \frac{{s}_{i(1)}}{\sqrt{{n}_{i(1)}}}\left({\displaystyle \frac{{Z}_{i}+{\mathrm{\Phi}}^{-1}\left({R}_{{\eta}_{i}}\right)\sqrt{{n}_{i(1)}}}{{U}_{i(1)}}}\right),}$$where Z_{i} ∼ N(0,1), ${U}_{i(1)}\sim \sqrt{{\chi}_{{n}_{i(1)}-1}^{2}/({n}_{i(1)}-1)}$, and i = 1,2,…,k.
The FGPQ for σ^{2}_{i} is
(15) $${R}_{{\sigma}_{i}^{2}}={\displaystyle \frac{({n}_{i(1)}-1){s}_{i(1)}^{2}}{{\chi}_{{n}_{i(1)}-1}^{2}},}$$
where χ^{2}_{ni(1)} – 1 is chi-squared distribution with n_{i(1)} − 1 degrees of freedom.
The FGPQ for μ_{i} is
(16) $${R}_{{\mu}_{i}}={\overline{y}}_{i(1)}-{\displaystyle \frac{{Z}_{i}}{\sqrt{{U}_{i(1)}}}\sqrt{{\displaystyle \frac{({n}_{i(1)}-1){s}_{i(1)}^{2}}{{n}_{i(1)}}}},}$$where Z_{i} denotes a standard normal distribution and U_{i(1)} denotes a chi-squared distribution with n_{i(1)} − 1 degrees of freedom.
Let ${B}_{{n}_{i(0)}+0.5,{n}_{i(1)}+0.5}$ be the beta random variable with shape parameters n_{i(0)} + 0.5 and n_{i(1)} + 0.5. Suppose that $k\phantom{\rule{thinmathspace}{0ex}}{R}_{{\delta}_{i}^{{}^{\prime}}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2$ has the probability distribution of the ${B}_{{n}_{i(0)}+0.5,{n}_{i(1)}+0.5}$ random variable whose values are bounded above by p. Let ϕ is the standard normal distribution function. Let
(17) $${\mathrm{\Phi}}^{-1}({R}_{{\eta}_{i}})={\mathrm{\Phi}}^{-1}\left({\displaystyle \frac{{p}_{i}-{R}_{{\delta}_{i}^{{}^{\prime}}}}{1-{R}_{{\delta}_{i}^{{}^{\prime}}}}}\right),$$where $\phantom{\rule{thinmathspace}{0ex}}{R}_{{\delta}_{i}^{{}^{\prime}}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2$ = H ^{−1}(V_{i}H(p_{i};n_{i(0)} + 0.5,n_{i(1)} + 0.5);n_{i(0)} + 0.5,n_{i(1)} +0.5), H(y;a,b) is the beta distribution function with shape parameters a and b, and V_{i} is uniform (0,1).
Therefore, the FGPQ for λ_{pi} is
(18) $${R}_{{\lambda}_{{p}_{i}}}={R}_{{\mu}_{i}}+{\displaystyle \frac{\sqrt{{R}_{{\sigma}_{i}^{2}}}}{\sqrt{{n}_{i(1)}}}\left({\displaystyle \frac{{Z}_{i}+{\mathrm{\Phi}}^{-1}({R}_{{\eta}_{i}})\sqrt{{n}_{i(1)}}}{\sqrt{{U}_{i(1)}}}}\right),}$$where U_{i(1)} denotes a chi-squared distribution with n_{i(1)} − 1 degrees of freedom, R_{σ2i} is defined in Eq. (15), R_{μi} is defined in Eq. (16), and ϕ^{−1}(R_{ηi}) is defined in Eq. (17).
The FGPQs for $Var({\hat{\mu}}_{i})$ and $Var({\hat{\sigma}}_{i}^{2})$ are
$${R}_{Var\left({\hat{\mu}}_{i}\right)}={\left(\mathrm{exp}\left({R}_{{\mu}_{i}}\right)\right)}^{2}\ast \left({\displaystyle \frac{{R}_{{\sigma}_{i}^{2}}}{{n}_{i(1)}}+1}\right)\ast {\left(\mathrm{exp}\left({\displaystyle \frac{\left({n}_{i(1)}-1\right){R}_{{\sigma}_{i}^{2}}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+3{R}_{{\sigma}_{i}^{2}}}}\right)\right)}^{2}$$
$$\ast \left({\displaystyle \frac{{\displaystyle \frac{2\left({n}_{i(1)}-1\right){R}_{{\sigma}_{i}^{2}}^{2}}{{n}_{i(1)}}}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+{\displaystyle \frac{6{R}_{{\sigma}_{i}^{2}}^{2}}{{n}_{i(1)}}}}+1}\right)$$
(19) $$-{\left(\mathrm{exp}\left({R}_{{\mu}_{i}}\right)\ast \mathrm{exp}\left({\displaystyle \frac{\left({n}_{i(1)}-1\right){R}_{{\sigma}_{i}^{2}}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+3{R}_{{\sigma}_{i}^{2}}}}\right)\right)}^{2}$$and
(20) $${R}_{Var\left({\hat{\sigma}}_{i}^{2}\right)}=\left({\displaystyle \frac{2{R}_{{\sigma}_{i}^{2}}}{{n}_{i(1)}}\ast \mathrm{exp}\left(4{R}_{{\mu}_{i}}+2{R}_{{\sigma}_{i}^{2}}\right)}\right)\ast \left(2{\left(\mathrm{exp}\left({R}_{{\sigma}_{i}^{2}}\right)-1\right)}^{2}+{R}_{{\sigma}_{i}^{2}}{\left(2\mathrm{exp}\left({R}_{{\sigma}_{i}^{2}}\right)-1\right)}^{2}\right).$$
Therefore, the FGPQ for $Var({\hat{\lambda}}_{{p}_{i}})$ is
(21) $${R}_{Var({\hat{\lambda}}_{{p}_{i}})}={R}_{{\lambda}_{{p}_{i}}}^{2}{R}_{Var({\hat{\mu}}_{i})}+{z}_{p}^{2}{R}_{Var({\hat{\sigma}}_{i}^{2})},$$where z_{p} is the 100p th percentile of the standard normal distribution and ${R}_{{\lambda}_{{p}_{i}}}$, ${R}_{Var({\hat{\mu}}_{i})}$, and ${R}_{Var({\hat{\sigma}}_{i}^{2})}$ are defined in Eqs. (18)–(20), respectively.
The weighted average of the FGPQ of λ_{pi} based on k individual sample is given by
(22) $${R}_{{\lambda}_{p}}=\sum _{i=1}^{k}{\displaystyle \frac{{R}_{{\lambda}_{{p}_{i}}}}{{R}_{Var\left({\hat{\lambda}}_{{p}_{i}}\right)}}/\sum _{i=1}^{k}{\displaystyle \frac{1}{{R}_{Var\left({\hat{\lambda}}_{{p}_{i}}\right)}},}}$$where ${R}_{{\lambda}_{{p}_{i}}}$ is defined in Eq. (18) and ${R}_{Var\left({\hat{\lambda}}_{{p}_{i}}\right)}$ is defined in Eq. (21).
The FGPQ in Eq. (22) satisfies two conditions of the definition given above. The FGCI is constructed using the quantiles of FGPQ defined in Eq. (22). Therefore, the 100(1 − α)% two-sided confidence interval for the weighted average of λ_{pi} based on the FGCI approach is
(23) $$[{L}_{\lambda .FGCI},{U}_{\lambda .FGCI}]=[{R}_{{\lambda}_{p}}(\alpha /2),{R}_{{\lambda}_{p}}(1-\alpha /2)],$$where R_{λp} (α/2) and R_{λp} (1 − α/2) denote the 100(α/2)-th and 100(1− α/2)-th percentiles of R_{λp} , respectively.
Therefore, the 100(1 − α)% two-sided confidence interval for the common percentile θ based on the FGCI approach is
(24) $$C{I}_{FGCI}=[{L}_{FGCI},{U}_{FGCI}]=[\mathrm{exp}({L}_{\lambda .FGCI}),\mathrm{exp}({U}_{\lambda .FGCI})],$$where L_{λ.FGCI} and U_{λ.FGCI} are defined in Eq. (23).
Algorithm 1 is used to construct the FGCI:
Adjusted MOVER approach
Hasan & Krishnamoorthy (2018) proposed the confidence interval for λ_{p}. Let l_{i} and u_{i} be the lower and upper limits of the confidence interval for λ_{pi} , where i = 1,2,…,k. Following Hasan & Krishnamoorthy (2018), the lower and upper limits of the confidence interval are given by
(25) $${l}_{i}={\overline{Y}}_{i(1)}-{t}_{{n}_{i(1)}-1;1-\alpha}({z}_{\hat{\eta}}\sqrt{{n}_{i(1)}}){\displaystyle \frac{{S}_{i(1)}}{\sqrt{{n}_{i(1)}}}}$$and
(26) $${u}_{i}={\overline{Y}}_{i(1)}+{t}_{{n}_{i(1)}-1;1-\alpha}({z}_{\hat{\eta}}\sqrt{{n}_{i(1)}}){\displaystyle \frac{{S}_{i(1)}}{\sqrt{{n}_{i(1)}}},}$$where ${z}_{\hat{\eta}}$ is the 100 $\hat{\eta}$ percentile of the standard normal distribution and ${t}_{{n}_{i(1)}-1;1-\alpha}({z}_{\hat{\eta}}\sqrt{{n}_{i(1)}})$ denotes the qth quantile of the noncentral t distribution with n_{i(1)} − 1 degrees of freedom and the noncentrality parameter ${z}_{\hat{\eta}}\sqrt{{n}_{i(1)}}$.
