Estimation of common percentile of rainfall datasets in Thailand using delta-lognormal distributions

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 $k⩾2$ 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 σ². The probability density function of V_i is defined as

(1) $$f(v_i;\mu ,{\sigma }^2)=\{\begin{array}{cc}{\displaystyle \frac{1}{v_i\sigma \sqrt{2\pi }}\exp \left(-{\displaystyle \frac{1}{2{\sigma }^2}{\left(\ln (v_i)-\mu \right)}^2}\right);}& v_i>0\\ 0;& otherwise\end{array}$$

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 σ², 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) $$G(x_i;\mu ,{\sigma }^2,{\delta }^{{}^{\prime }})=\{\begin{array}{cc}{\delta }^{{}^{\prime }};& x=0\\ {\delta }^{{}^{\prime }}+\delta F(x_i;\mu ,{\sigma }^2);& x>0\end{array},$$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 S² be the mean and variance based on log-transformed sample from log-normal distribution. Also, $\stackrel{\mathrm{\_}}{y}$ and s² be observed values of $\stackrel{\mathrm{\_}}{Y}$₍₁₎ and S², respectively. Let $\stackrel{\mathrm{\_}}{Y}$₍₁₎ and S²₍₁₎ be the mean and variance based on the log-transformed positive sample measurements. And let $\stackrel{\mathrm{\_}}{y}$₍₁₎ and s²₍₁₎ be observed values of $\stackrel{\mathrm{\_}}{Y}$₍₁₎ and S²₍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) $$q_p=\exp \left(\mu +{\mathrm{\Phi }}^{-1}\left({\displaystyle \frac{p-{\delta }^{{}^{\prime }}}{1-{\delta }^{{}^{\prime }}}}\right)\sigma \right),$$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,σ²_i), where X_i > 0. Let $\stackrel{\mathrm{\_}}{Y}$_i(1) and S²_i(1) be the mean and variance based on the log-transformed positive sample measurements, respectively. Also, let $\stackrel{\mathrm{\_}}{y}$_i(1) and s²_i(1) be the observed values of $\stackrel{\mathrm{\_}}{Y}$_i(1) and S²_i(1), respectively.

The estimator of the pth quantile of the delta-lognormal distribution is defined by

(5) $${\hat{q}}_{p_i}=\exp ({\hat{\lambda }}_{p_i}),$$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 S²_i(1) are

$$Var\left({\overline{Y}}_{i(1)}\right)={\left(\exp \left({\mu }_i\right)\right)}^2\ast \left({\displaystyle \frac{{\sigma }_i^2}{n_{i(1)}}+1}\right)\ast {\left(\exp \left({\displaystyle \frac{\left(n_{i(1)}-1\right){\sigma }_i^2}{2\left(n_{i(1)}+4\right)\left(n_{i(1)}-1\right)+3{\sigma }_i^2}}\right)\right)}^2$$

$$\ast \left({\displaystyle \frac{2\left(n_{i(1)}-1\right)\left({\sigma }_i^4/n_{i(1)}\right)}{2\left(n_{i(1)}+4\right)\left(n_{i(1)}-1\right)+\left(6{\sigma }_i^4/n_{i(1)}\right)}+1}\right)$$

(6) $$-{\left(\exp \left({\mu }_i\right)\ast \exp \left({\displaystyle \frac{\left(n_{i(1)}-1\right){\sigma }_i^2}{2\left(n_{i(1)}+4\right)\left(n_{i(1)}-1\right)+3{\sigma }_i^2}}\right)\right)}^2$$and

(7) $$Var\left(S_{i(1)}^2\right)=\left({\displaystyle \frac{2{\sigma }_i^2}{n_{i(1)}}\ast \exp \left(4{\mu }_i+2{\sigma }_i^2\right)}\right)\ast \left(2{\left(\exp \left({\sigma }_i^2\right)-1\right)}^2+{\sigma }_i^2{\left(2\exp \left({\sigma }_i^2\right)-1\right)}^2\right).$$

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(S²_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) $$\hat{\theta }=\exp ({\hat{\lambda }}_p),$$where ${\hat{\lambda }}_p$ is defined in Eq. (9).

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 μ₁ = μ₂ = … = 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 n₁,n₂,…,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.