For a given $\stackrel{\mathrm{\_}}{y}$_{i(1)}, s^{2}_{i(1)}, and n_{i(1)}, where i = 1,2,…,k |
For g = 1 to m, where m is number of generalized computations |
Generate ${\chi}_{{n}_{i(1)}-1}^{2}$ from chi-squared distribution with n_{i(1)} − 1 degrees of freedom |
Compute ${R}_{{\sigma}_{i}^{2}}$ from Eq. (15) |
Compute R_{μi} from Eq. (16) |
Compute ${R}_{{\lambda}_{{p}_{i}}}$ from Eq. (18) |
Compute ${R}_{Var({\hat{\mu}}_{i})}$ from Eq. (19) |
Compute ${R}_{Var({\hat{\sigma}}_{i}^{2})}$ from Eq. (20) |
Compute ${R}_{Var\left({\hat{\lambda}}_{{p}_{i}}\right)}$ from Eq. (21) |
Compute R_{λp} from Eq. (22) |
End g loop |
Compute ${R}_{{\lambda}_{p}}(\alpha /2)$ and ${R}_{{\lambda}_{p}}(1-\alpha /2)$ from Eq. (23) |
Compute L_{FGCI} and U_{FGCI} from Eq. (24) |
Let z_{α/2} be the (α/2)-th quantile of the standard normal distribution. According to Graybill & Deal (1959), the λ_{p} is weighted average of ${\hat{\lambda}}_{{p}_{i}}$ based on k individual samples. Therefore, the weighted average of λ_{pi} is defined by
(27) $${\hat{\lambda}}_{p}=\sum _{i=1}^{k}{\displaystyle \frac{{\hat{\lambda}}_{{p}_{i}}}{\hat{Var}\left({\hat{\lambda}}_{{p}_{i}}\right)}/\sum _{i=1}^{k}{\displaystyle \frac{1}{\hat{Var}\left({\hat{\lambda}}_{{p}_{i}}\right)},}}$$where
(28) $${\hat{\lambda}}_{{p}_{i}}={\overline{Y}}_{i(1)}+{\mathrm{\Phi}}^{-1}\left({\displaystyle \frac{{p}_{i}-{\delta}_{i}^{{}^{\prime}}}{1-{\delta}_{i}^{{}^{\prime}}}}\right){S}_{i(1)}$$and
(29) $$\hat{Var}\left({\hat{\lambda}}_{{p}_{i}}\right)={\displaystyle \frac{1}{2}\left({\displaystyle \frac{{\left({\hat{\lambda}}_{{p}_{i}}-{l}_{i}\right)}^{2}}{{z}_{\alpha /2}^{2}}+{\displaystyle \frac{{\left({u}_{i}-{\hat{\lambda}}_{{p}_{i}}\right)}^{2}}{{z}_{\alpha /2}^{2}}}}\right).}$$
Thangjai & Niwitpong (2020a) proposed the concept of adjusted MOVER approach. The lower and upper limits of the confidence interval for the weighted average of λ_{pi} based on adjusted MOVER approach are defined by
(30) $${L}_{\lambda .AM}={\hat{\lambda}}_{p}-\sqrt{\sum _{i=1}^{k}{\displaystyle \frac{{({\hat{\lambda}}_{{p}_{i}}-{l}_{i})}^{2}}{{(\hat{Var}({\hat{\lambda}}_{{p}_{{l}_{i}}}))}^{2}}/\sum _{i=1}^{k}{\displaystyle \frac{1}{{(\hat{Var}({\hat{\lambda}}_{{p}_{{l}_{i}}}))}^{2}}}}}$$and
(31) $${U}_{\lambda .AM}={\hat{\lambda}}_{p}+\sqrt{\sum _{i=1}^{k}{\displaystyle \frac{{({u}_{i}-{\hat{\lambda}}_{{p}_{i}})}^{2}}{{(\hat{Var}({\hat{\lambda}}_{{p}_{{u}_{i}}}))}^{2}}/\sum _{i=1}^{k}{\displaystyle \frac{1}{{(\hat{Var}({\hat{\lambda}}_{{p}_{{u}_{i}}}))}^{2}}}}},$$where
(32) $$\hat{Var}({\hat{\lambda}}_{{p}_{{l}_{i}}})={\displaystyle \frac{{({\hat{\lambda}}_{{p}_{i}}-{l}_{i})}^{2}}{{z}_{\alpha /2}^{2}}}$$and
(33) $$\hat{Var}({\hat{\lambda}}_{{p}_{{u}_{i}}})={\displaystyle \frac{{({u}_{i}-{\hat{\lambda}}_{{p}_{i}})}^{2}}{{z}_{\alpha /2}^{2}}.}$$
Substituting l_{i} is defined in Eq. (25) and u_{i} is defined in Eq. (26) back into Eqs. (30) and (31), the confidence interval for the weighted average of λ_{pi} based on the adjusted MOVER approach is obtained.
Therefore, the 100(1 − α)% two-sided confidence interval for the common percentile θ based on the adjusted MOVER approach is also obtained by
(34) $$C{I}_{AM}=[{L}_{AM},{U}_{AM}]=[\mathrm{exp}({L}_{\lambda .AM}),\mathrm{exp}({U}_{\lambda .AM})],$$where L_{λ.AM} and U_{λ.AM} are defined in Eqs. (30) and (31), respectively.
Bayesian approach
Here, the Bayesian approach is used to construct two confidence intervals for the common percentile of delta-lognormal distributions. First, the Bayesian confidence interval is constructed by using fiducial quantity. Second, the Bayesian confidence interval is constructed by using an approximate fiducial distribution.
Bayes (1763) introduced the Bayesian approach. The Bayesian approach uses the posterior distribution for constructing the Bayesian confidence interval. The posterior distribution is a conditional distribution and uses to make statements about the parameter.
First, the Bayesian confidence interval based on fiducial quantity is considered. According to Thangjai, Niwitpong & Niwitpong (2020b), the posterior distribution for ${\sigma}_{i}^{2}$ is inverse gamma distribution. It is defined by
(35) $${\sigma}_{i}^{2}|{y}_{i}\sim IG({\displaystyle \frac{{n}_{i(1)}-1}{2},{\displaystyle \frac{({n}_{i(1)}-1){s}_{i(1)}^{2}}{2}).}}$$
The conditional posterior distribution for ${\mu}_{i}$ given ${\sigma}_{i}^{2}$ and y_{i} is the normal distribution. It is defined by
(36) $${\mu}_{i}|{\sigma}_{i}^{2},{y}_{i}\sim N({\hat{\mu}}_{i},{\displaystyle \frac{{\sigma}_{i}^{2}}{{n}_{i(1)}}),}$$where ${\hat{\mu}}_{i}={\overline{y}}_{i(1)}$ and ${\sigma}_{i}^{2}$ is defined in Eq. (35).
Let ${B}_{{n}_{i(0)}+0.5,{n}_{i(1)}+0.5}$ be the beta random variable with shape parameters n_{i(0)} + 0.5 and n_{i(1)} + 0.5. Also, let Q_{δ′i} be the probability distribution of the ${B}_{{n}_{i(0)}+0.5,{n}_{i(1)}+0.5}$ random variable. Suppose that
(37) $${Q}_{{\eta}_{i}}={\displaystyle \frac{{p}_{i}-{Q}_{{\delta}_{i}^{{}^{\prime}}}}{1-{Q}_{{\delta}_{i}^{{}^{\prime}}}},}$$where ${Q}_{{\delta}_{i}^{\prime}}$ = H ^{−1}(V_{i}H(p_{i};n_{i(0)} + 0.5,n_{i(1)} + 0.5);n_{i(0)} + 0.5,n_{i(1)} + 0.5), V_{i} denotes uniform(0,1), and H(p_{i} ; n_{i(0)} + 0.5,n_{i(1)} + 0.5) denotes the beta distribution function with shape parameters n_{i(0)} + 0.5 and n_{i(1)} + 0.5.
The posterior distribution of λ_{pi} is defined by
(38) $${\lambda}_{{p}_{i}}={\mu}_{i}+{\displaystyle \frac{\sqrt{{\sigma}_{i}^{2}}}{\sqrt{{n}_{i(1)}}}\left({\displaystyle \frac{{Z}_{i}+{\mathrm{\Phi}}^{-1}\left({Q}_{{\eta}_{i}}\right)\sqrt{{n}_{i(1)}}}{\sqrt{{U}_{i(1)}}}}\right),}$$where ϕ denotes the standard normal distribution function, U_{i(1)} denotes a chi-squared distribution with n_{i(1)} − 1 degrees of freedom, and ${\sigma}_{i}^{2}$ and μ_{i} are defined in Eqs. (35) and (36), respectively.
The variance of $\stackrel{\mathrm{\_}}{Y}$_{i(1)} is
$$Var\left({\overline{Y}}_{i(1)}\right)={\left(\mathrm{exp}\left({\mu}_{i}\right)\right)}^{2}\ast \left({\displaystyle \frac{{\sigma}_{i}^{2}}{{n}_{i(1)}}+1}\right)\ast {\left(\mathrm{exp}\left({\displaystyle \frac{\left({n}_{i(1)}-1\right){\sigma}_{i}^{2}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+3{\sigma}_{i}^{2}}}\right)\right)}^{2}$$
$$\ast \left({\displaystyle \frac{{\displaystyle \frac{2\left({n}_{i(1)}-1\right){\sigma}_{i}^{4}}{{n}_{i(1)}}}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+{\displaystyle \frac{6{\sigma}_{i}^{4}}{{n}_{i(1)}}}}+1}\right)$$
(39) $$-{\left(\mathrm{exp}\left({\mu}_{i}\right)\ast \mathrm{exp}\left({\displaystyle \frac{\left({n}_{i(1)}-1\right){\sigma}_{i}^{2}}{2\left({n}_{i(1)}+4\right)\left({n}_{i(1)}-1\right)+3{\sigma}_{i}^{2}}}\right)\right)}^{2},$$where ${\sigma}_{i}^{2}$ and μ_{i} are defined in Eqs. (35) and (36), respectively.
The variance of S^{2}_{i(1)} is
(40) $$Var\left({S}_{i(1)}^{2}\right)=\left(\left({\displaystyle \frac{2{\sigma}_{i}^{2}}{{n}_{i(1)}}}\right)\mathrm{exp}\left(4{\mu}_{i}+2{\sigma}_{i}^{2}\right)\right)\ast \left(2{\left(\mathrm{exp}\left({\sigma}_{i}^{2}\right)-1\right)}^{2}+{\sigma}_{i}^{2}{\left(2\mathrm{exp}\left({\sigma}_{i}^{2}\right)-1\right)}^{2}\right).$$
Therefore, the variance of ${\hat{\lambda}}_{{p}_{i}}$ is
(41) $$Var({\hat{\lambda}}_{{p}_{i}})={\lambda}_{{p}_{i}}^{2}Var({\overline{Y}}_{i(1)})+{z}_{p}^{2}Var({S}_{i(1)}^{2}),$$where z_{p} is the 100p percentile of the standard normal distribution and λ_{pi} , Var( $\stackrel{\mathrm{\_}}{Y}$_{i(1)}), and Var(S^{2}_{i(1)}) are defined in Eqs. (38)–(40), respectively.
For a given $\stackrel{\mathrm{\_}}{y}$_{i(1)}, s^{2}_{i(1)}, and n_{i(1)}, where i = 1,2,…,k |
For g = 1 to m |
Generate ${\sigma}_{i}^{2}$|y_{i} from Eq. (35) |
Generate μ_{i}| ${\sigma}_{i}^{2}$,y_{i} from Eq. (36) |
Compute Q_{ηi} from Eq. (37) |
Compute λ_{pi} from Eq. (38) |
Compute $\stackrel{\mathrm{\_}}{Y}$ from Eq. (39) |
Compute Var(S^{2}_{i(1)}) from Eq. (40) |
Compute $Var({\hat{\lambda}}_{{p}_{i}})$ from Eq. (41) |
Compute λ_{p} from Eq. (42) |
End g loop |
Compute L_{λ.BS1} and U_{λ.BS1} |
Compute L_{BS1} and U_{BS1} |
Therefore, the weighted average of λ_{pi} is defined by
(42) $${\lambda}_{p}=\sum _{i=1}^{k}{\displaystyle \frac{{\lambda}_{{p}_{i}}}{Var\left({\hat{\lambda}}_{{p}_{i}}\right)}/\sum _{i=1}^{k}{\displaystyle \frac{1}{Var\left({\hat{\lambda}}_{{p}_{i}}\right)},}}$$where λ_{pi} and $Var({\hat{\lambda}}_{{p}_{i}})$ are defined in Eqs. (38) and (41), respectively.
According to Gelman et al. (2013), the highest posterior density interval is used to construct the Bayesian confidence interval. Therefore, the confidence interval for the weighted average of λ_{pi} based on Bayesian approach using fiducial quantity are defined by
(43) $$C{I}_{\lambda .BS1}=[{L}_{\lambda .BS1},{U}_{\lambda .BS1}],$$where L_{λ.BS1} and U_{λ.BS1} are the lower and upper limits of the shortest 100(1 − α)% highest posterior density interval of λ_{p}, respectively.
Therefore, the 100(1 − α)% two-sided confidence interval for the common percentile θ based on the Bayesian approach using fiducial quantity is also obtained by
(44) $$C{I}_{BS1}=[{L}_{BS1},{U}_{BS1}]=[\mathrm{exp}({L}_{\lambda .BS1}),\mathrm{exp}({U}_{\lambda .BS1})],$$where L_{λ.BS1} and U_{λ.BS1} are defined in Eq. (43).
The Algorithm 2 is used to construct the Bayesian confidence interval using fiducial quantity:
Second, the Bayesian confidence interval using an approximate fiducial distribution is considered. The posterior distribution of λ_{pi} is defined by
(45) $${\lambda}_{{p}_{i}}={\mu}_{i}+{\displaystyle \frac{\sqrt{{\sigma}_{i}^{2}}}{\sqrt{{n}_{i(1)}}}{t}_{{n}_{i(1)}-1;1-\alpha}({z}_{\hat{\eta}}\sqrt{{n}_{i(1)}}),}$$where ${\sigma}_{i}^{2}$ and μ_{i} are defined in Eqs. (35) and (36), respectively, ${t}_{{n}_{i(1)}-1;1-\alpha}({z}_{\hat{\eta}}\sqrt{{n}_{i(1)}})$ is the (1 − α)th quantile of the noncentral t distribution with n_{i(1)} − 1 degrees of freedom and the noncentrality parameter ${z}_{\hat{\eta}}\sqrt{{n}_{i(1)}}$.
For a given $\stackrel{\mathrm{\_}}{y}$_{i(1)}, s^{2}_{i(1)}, and n_{i(1)}, where i = 1,2,…,k |
For g = 1 to m |
Generate ${\sigma}_{i}^{2}$|y_{i} from Eq. (35) |
Generate μ_{i}| ${\sigma}_{i}^{2}$,y_{i} from Eq. (36) |
Compute λ_{pi} from Eq. (45) |
Compute $\stackrel{\mathrm{\_}}{Y}$ from Eq. (39) |
Compute Var(S^{2}_{i(1)}) from Eq. (40) |
Compute $Var({\hat{\lambda}}_{{p}_{i}})$ from Eq. (46) |
Compute λ_{p} from Eq. (47) |
End g loop |
Compute L_{λ.BS2} and U_{λ.BS2} |
Compute L_{BS2} and U_{BS2} |
Therefore, the variance of ${\hat{\lambda}}_{{p}_{i}}$ is
(46) $$Var({\hat{\lambda}}_{{p}_{i}})={\lambda}_{{p}_{i}}^{2}Var({\overline{Y}}_{i(1)})+{z}_{p}^{2}Var({S}_{i(1)}^{2}),$$where z_{p} is the 100p percentile of the standard normal distribution and Var( $\stackrel{\mathrm{\_}}{Y}$_{i(1)}), Var(S^{2}_{i(1)}), and λ_{pi} are defined in Eqs. (39), (40) and (45), respectively.
Therefore, the weighted average of λ_{pi} is defined by
(47) $${\lambda}_{p}=\sum _{i=1}^{k}{\displaystyle \frac{{\lambda}_{{p}_{i}}}{Var\left({\hat{\lambda}}_{{p}_{i}}\right)}/\sum _{i=1}^{k}{\displaystyle \frac{1}{Var\left({\hat{\lambda}}_{{p}_{i}}\right)},}}$$where λ_{pi} and $Var({\hat{\lambda}}_{{p}_{i}})$ are defined in Eqs. (45) and (46), respectively.
Therefore, the confidence interval for the weighted average of λ_{pi} based on Bayesian approach using an approximate fiducial distribution are defined by
(48) $$C{I}_{\lambda .BS2}=[{L}_{\lambda .BS2},{U}_{\lambda .BS2}],$$where L_{λ.BS2} and U_{λ.BS2} are the lower and upper limits of the shortest 100(1 − α)% highest posterior density interval of λ_{p}, respectively.
Therefore, the 100(1 − α)% two-sided confidence interval for the common percentile θ based on the Bayesian approach using an approximate fiducial distribution is also obtained by
(49) $$C{I}_{BS2}=[{L}_{BS2},{U}_{BS2}]=[\mathrm{exp}({L}_{\lambda .BS2}),\mathrm{exp}({U}_{\lambda .BS2})],$$where L_{λ.BS2} and U_{λ.BS2} are defined in Eq. (48).
For a given (n_{1},n_{2},…,n_{k}), (μ_{1},μ_{2},…,μ_{k}), ${\sigma}_{1}^{2},{\sigma}_{2}^{2},\mathrm{...},{\sigma}_{k}^{2}$, ${\delta}_{1}^{\prime},{\delta}_{2}^{\prime},\mathrm{...},{\delta}_{k}^{\prime}$ and θ |
For h = 1 to M |
Generate x_{ij} from DLN(${\mu}_{i},{\sigma}_{i}^{2},{\delta}_{i}^{\prime}$), where i = 1,2,…,k and j = 1,2,…,n_{i} |
Compute y_{ij} = ln(x_{ij}), where i = 1,2,…,k and j = 1,2,…,n_{i} |
Compute $\stackrel{\mathrm{\_}}{y}$_{i(1)}, s^{2}_{i(1)}, and n_{i(1)}, where i = 1,2,…,k |
Use Algorithm 1 with m = 1,000 to construct CI_{FGCI(h)} = [L_{FGCI(h)},U_{FGCI(h)}] |
Use Eq. (34) to construct CI_{AM(h)} = [L_{AM(h)},U_{AM(h)}] |
Use Algorithm 2 with m = 1,000 to construct CI_{BS1(h)} = [L_{BS1(h)},U_{BS1(h)}] |
Use Algorithm 3 with m = 1,000 to construct CI_{BS2(h)} = [L_{BS2(h)},U_{BS2(h)}] |
If $[{L}_{(h)}\u2a7d\theta \u2a7d{U}_{(h)}]$, set p_{(h)} = 1; else set p_{(h)} = 0 |
Compute U_{(h)} − L_{(h)} |
End h loop |
Compute mean of p_{(h)} defined by the coverage probability |
Compute mean of U_{(h)} − L_{(h)} defined by the average length |
The Algorithm 3 is used to construct the Bayesian confidence interval using an approximate fiducial distribution:
Results
FGCI, adjusted MOVER, and two Bayesian approaches were used to construct four confidence intervals for the common percentile of delta-lognormal distributions denoted as CI_{FGCI}, CI_{AM}, and CI_{BS1} and CI_{BS2}, respectively. The performances of the confidence intervals were compared by computing their coverage probabilities and average lengths. The coverage probability of the 100(1 − α)% confidence level is $c\pm {z}_{\alpha /2}\sqrt{{\displaystyle \frac{c(1-c)}{M}}}$, where c is the nominal confidence level, z_{α/2} is the (α/2)-th quantile of a standard normal distribution, and M is the number of simulation runs. At the 95% confidence level, the most appropriate confidence interval will have a coverage probability in the range [0.9440,0.9560] with the shortest average length. For the FGCI and Bayesian approaches, the critical value was evaluated for 1,000 samples. The simulation results were obtained from 5,000 runs.
The Algorithm 4 is used to evaluate the coverage probability and average length:
In this simulation study, the nominal confidence level was chosen as 0.95. The variables in the simulation study were set as number of samples k = 3 or k = 6; population means μ_{1} = μ_{2} = … = μ_{k} = 1; population variances ${\sigma}_{1}^{2},{\sigma}_{2}^{2},\mathrm{...},{\sigma}_{k}^{2}$; the probabilities of obtaining zero observations ${\delta}_{1}^{\prime},{\delta}_{2}^{\prime},\mathrm{...},{\delta}_{k}^{\prime}$; and sample sizes n_{1},n_{2},…,n_{k}.
The performances of the proposed confidence intervals for k = 3 and k = 6 are given in Tables 1 and 2 and displayed in Figs. 1–6. For k = 3, the coverage probabilities of CI_{FGCI} were greater than the nominal confidence level of 0.95 for all cases whereas those of CI_{AM} were less than 0.95 when the sample sizes were small but close otherwise. For the Bayesian methods, the coverage probabilities of CI_{BS1} were greater than 0.95 for some cases whereas those of CI_{BS2} were less than 0.95 for all cases. Meanwhile, the average lengths of were shorter than those of CI_{FGCI}.
(n_{1}, n_{2}, n_{3}) | (μ_{1}, μ_{2}, μ_{3}) | (δ^{′}_{1}, δ^{′}_{2}, δ^{′}_{3}) | (σ^{2}_{1}, σ^{2}_{2}, σ^{2}_{3}) | CP (AL) | |||
---|---|---|---|---|---|---|---|
CI_{FGCI} | CI_{AM} | CI_{BS1} | CI_{BS2} | ||||
(10, 10, 10) | (1, 1, 1) | (0.1, 0.1, 0.1) | (0.5, 0.5, 0.5) | 0.9616 | 0.9188 | 0.9302 | 0.9122 |
(13.1225) | (13.5253) | (11.1999) | (16.5684) | ||||
(0.5, 0.5, 1) | 0.9788 | 0.9464 | 0.9614 | 0.8578 | |||
(16.7407) | (16.1035) | (14.1176) | (21.7400) | ||||
(1, 1, 1) | 0.9656 | 0.9276 | 0.9362 | 0.9108 | |||
(35.3099) | (29.9111) | (28.3798) | (49.9308) | ||||
(1, 1, 2) | 0.9794 | 0.9480 | 0.9602 | 0.8580 | |||
(46.7255) | (36.5878) | (36.8141) | (67.6506) | ||||
(2, 2, 2) | 0.9684 | 0.9238 | 0.9428 | 0.9240 | |||
(129.2134) | (84.0922) | (93.1638) | (201.9722) | ||||
(0.1, 0.1, 0.3) | (0.5, 0.5, 0.5) | 0.9626 | 0.9180 | 0.9352 | 0.8906 | ||
(14.0448) | (13.4411) | (11.9074) | (18.7175) | ||||
(0.5, 0.5, 1) | 0.9754 | 0.9248 | 0.9550 | 0.8696 | |||
(16.9407) | (15.0689) | (14.2855) | (22.2965) | ||||
(1, 1, 1) | 0.9650 | 0.9104 | 0.9366 | 0.9036 | |||
(36.3109) | (28.2836) | (28.9786) | (55.5862) | ||||
(1, 1, 2) | 0.9760 | 0.9252 | 0.9550 | 0.8694 | |||
(47.6926) | (33.6305) | (37.6027) | (71.4143) | ||||
(2, 2, 2) | 0.9656 | 0.9140 | 0.9406 | 0.8912 | |||
(148.9675) | (82.9091) | (103.9126) | (270.3032) | ||||
(0.3, 0.3, 0.3) | (0.5, 0.5, 0.5) | 0.9656 | 0.8762 | 0.9404 | 0.9046 | ||
(15.5403) | (12.1970) | (12.8115) | (19.1430) | ||||
(0.5, 0.5, 1) | 0.9772 | 0.9066 | 0.9586 | 0.8540 | |||
(20.0826) | (14.4529) | (16.1167) | (25.2621) | ||||
(1, 1, 1) | 0.9658 | 0.8752 | 0.9418 | 0.9110 | |||
(42.2880) | (25.1633) | (31.9548) | (56.9812) | ||||
(1, 1, 2) | 0.9800 | 0.9082 | 0.9662 | 0.8656 | |||
(59.9388) | (31.0398) | (43.3691) | (83.2374) | ||||
(2, 2, 2) | 0.9672 | 0.8722 | 0.9356 | 0.9114 | |||
(190.4600) | (70.9535) | (118.5987) | (269.8711) | ||||
(0.3, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9672 | 0.8810 | 0.9456 | 0.8878 | ||
(15.8362) | (11.7858) | (12.9445) | (19.0324) | ||||
(0.5, 0.5, 1) | 0.9758 | 0.8934 | 0.9584 | 0.8794 | |||
(20.2488) | (13.5859) | (16.2743) | (24.4542) | ||||
(1, 1, 1) | 0.9730 | 0.8826 | 0.9534 | 0.8912 | |||
(43.8636) | (24.3271) | (32.7406) | (57.1125) | ||||
(1, 1, 2) | 0.9802 | 0.8916 | 0.9596 | 0.8790 | |||
(60.3466) | (29.5326) | (43.3793) | (79.5503) | ||||
(2, 2, 2) | 0.9732 | 0.8810 | 0.9510 | 0.8938 | |||
(197.5581) | (67.5715) | (122.5223) | (268.6405) | ||||
(0.5, 0.5, 0.5) | (0.5, 0.5, 0.5) | 0.9774 | 0.8736 | 0.9588 | 0.8836 | ||
(16.7548) | (10.9530) | (13.3976) | (17.8895) | ||||
(0.5, 0.5, 1) | 0.9864 | 0.9016 | 0.9732 | 0.8486 | |||
(21.9578) | (13.0428) | (17.0648) | (23.6756) | ||||
(1, 1, 1) | 0.9780 | 0.8732 | 0.9562 | 0.8874 | |||
(48.3134) | (22.2962) | (34.3590) | (52.5244) | ||||
(1, 1, 2) | 0.9848 | 0.8948 | 0.9684 | 0.8584 | |||
(70.4925) | (27.6249) | (47.5502) | (75.9134) | ||||
(2, 2, 2) | 0.9774 | 0.8676 | 0.9552 | 0.8978 | |||
(227.3056) | (58.6553) | (129.4709) | (233.7485) | ||||
(0.1, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9712 | 0.9148 | 0.9508 | 0.8554 | ||
(15.2138) | (12.7495) | (12.6556) | (20.9990) | ||||
(0.5, 0.5, 1) | 0.9744 | 0.9112 | 0.9568 | 0.8492 | |||
(18.5054) | (14.1463) | (15.3719) | (25.5513) | ||||
(1, 1, 1) | 0.9738 | 0.9138 | 0.9522 | 0.8634 | |||
(40.5824) | (26.4336) | (31.3253) | (65.7662) | ||||
(1, 1, 2) | 0.9758 | 0.9142 | 0.9598 | 0.8592 | |||
(53.6772) | (31.1696) | (40.8956) | (85.5507) | ||||
(2, 2, 2) | 0.9726 | 0.9078 | 0.9466 | 0.8586 | |||
(166.2144) | (74.6314) | (111.7168) | (316.6658) | ||||
(30, 30, 30) | (1, 1, 1) | (0.1, 0.1, 0.1) | (0.5, 0.5, 0.5) | 0.9616 | 0.9998 | 0.9480 | 0.3428 |
(6.5534) | (15.5421) | (6.2191) | (11.3560) | ||||
(0.5, 0.5, 1) | 0.9832 | 0.9994 | 0.9764 | 0.2838 | |||
(7.7955) | (18.0345) | (7.3614) | (13.9841) | ||||
(1, 1, 1) | 0.9662 | 0.9999 | 0.9500 | 0.3958 | |||
(14.7780) | (33.5906) | (13.9057) | (31.7637) | ||||
(1, 1, 2) | 0.9842 | 0.9996 | 0.9768 | 0.2916 | |||
(18.1783) | (41.1211) | (16.9299) | (40.8851) | ||||
(2, 2, 2) | 0.9682 | 0.9996 | 0.9474 | 0.3946 | |||
(41.9636) | (98.4243) | (38.5704) | (121.0134) | ||||
(0.1, 0.1, 0.3) | (0.5, 0.5, 0.5) | 0.9664 | 0.9992 | 0.9538 | 0.2796 | ||
(6.6627) | (15.3912) | (6.3104) | (12.6293) | ||||
(0.5, 0.5, 1) | 0.9798 | 0.9996 | 0.9696 | 0.3076 | |||
(7.7158) | (16.9017) | (7.2933) | (14.1194) | ||||
(1, 1, 1) | 0.9622 | 0.9994 | 0.9478 | 0.3154 | |||
(14.7796) | (32.9862) | (13.8794) | (36.2047) | ||||
(1, 1, 2) | 0.9842 | 0.9994 | 0.9750 | 0.3334 | |||
(18.0534) | (37.8567) | (16.8752) | (41.6387) | ||||
(2, 2, 2) | 0.9740 | 0.9996 | 0.9556 | 0.3324 | |||
(41.3229) | (96.0415) | (38.0324) | (146.8337) | ||||
(0.3, 0.3, 0.3) | (0.5, 0.5, 0.5) | 0.9626 | 0.9984 | 0.9472 | 0.4062 | ||
(6.6455) | (14.1889) | (6.2642) | (11.9336) | ||||
(0.5, 0.5, 1) | 0.9822 | 0.9996 | 0.9728 | 0.3346 | |||
(7.8844) | (16.4172) | (7.3760) | (14.7705) | ||||
(1, 1, 1) | 0.9620 | 0.9978 | 0.9472 | 0.4338 | |||
(14.5852) | (29.7677) | (13.5198) | (32.8589) | ||||
(1, 1, 2) | 0.9834 | 0.9992 | 0.9714 | 0.3410 | |||
(17.9405) | (36.5177) | (16.5369) | (43.0495) | ||||
(2, 2, 2) | 0.9656 | 0.9984 | 0.9480 | 0.4668 | |||
(40.0606) | (83.4089) | (36.2743) | (125.1631) | ||||
(0.3, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9706 | 0.9992 | 0.9570 | 0.3560 | ||
(6.6983) | (13.8889) | (6.2906) | (12.5125) | ||||
(0.5, 0.5, 1) | 0.9756 | 0.9972 | 0.9660 | 0.3978 | |||
(7.7910) | (15.2157) | (7.3026) | (14.3137) | ||||
(1, 1, 1) | 0.9692 | 0.9992 | 0.9556 | 0.3772 | |||
(14.5416) | (28.9139) | (13.4969) | (35.2043) | ||||
(1, 1, 2) | 0.9814 | 0.9986 | 0.9724 | 0.4040 | |||
(17.9096) | (33.2614) | (16.5173) | (41.8378) | ||||
(2, 2, 2) | 0.9766 | 0.9992 | 0.9596 | 0.3980 | |||
(39.5256) | (79.6347) | (35.6352) | (136.1955) | ||||
(0.5, 0.5, 0.5) | (0.5, 0.5, 0.5) | 0.9776 | 0.9974 | 0.9678 | 0.4478 | ||
(6.7086) | (12.7519) | (6.2497) | (11.3794) | ||||
(0.5, 0.5, 1) | 0.9876 | 0.9990 | 0.9818 | 0.3698 | |||
(7.8981) | (14.6321) | (7.3269) | (13.9327) | ||||
(1, 1, 1) | 0.9808 | 0.9978 | 0.9704 | 0.4718 | |||
(14.4210) | (25.9801) | (13.2412) | (30.2652) | ||||
(1, 1, 2) | 0.9906 | 0.9986 | 0.9856 | 0.3730 | |||
(17.8024) | (31.4804) | (16.1841) | (39.4965) | ||||
(2, 2, 2) | 0.9822 | 0.9978 | 0.9694 | 0.4786 | |||
(38.0542) | (67.9039) | (33.8151) | (106.6205) | ||||
(0.1, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9742 | 0.9996 | 0.9646 | 0.1876 | ||
(6.8272) | (14.8090) | (6.4405) | (14.7957) | ||||
(0.5, 0.5, 1) | 0.9818 | 0.9992 | 0.9720 | 0.2618 | |||
(7.8849) | (16.0906) | (7.4282) | (16.3482) | ||||
(1, 1, 1) | 0.9764 | 0.9996 | 0.9590 | 0.2104 | |||
(14.9332) | (31.3658) | (13.9415) | (44.6454) | ||||
(1, 1, 2) | 0.9812 | 0.9996 | 0.9710 | 0.2754 | |||
(18.1263) | (35.2511) | (16.9290) | (50.0834) | ||||
(2, 2, 2) | 0.9764 | 0.9998 | 0.9650 | 0.2354 | |||
(40.6140) | (88.0470) | (37.2159) | (189.3741) | ||||
(50, 50, 50) | (1, 1, 1) | (0.1, 0.1, 0.1) | (0.5, 0.5, 0.5) | 0.9638 | 0.9999 | 0.9546 | 0.0812 |
(5.0444) | (15.8510) | (4.8682) | (9.2761) | ||||
(0.5, 0.5, 1) | 0.9878 | 0.9999 | 0.9836 | 0.0626 | |||
(5.9532) | (18.3695) | (5.7275) | (11.2863) | ||||
(1, 1, 1) | 0.9646 | 0.9999 | 0.9538 | 0.0978 | |||
(11.3314) | (34.3475) | (10.8900) | (26.3688) | ||||
(1, 1, 2) | 0.9842 | 0.9999 | 0.9770 | 0.0764 | |||
(13.5810) | (41.7624) | (13.0107) | (32.6945) | ||||
(2, 2, 2) | 0.9660 | 0.9999 | 0.9570 | 0.1064 | |||
(31.0233) | (100.0403) | (29.5751) | (99.6549) | ||||
(0.1, 0.1, 0.3) | (0.5, 0.5, 0.5) | 0.9578 | 0.9999 | 0.9514 | 0.0614 | ||
(5.1143) | (15.7165) | (4.9284) | (10.3754) | ||||
(0.5, 0.5, 1) | 0.9852 | 0.9999 | 0.9792 | 0.0848 | |||
(5.8783) | (17.2641) | (5.6547) | (11.3551) | ||||
(1, 1, 1) | 0.9626 | 0.9999 | 0.9506 | 0.0690 | |||
(11.3605) | (34.0661) | (10.9094) | (30.4236) | ||||
(1, 1, 2) | 0.9816 | 0.9999 | 0.9740 | 0.0922 | |||
(13.4399) | (38.3581) | (12.8614) | (32.8323) | ||||
(2, 2, 2) | 0.9674 | 0.9999 | 0.9524 | 0.0748 | |||
(30.8706) | (99.3191) | (29.4213) | (122.6326) | ||||
(0.3, 0.3, 0.3) | (0.5, 0.5, 0.5) | 0.9642 | 0.9999 | 0.9542 | 0.1258 | ||
(5.0770) | (14.5207) | (4.8695) | (9.6186) | ||||
(0.5, 0.5, 1) | 0.9852 | 0.9998 | 0.9804 | 0.1042 | |||
(5.9510) | (16.7797) | (5.6947) | (11.7019) | ||||
(1, 1, 1) | 0.9634 | 0.9999 | 0.9518 | 0.1432 | |||
(11.0306) | (30.7406) | (10.5358) | (26.8617) | ||||
(1, 1, 2) | 0.9860 | 0.9999 | 0.9774 | 0.1090 | |||
(13.2021) | (37.2641) | (12.5360) | (33.7078) | ||||
(2, 2, 2) | 0.9654 | 0.9999 | 0.9530 | 0.1632 | |||
(28.8237) | (85.2569) | (27.2393) | (98.3914) | ||||
(0.3, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9722 | 0.9999 | 0.9624 | 0.0854 | ||
(5.1636) | (14.3764) | (4.9468) | (10.4763) | ||||
(0.5, 0.5, 1) | 0.9824 | 0.9999 | 0.9758 | 0.1360 | |||
(5.9178) | (15.6579) | (5.6638) | (11.6016) | ||||
(1, 1, 1) | 0.9744 | 0.9999 | 0.9610 | 0.1060 | |||
(10.9913) | (29.9042) | (10.4848) | (29.3116) | ||||
(1, 1, 2) | 0.9828 | 0.9999 | 0.9758 | 0.1372 | |||
(13.1756) | (33.8248) | (12.5288) | (33.0561) | ||||
(2, 2, 2) | 0.9754 | 0.9999 | 0.9658 | 0.1096 | |||
(28.6218) | (83.3013) | (26.9484) | (112.8997) | ||||
(0.5, 0.5, 0.5) | (0.5, 0.5, 0.5) | 0.9806 | 0.9999 | 0.9738 | 0.1624 | ||
(5.0797) | (13.0450) | (4.8474) | (9.3196) | ||||
(0.5, 0.5, 1) | 0.9888 | 0.9999 | 0.9826 | 0.1290 | |||
(5.9101) | (14.9031) | (5.6202) | (11.2249) | ||||
(1, 1, 1) | 0.9800 | 0.9998 | 0.9690 | 0.1740 | |||
(10.6122) | (26.4904) | (10.0587) | (24.5042) | ||||
(1, 1, 2) | 0.9914 | 0.9999 | 0.9868 | 0.1364 | |||
(12.7123) | (31.7257) | (11.9502) | (30.8441) | ||||
(2, 2, 2) | 0.9832 | 0.9998 | 0.9746 | 0.1872 | |||
(27.0649) | (70.5521) | (25.3041) | (86.2946) | ||||
(0.1, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9826 | 0.9999 | 0.9752 | 0.0226 | ||
(5.2820) | (15.2200) | (5.0773) | (12.6177) | ||||
(0.5, 0.5, 1) | 0.9836 | 0.9999 | 0.9782 | 0.0502 | |||
(5.9733) | (16.4098) | (5.7396) | (13.2880) | ||||
(1, 1, 1) | 0.9756 | 0.9999 | 0.9686 | 0.0300 | |||
(11.4649) | (32.4412) | (10.9935) | (38.9225) | ||||
(1, 1, 2) | 0.9826 | 0.9999 | 0.9764 | 0.0622 | |||
(13.5286) | (35.7961) | (12.9186) | (40.2835) | ||||
(2, 2, 2) | 0.9772 | 0.9999 | 0.9642 | 0.0326 | |||
(29.9892) | (91.7238) | (28.5242) | (168.0056) | ||||
(100, 100, 100) | (1, 1, 1) | (0.1, 0.1, 0.1) | (0.5, 0.5, 0.5) | 0.9714 | 0.9999 | 0.9700 | 0.0012 |
(3.5528) | (16.0693) | (3.4567) | (6.8105) | ||||
(0.5, 0.5, 1) | 0.9862 | 0.9999 | 0.9810 | 0.0022 | |||
(4.1673) | (18.4836) | (4.0644) | (8.1855) | ||||
(1, 1, 1) | 0.9596 | 0.9999 | 0.9600 | 0.0030 | |||
(8.0642) | (34.8008) | (7.8775) | (20.0550) | ||||
(1, 1, 2) | 0.9880 | 0.9999 | 0.9826 | 0.0008 | |||
(9.4937) | (42.1608) | (9.2431) | (23.9587) | ||||
(2, 2, 2) | 0.9698 | 0.9999 | 0.9628 | 0.0042 | |||
(22.0811) | (101.6958) | (21.4862) | (77.4539) | ||||
(0.1, 0.1, 0.3) | (0.5, 0.5, 0.5) | 0.9724 | 0.9999 | 0.9690 | 0.0002 | ||
(3.6095) | (16.0508) | (3.5168) | (7.7891) | ||||
(0.5, 0.5, 1) | 0.9896 | 0.9999 | 0.9844 | 0.0014 | |||
(4.1229) | (17.4908) | (4.0160) | (8.2179) | ||||
(1, 1, 1) | 0.9726 | 0.9999 | 0.9684 | 0.0012 | |||
(8.1324) | (34.6864) | (7.9376) | (23.5548) | ||||
(1, 1, 2) | 0.9856 | 0.9999 | 0.9826 | 0.0014 | |||
(9.4391) | (39.1547) | (9.1788) | (23.8943) | ||||
(2, 2, 2) | 0.9730 | 0.9999 | 0.9684 | 0.0020 | |||
(21.9251) | (101.2034) | (21.4158) | (96.7689) | ||||
(0.3, 0.3, 0.3) | (0.5, 0.5, 0.5) | 0.9720 | 0.9999 | 0.9684 | 0.0040 | ||
(3.5729) | (14.7932) | (3.4684) | (7.0743) | ||||
(0.5, 0.5, 1) | 0.9858 | 0.9999 | 0.9836 | 0.0056 | |||
(4.1667) | (16.9502) | (4.0517) | (8.4757) | ||||
(1, 1, 1) | 0.9652 | 0.9999 | 0.9590 | 0.0068 | |||
(7.7922) | (31.1248) | (7.5794) | (20.0571) | ||||
(1, 1, 2) | 0.9874 | 0.9999 | 0.9822 | 0.0044 | |||
(9.1701) | (37.4998) | (8.9057) | (24.2793) | ||||
(2, 2, 2) | 0.9668 | 0.9999 | 0.9584 | 0.0070 | |||
(20.1903) | (86.8276) | (19.5881) | (74.1374) | ||||
(0.3, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9806 | 0.9999 | 0.9772 | 0.0012 | ||
(3.6457) | (14.6436) | (3.5429) | (7.9609) | ||||
(0.5, 0.5, 1) | 0.9856 | 0.9999 | 0.9830 | 0.0060 | |||
(4.1307) | (15.8826) | (4.0104) | (8.4596) | ||||
(1, 1, 1) | 0.9786 | 0.9999 | 0.9762 | 0.0024 | |||
(7.8435) | (30.6015) | (7.6256) | (23.0574) | ||||
(1, 1, 2) | 0.9840 | 0.9999 | 0.9796 | 0.0056 | |||
(9.0767) | (34.0380) | (8.8051) | (23.6508) | ||||
(2, 2, 2) | 0.9750 | 0.9999 | 0.9694 | 0.0040 | |||
(20.0959) | (85.2629) | (19.5149) | (90.5855) | ||||
(0.5, 0.5, 0.5) | (0.5, 0.5, 0.5) | 0.9862 | 0.9999 | 0.9836 | 0.0112 | ||
(3.5578) | (13.1718) | (3.4480) | (6.9213) | ||||
(0.5, 0.5, 1) | 0.9916 | 0.9999 | 0.9898 | 0.0098 | |||
(4.1245) | (14.9820) | (3.9938) | (8.2568) | ||||
(1, 1, 1) | 0.9826 | 0.9999 | 0.9800 | 0.0120 | |||
(7.4301) | (26.7038) | (7.2045) | (18.5641) | ||||
(1, 1, 2) | 0.9916 | 0.9999 | 0.9874 | 0.0102 | |||
(8.6974) | (31.8035) | (8.3988) | (22.4537) | ||||
(2, 2, 2) | 0.9856 | 0.9999 | 0.9806 | 0.0130 | |||
(18.2724) | (70.2128) | (17.6212) | (64.1559) | ||||
(0.1, 0.3, 0.5) | (0.5, 0.5, 0.5) | 0.9850 | 0.9999 | 0.9838 | 0.0002 | ||
(3.7613) | (15.5088) | (3.6583) | (9.8258) | ||||
(0.5, 0.5, 1) | 0.9896 | 0.9999 | 0.9850 | 0.0008 | |||
(4.1564) | (16.6099) | (4.0451) | (9.7186) | ||||
(1, 1, 1) | 0.9804 | 0.9999 | 0.9788 | 0.0004 | |||
(8.2611) | (32.8864) | (8.0504) | (31.4164) | ||||
(1, 1, 2) | 0.9838 | 0.9999 | 0.9812 | 0.0012 | |||
(9.4614) | (36.4256) | (9.2045) | (29.7488) | ||||
(2, 2, 2) | 0.9802 | 0.9999 | 0.9750 | 0.0004 | |||
(21.6124) | (94.5200) | (21.1194) | (143.6208) |
(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) | (μ_{1}, μ_{2}, μ_{3}, μ_{4}, μ_{5}, μ_{6}) | (δ^{′}_{1}, δ^{′}_{2}, δ^{′}_{3}, δ^{′}_{4}, δ^{′}_{5}, δ^{′}_{6}) | (σ^{2}_{1}, σ^{2}_{2}, σ^{2}_{3}, σ^{2}_{4}, σ^{2}_{5}, σ^{2}_{6}) | CP (AL) | |||
---|---|---|---|---|---|---|---|
CI_{FGCI} | CI_{AM} | CI_{BS1} | CI_{BS2} | ||||
(10, 10, 10, 10, 10, 10) | (1, 1, 1, 1, 1, 1) | (0.1, 0.1, 0.1, 0.1, 0.1, 0.1) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8016 | 0.9406 | 0.7298 | 0.9620 |
(7.6250) | (12.6488) | (6.8578) | (10.0509) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9300 | 0.9776 | 0.8918 | 0.9348 | |||
(10.2607) | (16.3065) | (9.1466) | (13.9621) | ||||
(1, 1, 1, 1, 1, 1) | 0.8328 | 0.9534 | 0.7564 | 0.9688 | |||
(16.5917) | (26.4428) | (14.5405) | (24.5432) | ||||
(1, 1, 1, 2, 2, 2) | 0.9314 | 0.9712 | 0.8906 | 0.9422 | |||
(23.7564) | (36.4290) | (20.4332) | (36.4737) | ||||
(2, 2, 2, 2, 2, 2) | 0.8450 | 0.9496 | 0.7682 | 0.9664 | |||
(45.5142) | (70.4158) | (37.9084) | (76.2348) | ||||
(0.1, 0.1, 0.1, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8306 | 0.9338 | 0.7700 | 0.9566 | ||
(8.0197) | (12.3650) | (7.1669) | (11.2953) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9064 | 0.9494 | 0.8652 | 0.9466 | |||
(10.2536) | (14.6196) | (9.1805) | (14.3067) | ||||
(1, 1, 1, 1, 1, 1) | 0.8454 | 0.9430 | 0.7788 | 0.9612 | |||
(16.9729) | (25.0168) | (14.8551) | (28.0269) | ||||
(1, 1, 1, 2, 2, 2) | 0.9172 | 0.9500 | 0.8754 | 0.9462 | |||
(23.6133) | (31.8925) | (20.6296) | (38.2905) | ||||
(2, 2, 2, 2, 2, 2) | 0.8698 | 0.9372 | 0.7942 | 0.9648 | |||
(47.5882) | (66.9459) | (39.1228) | (95.2103) | ||||
(0.3, 0.3, 0.3, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8292 | 0.8986 | 0.7616 | 0.9526 | ||
(8.0671) | (11.1569) | (7.1500) | (10.8082) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9262 | 0.9446 | 0.8912 | 0.9316 | |||
(10.8288) | (14.3171) | (9.5091) | (15.1668) | ||||
(1, 1, 1, 1, 1, 1) | 0.8450 | 0.9026 | 0.7812 | 0.9578 | |||
(17.2150) | (22.4168) | (14.8647) | (26.5081) | ||||
(1, 1, 1, 2, 2, 2) | 0.9316 | 0.9406 | 0.8936 | 0.9310 | |||
(25.0334) | (30.8319) | (20.9536) | (40.4152) | ||||
(2, 2, 2, 2, 2, 2) | 0.8696 | 0.9102 | 0.8010 | 0.9598 | |||
(48.9337) | (58.5179) | (39.0391) | (86.8973) | ||||
(0.3, 0.3, 0.3, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8694 | 0.9056 | 0.8156 | 0.9454 | ||
(8.2082) | (10.7620) | (7.2262) | (10.8717) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9082 | 0.9132 | 0.8642 | 0.9414 | |||
(10.6453) | (12.9390) | (9.3519) | (14.2133) | ||||
(1, 1, 1, 1, 1, 1) | 0.8786 | 0.9076 | 0.8226 | 0.9434 | |||
(17.1771) | (21.1382) | (14.6899) | (25.9780) | ||||
(1, 1, 1, 2, 2, 2) | 0.9248 | 0.9186 | 0.8838 | 0.9364 | |||
(24.9900) | (27.5973) | (20.9123) | (38.3552) | ||||
(2, 2, 2, 2, 2, 2) | 0.8968 | 0.9100 | 0.8360 | 0.9498 | |||
(48.8431) | (53.4199) | (38.1265) | (84.5829) | ||||
(0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8952 | 0.8970 | 0.8436 | 0.9408 | ||
(8.2055) | (9.9677) | (7.1538) | (9.9022) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9476 | 0.9296 | 0.9196 | 0.9152 | |||
(11.0288) | (12.6493) | (9.4979) | (13.6809) | ||||
(1, 1, 1, 1, 1, 1) | 0.8942 | 0.8856 | 0.8432 | 0.9466 | |||
(16.9730) | (18.9134) | (14.3020) | (22.7060) | ||||
(1, 1, 1, 2, 2, 2) | 0.9516 | 0.9354 | 0.9220 | 0.9248 | |||
(25.2937) | (26.0932) | (20.7248) | (34.8531) | ||||
(2, 2, 2, 2, 2, 2) | 0.9090 | 0.8974 | 0.8518 | 0.9478 | |||
(48.6057) | (47.0657) | (36.9784) | (69.3067) | ||||
(0.1, 0.1, 0.3, 0.3, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8618 | 0.9338 | 0.8062 | 0.9398 | ||
(8.0785) | (11.8067) | (7.1875) | (11.8587) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9074 | 0.9450 | 0.8684 | 0.9362 | |||
(10.4975) | (14.0510) | (9.3552) | (15.3534) | ||||
(1, 1, 1, 1, 1, 1) | 0.8708 | 0.9336 | 0.8134 | 0.9498 | |||
(17.0284) | (23.6226) | (14.7665) | (29.7163) | ||||
(1, 1, 1, 2, 2, 2) | 0.9086 | 0.9372 | 0.8658 | 0.9400 | |||
(23.8516) | (29.9712) | (20.6328) | (41.4774) | ||||
(2, 2, 2, 2, 2, 2) | 0.8848 | 0.9292 | 0.8248 | 0.9496 | |||
(47.6110) | (62.2276) | (38.6294) | (101.9620) | ||||
(30, 30, 30, 30, 30, 30) | (1, 1, 1, 1, 1, 1) | (0.1, 0.1, 0.1, 0.1, 0.1, 0.1) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8136 | 0.9999 | 0.7924 | 0.4276 |
(4.8154) | (15.4392) | (4.6005) | (8.5477) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9566 | 0.9999 | 0.9432 | 0.3178 | |||
(6.1127) | (19.5638) | (5.8366) | (11.3839) | ||||
(1, 1, 1, 1, 1, 1) | 0.7978 | 0.9998 | 0.7678 | 0.5198 | |||
(10.1940) | (33.0285) | (9.7368) | (22.2396) | ||||
(1, 1, 1, 2, 2, 2) | 0.9566 | 0.9999 | 0.9408 | 0.3356 | |||
(13.7028) | (46.1774) | (13.0098) | (31.6191) | ||||
(2, 2, 2, 2, 2, 2) | 0.8108 | 0.9999 | 0.7760 | 0.5912 | |||
(25.8690) | (94.2322) | (24.5752) | (73.8401) | ||||
(0.1, 0.1, 0.1, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8222 | 0.9999 | 0.7992 | 0.3026 | ||
(4.8871) | (15.3067) | (4.6439) | (9.7968) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9426 | 0.9999 | 0.9254 | 0.3602 | |||
(6.0826) | (17.8290) | (5.7915) | (11.6771) | ||||
(1, 1, 1, 1, 1, 1) | 0.8142 | 0.9999 | 0.7898 | 0.3886 | |||
(10.1200) | (32.5466) | (9.6468) | (26.1704) | ||||
(1, 1, 1, 2, 2, 2) | 0.9460 | 0.9999 | 0.9286 | 0.4108 | |||
(13.4001) | (40.1522) | (12.7380) | (31.9072) | ||||
(2, 2, 2, 2, 2, 2) | 0.8304 | 0.9999 | 0.7912 | 0.4350 | |||
(25.0569) | (92.4834) | (23.7630) | (92.8136) | ||||
(0.3, 0.3, 0.3, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8066 | 0.9998 | 0.7786 | 0.4798 | ||
(4.8037) | (14.2032) | (4.5502) | (8.9447) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9442 | 0.9999 | 0.9282 | 0.3554 | |||
(6.0678) | (17.9521) | (5.7493) | (11.9905) | ||||
(1, 1, 1, 1, 1, 1) | 0.7986 | 0.9999 | 0.7630 | 0.5626 | |||
(9.7316) | (29.5119) | (9.2288) | (22.7197) | ||||
(1, 1, 1, 2, 2, 2) | 0.9494 | 0.9998 | 0.9282 | 0.3774 | |||
(13.0084) | (40.9760) | (12.2616) | (32.8189) | ||||
(2, 2, 2, 2, 2, 2) | 0.8380 | 0.9998 | 0.7972 | 0.6042 | |||
(23.8306) | (81.3217) | (22.3862) | (75.2510) | ||||
(0.3, 0.3, 0.3, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8732 | 0.9999 | 0.8478 | 0.3694 | ||
(4.8236) | (13.8424) | (4.5534) | (9.5585) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9384 | 0.9996 | 0.9208 | 0.4376 | |||
(6.0583) | (16.1432) | (5.7323) | (11.7522) | ||||
(1, 1, 1, 1, 1, 1) | 0.8742 | 0.9999 | 0.8426 | 0.4332 | |||
(9.6368) | (28.4826) | (9.1113) | (24.5225) | ||||
(1, 1, 1, 2, 2, 2) | 0.9448 | 0.9998 | 0.9234 | 0.4502 | |||
(12.9321) | (35.2431) | (12.2181) | (31.4213) | ||||
(2, 2, 2, 2, 2, 2) | 0.8852 | 0.9999 | 0.8526 | 0.4724 | |||
(23.0537) | (77.7036) | (21.6017) | (82.8702) | ||||
(0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8842 | 0.9999 | 0.8598 | 0.5324 | ||
(4.7101) | (12.6219) | (4.4268) | (8.3001) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9674 | 0.9999 | 0.9548 | 0.3984 | |||
(5.9720) | (15.9091) | (5.6081) | (11.1150) | ||||
(1, 1, 1, 1, 1, 1) | 0.8932 | 0.9998 | 0.8646 | 0.5998 | |||
(9.2881) | (25.5004) | (8.7411) | (20.1494) | ||||
(1, 1, 1, 2, 2, 2) | 0.9688 | 0.9996 | 0.9558 | 0.4210 | |||
(12.4083) | (34.7170) | (11.5961) | (28.9112) | ||||
(2, 2, 2, 2, 2, 2) | 0.9084 | 0.9998 | 0.8784 | 0.6178 | |||
(21.7684) | (66.3863) | (20.2199) | (62.2931) | ||||
(0.1, 0.1, 0.3, 0.3, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8712 | 0.9999 | 0.8524 | 0.2040 | ||
(4.9587) | (14.9867) | (4.7059) | (11.0931) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9378 | 0.9999 | 0.9246 | 0.2912 | |||
(6.1725) | (17.2994) | (5.8622) | (13.0781) | ||||
(1, 1, 1, 1, 1, 1) | 0.8688 | 0.9999 | 0.8436 | 0.2546 | |||
(10.0836) | (31.5761) | (9.6026) | (30.5581) | ||||
(1, 1, 1, 2, 2, 2) | 0.9418 | 0.9999 | 0.9214 | 0.3374 | |||
(13.4034) | (38.5120) | (12.7487) | (37.3294) | ||||
(2, 2, 2, 2, 2, 2) | 0.8832 | 0.9999 | 0.8506 | 0.2834 | |||
(24.5153) | (89.6123) | (23.2187) | (114.9333) | ||||
(50, 50, 50, 50, 50, 50) | (1, 1, 1, 1, 1, 1) | (0.1, 0.1, 0.1, 0.1, 0.1, 0.1) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8252 | 0.9999 | 0.8160 | 0.0760 |
(3.8541) | (15.8903) | (3.7075) | (7.2803) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9652 | 0.9999 | 0.9556 | 0.0696 | |||
(4.9129) | (20.1920) | (4.7472) | (9.7267) | ||||
(1, 1, 1, 1, 1, 1) | 0.8024 | 0.9999 | 0.7922 | 0.1286 | |||
(8.3897) | (34.3204) | (8.1175) | (19.9112) | ||||
(1, 1, 1, 2, 2, 2) | 0.9632 | 0.9999 | 0.9540 | 0.0850 | |||
(10.9142) | (47.4026) | (10.5511) | (27.1045) | ||||
(2, 2, 2, 2, 2, 2) | 0.8206 | 0.9999 | 0.7954 | 0.1518 | |||
(21.6198) | (99.4624) | (20.9268) | (69.1200) | ||||
(0.1, 0.1, 0.1, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8386 | 0.9999 | 0.8338 | 0.0434 | ||
(3.9282) | (15.8730) | (3.7744) | (8.5329) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9476 | 0.9999 | 0.9386 | 0.0852 | |||
(4.8147) | (18.3732) | (4.6415) | (9.8208) | ||||
(1, 1, 1, 1, 1, 1) | 0.8316 | 0.9999 | 0.8234 | 0.0648 | |||
(8.3834) | (34.3113) | (8.1010) | (24.1798) | ||||
(1, 1, 1, 2, 2, 2) | 0.9520 | 0.9999 | 0.9454 | 0.1002 | |||
(10.7859) | (41.8454) | (10.4130) | (27.4038) | ||||
(2, 2, 2, 2, 2, 2) | 0.8306 | 0.9999 | 0.8126 | 0.0860 | |||
(21.0155) | (99.3153) | (20.3178) | (89.3190) | ||||
(0.3, 0.3, 0.3, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8204 | 0.9999 | 0.8144 | 0.1090 | ||
(3.8682) | (14.8042) | (3.7075) | (7.6852) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9576 | 0.9999 | 0.9480 | 0.0946 | |||
(4.8456) | (18.5610) | (4.6606) | (10.1607) | ||||
(1, 1, 1, 1, 1, 1) | 0.8036 | 0.9999 | 0.7884 | 0.1670 | |||
(8.0010) | (30.8982) | (7.7037) | (20.1601) | ||||
(1, 1, 1, 2, 2, 2) | 0.9606 | 0.9999 | 0.9508 | 0.1088 | |||
(10.4571) | (42.7048) | (10.0499) | (28.0026) | ||||
(2, 2, 2, 2, 2, 2) | 0.8290 | 0.9999 | 0.8018 | 0.2020 | |||
(19.6656) | (86.7010) | (18.9539) | (68.5778) | ||||
(0.3, 0.3, 0.3, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8762 | 0.9999 | 0.8740 | 0.0820 | ||
(3.8893) | (14.4091) | (3.7216) | (8.4659) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9470 | 0.9999 | 0.9374 | 0.1240 | |||
(4.7961) | (16.6455) | (4.6096) | (9.9949) | ||||
(1, 1, 1, 1, 1, 1) | 0.8826 | 0.9999 | 0.8700 | 0.0966 | |||
(7.9530) | (30.2705) | (7.6625) | (22.9065) | ||||
(1, 1, 1, 2, 2, 2) | 0.9530 | 0.9999 | 0.9418 | 0.1488 | |||
(10.3143) | (36.6303) | (9.9176) | (27.0717) | ||||
(2, 2, 2, 2, 2, 2) | 0.8848 | 0.9999 | 0.8646 | 0.116 | |||
(18.8334) | (83.0011) | (18.1105) | (79.5545) | ||||
(0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.9070 | 0.9999 | 0.8962 | 0.1574 | ||
(3.8144) | (13.1624) | (3.6426) | (7.2554) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9786 | 0.9999 | 0.9720 | 0.1226 | |||
(4.7384) | (16.3421) | (4.5340) | (9.5195) | ||||
(1, 1, 1, 1, 1, 1) | 0.8930 | 0.9999 | 0.8780 | 0.2186 | |||
(7.5547) | (26.6011) | (7.2500) | (18.0184) | ||||
(1, 1, 1, 2, 2, 2) | 0.9754 | 0.9999 | 0.9672 | 0.1400 | |||
(9.8074) | (35.9742) | (9.3638) | (24.9016) | ||||
(2, 2, 2, 2, 2, 2) | 0.9104 | 0.9999 | 0.8920 | 0.2498 | |||
(17.6016) | (70.1342) | (16.8252) | (56.8175) | ||||
(0.1, 0.1, 0.3, 0.3, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8938 | 0.9999 | 0.8886 | 0.0200 | ||
(4.0160) | (15.5456) | (3.8493) | (9.9117) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9526 | 0.9999 | 0.9446 | 0.0608 | |||
(4.8840) | (17.7752) | (4.6987) | (11.1697) | ||||
(1, 1, 1, 1, 1, 1) | 0.8756 | 0.9999 | 0.8678 | 0.0316 | |||
(8.3704) | (33.2824) | (8.0893) | (29.0916) | ||||
(1, 1, 1, 2, 2, 2) | 0.9488 | 0.9999 | 0.9402 | 0.0792 | |||
(10.7093) | (39.7266) | (10.3501) | (31.9445) | ||||
(2, 2, 2, 2, 2, 2) | 0.8808 | 0.9999 | 0.8704 | 0.0376 | |||
(20.3156) | (95.13447) | (19.6846) | (115.0963) | ||||
(100, 100, 100, 100, 100, 100) | (1, 1, 1, 1, 1, 1) | (0.1, 0.1, 0.1, 0.1, 0.1, 0.1) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8732 | 0.9999 | 0.8772 | 0.0012 |
(2.7329) | (16.1904) | (2.6471) | (5.4062) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9700 | 0.9999 | 0.9648 | 0.0008 | |||
(3.5414) | (20.3978) | (3.4494) | (7.2442) | ||||
(1, 1, 1, 1, 1, 1) | 0.8348 | 0.9999 | 0.8354 | 0.0030 | |||
(6.2650) | (35.0941) | (6.0971) | (15.9937) | ||||
(1, 1, 1, 2, 2, 2) | 0.9638 | 0.9999 | 0.9616 | 0.0020 | |||
(8.0178) | (48.2604) | (7.8326) | (20.8600) | ||||
(2, 2, 2, 2, 2, 2) | 0.8170 | 0.9999 | 0.8160 | 0.0022 | |||
(16.7047) | (102.2348) | (16.3612) | (59.3653) | ||||
(0.1, 0.1, 0.1, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8846 | 0.9999 | 0.8918 | 0.0000 | ||
(2.8196) | (16.3093) | (2.7261) | (6.5520) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9660 | 0.9999 | 0.9618 | 0.0010 | |||
(3.4680) | (18.7847) | (3.3805) | (7.3270) | ||||
(1, 1, 1, 1, 1, 1) | 0.8520 | 0.9999 | 0.8582 | 0.0010 | |||
(6.2984) | (35.2816) | (6.1329) | (19.9033) | ||||
(1, 1, 1, 2, 2, 2) | 0.9616 | 0.9999 | 0.9600 | 0.0028 | |||
(7.9279) | (42.9625) | (7.7340) | (20.7974) | ||||
(2, 2, 2, 2, 2, 2) | 0.8448 | 0.9999 | 0.8444 | 0.0010 | |||
(16.3443) | (103.2218) | (16.0569) | (79.6386) | ||||
(0.3, 0.3, 0.3, 0.3, 0.3, 0.3) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.8578 | 0.9999 | 0.8664 | 0.0026 | ||
(2.7683) | (15.0435) | (2.6748) | (5.7390) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9690 | 0.9999 | 0.9646 | 0.0024 | |||
(3.5230) | (18.8232) | (3.4233) | (7.6004) | ||||
(1, 1, 1, 1, 1, 1) | 0.8214 | 0.9999 | 0.8286 | 0.0072 | |||
(6.0201) | (31.7049) | (5.8552) | (16.1664) | ||||
(1, 1, 1, 2, 2, 2) | 0.9644 | 0.9999 | 0.9596 | 0.0038 | |||
(7.6498) | (43.1671) | (7.4550) | (21.2596) | ||||
(2, 2, 2, 2, 2, 2) | 0.8132 | 0.9999 | 0.8148 | 0.0082 | |||
(15.0719) | (88.6894) | (14.7550) | (57.1714) | ||||
(0.3, 0.3, 0.3, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.9180 | 0.9999 | 0.9234 | 0.0006 | ||
(2.8449) | (14.8638) | (2.7437) | (6.7390) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9618 | 0.9999 | 0.9598 | 0.0038 | |||
(3.4498) | (16.9168) | (3.3505) | (7.5390) | ||||
(1, 1, 1, 1, 1, 1) | 0.8982 | 0.9999 | 0.9022 | 0.0020 | |||
(6.0485) | (31.1951) | (5.8747) | (19.4836) | ||||
(1, 1, 1, 2, 2, 2) | 0.9620 | 0.9999 | 0.9538 | 0.0082 | |||
(7.5600) | (37.2542) | (7.3573) | (20.7136) | ||||
(2, 2, 2, 2, 2, 2) | 0.8984 | 0.9999 | 0.8926 | 0.0022 | |||
(14.6875) | (86.9716) | (14.3815) | (73.0262) | ||||
(0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.9250 | 0.9999 | 0.9244 | 0.0094 | ||
(2.7656) | (13.3827) | (2.6605) | (5.6144) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9828 | 0.9999 | 0.9804 | 0.0060 | |||
(3.4635) | (16.5766) | (3.3548) | (7.3430) | ||||
(1, 1, 1, 1, 1, 1) | 0.9070 | 0.9999 | 0.9082 | 0.0114 | |||
(5.6762) | (27.1235) | (5.4975) | (14.7226) | ||||
(1, 1, 1, 2, 2, 2) | 0.9806 | 0.9999 | 0.9746 | 0.0076 | |||
(7.2151) | (36.3350) | (7.0047) | (19.4473) | ||||
(2, 2, 2, 2, 2, 2) | 0.9242 | 0.9999 | 0.9154 | 0.0130 | |||
(13.4330) | (71.8551) | (13.0897) | (48.3645) | ||||
(0.1, 0.1, 0.3, 0.3, 0.5, 0.5) | (0.5, 0.5, 0.5, 0.5, 0.5, 0.5) | 0.9394 | 0.9999 | 0.9410 | 0.0002 | ||
(2.9524) | (16.0641) | (2.8508) | (8.0282) | ||||
(0.5, 0.5, 0.5, 1, 1, 1) | 0.9658 | 0.9999 | 0.9632 | 0.0010 | |||
(3.5098) | (18.1987) | (3.4133) | (8.4494) | ||||
(1, 1, 1, 1, 1, 1) | 0.9094 | 0.9999 | 0.9152 | 0.0000 | |||
(6.4386) | (34.4869) | (6.2743) | (25.4436) | ||||
(1, 1, 1, 2, 2, 2) | 0.9642 | 0.9999 | 0.9590 | 0.0006 | |||
(7.9464) | (41.02513) | (7.7511) | (24.8286) | ||||
(2, 2, 2, 2, 2, 2) | 0.9058 | 0.9999 | 0.9016 | 0.0000 | |||
(16.2537) | (100.0387) | (15.9633) | (111.3212) |
For k = 6, the performances of CI_{AM} were better than those of the others for small sample sizes, but its coverage probabilities were equal to 1 for large sample sizes. Meanwhile, those of CI_{FGCI} and CI_{BS1} were greater than the nominal confidence level of 0.95 for large sample sizes but the average lengths of CI_{BS1} were shorter than those of CI_{FGCI}. Moreover, the coverage probabilities of CI_{BS2} were less than the nominal confidence level of 0.95.
In Fig. 1, the coverage probabilities of CI_{BS2} were smaller when the sample sizes are larger because its average lengths were shorter which was effect to the proportion of the time that the interval contains the true value. In addition, the prior distribution was effect to the coverage probabilities; see Yosboonruang, Niwitpong & Niwitpong (2020). Moreover, in Fig. 4, the average lengths of CI_{AM} were wider when the sample sizes are larger because the confidence interval uses the concepts of the central limit theorem and is based on the initial confidence interval for the single parameter. The initial confidence interval was effect to the coverage probabilities and the average lengths; see Thangjai, Niwitpong & Niwitpong (2020b).
Empirical application
The efficacies of the proposed confidence intervals are illustrated using real datasets. Beforehand, a Monte Carlo simulation consisting of 1,000 repetitions was run to compute the FGCI and Bayesian confidence intervals.
The datasets comprise rainfall data on 1 September 2021 for five regions in Thailand reported by the Thai Meteorological Department. The data are reported in Table 3 and the statistics in Table 4. The densities of these datasets are presented in Fig. 7. First, the possible distributions for the positive rainfall data were considered by applying the minimum Akaike information criterion (AIC), the results for which are reported in Table 5. It can be seen that the log-normal distribution fits the positive rainfall data for the northern, northeastern, and eastern regions of Thailand. Moreover, histograms of the rainfall data series and QQ-plots of the log-transformed rainfall data series in Figs. 8 and 9, respectively, follow normal distributions. Next, the zero values in the rainfall datasets fit binomial distributions, and so overall, the rainfall datasets follow delta-lognormal distributions.
Northern | Northeastern | Central | Eastern | Southern | |||||
---|---|---|---|---|---|---|---|---|---|
17.5 | 0.0 | 1.2 | 62.8 | 2.0 | 4.4 | 16.4 | 0.3 | 0.1 | 8.4 |
6.1 | 0.0 | 0.2 | 0.0 | 9.3 | 31.2 | 1.3 | 12.9 | 4.2 | 0.5 |
1.6 | 1.2 | 0.0 | 4.4 | 0.0 | 8.6 | 29.1 | 12.6 | 19.7 | 8.4 |
0.0 | 5.6 | 25.1 | 0.0 | 6.0 | 11.7 | 11.4 | 0.0 | 0.7 | 31.3 |
0.2 | 0.4 | 39.0 | 4.5 | 3.0 | 1.2 | 118.3 | 6.7 | 0.0 | 0.3 |
3.6 | 0.4 | 0.8 | 0.5 | 9.6 | 1.4 | 7.8 | 4.2 | 0.3 | 1.6 |
0.2 | 0.3 | 0.0 | 19.6 | 21.4 | 8.7 | 1.0 | 0.2 | 1.5 | 8.1 |
0.0 | 3.5 | 6.8 | 0.6 | 1.2 | 2.2 | 1.3 | 0.0 | 1.0 | |
0.8 | 0.0 | 1.2 | 27.6 | 35.9 | 11.3 | 0.0 | 5.5 | ||
0.3 | 2.5 | 0.6 | 0.4 | 7.6 | 8.8 | 0.0 | 0.0 | ||
7.0 | 5.6 | 7.2 | 0.0 | 1.8 | 0.4 | 0.0 | 145.2 | ||
0.8 | 0.2 | 0.0 | 0.0 | 65.8 | 23.2 | ||||
3.3 | 4.1 | 0.0 | 0.0 | 2.4 | 7.6 | ||||
18.4 | 15.6 | 0.8 | 0.0 | 24.6 | 33.8 | ||||
0.0 | 12.0 |
Note:
Source, Thai Meteorological Department (https://www.tmd.go.th/climate/climate.php).
Statistics | Northern | Northeastern | Central | Eastern | Southern |
---|---|---|---|---|---|
n_{i} | 29 | 28 | 22 | 15 | 29 |
n_{i(1)} | 23 | 18 | 21 | 14 | 16 |
n_{i(0)} | 6 | 10 | 1 | 1 | 13 |
$\stackrel{\mathrm{\_}}{y}$_{i(1)} | 0.56 | 1.10 | 1.63 | 1.54 | 2.07 |
s^{2}_{i(1)} | 2.28 | 3.26 | 1.38 | 3.21 | 3.66 |
Distribution | Northern | Northeastern | Central | Eastern | Southern |
---|---|---|---|---|---|
Normal | 146.9695 | 156.6773 | 157.6655 | 138.4266 | 163.5237 |
Log-normal | 113.0194 | 115.0962 | 137.7288 | 102.1466 | 135.4856 |
Gamma | 114.4763 | 118.5788 | 137.9854 | 104.0756 | 133.4562 |
Exponential | 114.2356 | 124.2752 | 134.9935 | 106.5699 | 135.0211 |
The 95% two-sided confidence intervals for the common percentile of delta-lognormal distributions based on the four methods were CI_{FGCI} = [5.7849,86.7576] with an interval length of 80.9727, CI_{AM} = [0.0347,161.3430] with an interval length of 161.3083, CI_{BS1} = [5.3282,66.9045] with an interval length of 61.5763, and CI_{BS2} = [23.5146,397.8828] with an interval length of 374.3682. Thus, the results of the proposed approaches to construct the confidence intervals are in accordance with those from the simulation study for a small sample size. The Bayesian confidence interval based on fiducial quantity had the shortest average length.
In this article, we are interested in the 95% confidence interval for the common percentile of rainfall data for the p = 0.95 percentile. The common percentile of rainfall data is weighted average of the percentiles for rainfall data of five regions. Generally, the p-th percentile is any number between 0 and 100. For p = 0.95, the p-th percentile is in range between 2.50 and 97.50. Therefore, the lower and upper limits of the confidence interval are the 2.50-th and 97.50-th percentiles of the common percentile for rainfall data. It can be confirmed that the lower and upper limits of the common percentile for rainfall data will work. The confidence interval therefore is in range between the lower limit and the upper limit. In interpreting practice, it was supposed to have at least a 95% chance of covering the 95th percentile. If the percentile is less than the lower limit, then we will have the observed 2.50 or fewer values in rainfall data that are below the 95th percentile. If the percentile exceeds the upper limit, then we wilhave the observed 97.50 or more out of 100 values in rainfall data that are below the 95th percentile.
Discussion
This paper proposed the two Bayesian approaches using the fiducial quantity and the approximate fiducial distribution. The prior distributions for λ_{pi} of two Bayesian approaches are difference. Then, the performances of the two Bayesian approaches are difference.
As a limitation of this paper, the FGCI approach can be used to construct the confidence interval for complex parameters, but the FGCI approach is based on the simulation techniques. Moreover, the numerical simulation is based on the maximum likelihood estimate only. The adjusted MOVER approach is easy to use the exact formula for constructing the confidence interval, but the adjusted MOVER approach requires initial confidence interval of single parameter. The Bayesian approach can be used to estimate the confidence interval for complex parameters, but the Bayesian approach is based on the simulation techniques. Moreover, the numerical simulation is based on the prior distribution.
For the sake of saving space, the simulation results for k = 4, k = 5, and k > 6 are not displayed. The results for k = 4 and k = 5 are similarly the results for k = 3. As the sample case increased, the results for k > 6 are similarly the results for k = 6.
Considering sample sizes, the performance of the proposed approaches when the sample size in each group is different are similarly the performance of the proposed approaches when the sample size in each group is equal.
The FGCI and Bayesian approaches were derived using a simulation technique whereas the adjusted MOVER approach was derived using formulas. The results in this investigation are similar to those of Harvey & van der Merwe (2012); Rao & D’Cunha (2016), Thangjai & Niwitpong (2020b), Thangjai, Niwitpong & Niwitpong (2021a) and Thangjai, Niwitpong & Niwitpong (2021b).
Conclusion
We provided confidence intervals for the common percentile of several delta-lognormal distributions using FGCI, adjusted MOVER, and two Bayesian approaches. The results indicate that CI_{BS1} was better than the others for k = 3 or k = 6. Overall, the Bayesian approach was superior to the classical approach for constructing confidence intervals for the common percentile of several delta-lognormal distributions. In future research, we will consider the statistical inference for percentiles of delta-lognormal distribution and other distributions.
Supplemental Information
The rainfall data of five regions (mm).
Northern Northeastern Central Eastern Southern.