Estimation methods for the ratio of medians of threeparameter lognormal distributions containing zero values and their application to wind speed data from northern Thailand
 Published
 Accepted
 Received
 Academic Editor
 Haider Mahmood
 Subject Areas
 Statistics, Computational Science, Natural Resource Management, Environmental Contamination and Remediation, Environmental Impacts
 Keywords
 Wind speed, Particulate matter, Confidence intervals, Ratio of medians, Fiducial, Normal approximation, Bayesian method
 Copyright
 © 2022 Maneerat 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 methods for the ratio of medians of threeparameter lognormal distributions containing zero values and their application to wind speed data from northern Thailand. PeerJ 10:e14194 https://doi.org/10.7717/peerj.14194
Abstract
Wind speed has an important impact on the formation and dispersion of fine particulate matter (PM), which can cause several health problems. During the transition from the winter to the summer season in northern Thailand, the wind speed has been low for longer than usual, which has resulted in fine PM accumulating in the air. Motivated by this, we have identified a need to investigate wind speed due to its effect on PM formation and dispersion and to raise awareness among the general public. The hourly windspeed can be approximated by using confidence intervals for the ratio of the medians of threeparameter lognormal distributions containing zero values. Thus, we constructed them by using fiducial, normal approximation, and Bayesian methods. By way of comparison, the performance measures for all ofthe proposed methods (the coverage percentage, lower and upper error probabilities (LEP and UEP,respectively), and expected length) were assessed via Monte Carlo simulation. The results of Monte Carlo simulation studies show that the Bayesian method provided coverage percentages close to the nominal confidence level and shorter intervals than the other methods. Importantly, it maintained a good balance between LEP and UEP even for large variation and percentage of zerovalued observations. To illustrate the efficacy of our proposed methods, we applied them to hourly wind speed data from northern Thailand.
Introduction
Oxygen in the air is necessary for the survival of humans and other animals. How fast the air moves past a certain point is known as wind speed (measured in km/h), which is an important phenomenon in meteorology for monitoring and predicting weather patterns in a given area. It is reported daily along with the temperature, precipitation, and humidity for all provinces on the Thai Meteorological Department website. Importantly, low wind speeds during the transition from the winter to the summer season lead to increased fine particulate matter (PM) levels. In 2020, high PM_{2.5} (PM ≤ 2.5 µm) increased the incidences of several ailments (respiratory illness, allergic reactions in the eyes and nasal passages, etc.) in almost all of the regions in Thailand, especially in the northern region (Tanraksa & Kendall, 2020). Spikes in PM_{2.5} levels in the northern and northeastern regions of Thailand occur during the transition from the winter to the summer season (January–March) (Wipatayotin & Tangprasert, 2021). These reasons have led to our interest in estimating wind speeds to provide essential information on current PM_{2.5} levels based on historic data. The hourly wind speed data for Phitsanulok and Phayao provinces located in the northern region follow the assumptions for a threeparameter lognormal (TPLN) distribution containing zero values indicating no wind. By way of comparison, estimating the ratio of the median wind speeds in two areas is used as a starting point in this study. The median, a measure of central tendency, is the central value of a dataset (the midpoint of a distribution). Moreover, it is more efficacious to use the median than the mean in analyses when the distribution of data is skewed. In addition, the ratio of the medians of two datasets can be used to measure the difference between them.
A lognormal distribution is used to represent rightskewed data when the threshold parameter (the lower bound of the data) is equal to zero. The TPLN distribution first introduced by Aitchison & Brown (1963) is suitable for highly rightskewed data that do not fit a lognormal distribution because the threshold parameter is greater than zero. It has been used in hydrology (Burges, Lettenmaier & Bates, 1975; Charbeneau, 1978) and for the analysis of flood frequency (Singh & Rajagopal, 1986; Singh & Singh, 1987). In this study, zero values are included among both simulated and wind speed data that follow a TPLN distribution (i.e., a TPLN distribution containing zero values).
One of statistical inference methods is the parameter estimations, including the point and interval estimations with the bestknown example of the latter being a confidence interval (CI). According to Casella & Berger (2002), the CI is a range of numbers containing the parameter of interest with the desirable level of confidence which is better than a point estimator. For this reason, the CI is focused in this study. The point parameter estimations of the TPLN distribution have been formulated and discussed by various authors. Cohen & Whitten (1980) modified the local maximum likelihood and the moment estimators for the mean, variance and threshold parameters by using the first, second and third order statistics. Next, the moment estimation has been developed by replacing the third moment in a function of the first order statistics, as previously described in Cohen, Whitten & Ding (1985). Later, Singh, Cruise & Ma (1990) compared the five methods, including the regular method of moments, the modified method of moment (Cohen & Whitten, 1980), the regular maximum likelihood estimate (MLE), the modified MLE, and entropy to estimate the parameters and the quantiles of the TPLN via Monte Carlo simulation.
In particular, some researchers have formulated methods for estimating the CIs for the parameters of the TPLN. Royston (1992) used the zero skewness method to estimate the threshold parameter and its certain functions, motivated by Griffiths (1980). After that, Pang et al. (2005) presented the Bayesian estimation using Markov chain Monte Carlo to approximate the coefficient of variation of threeparameter Weibull, lognormal and gamma distributions. Later, Basak, Basak & Balakrishnan (2009) made use of numerical methods (Newton–Raphson and EM algorithms) based on progressively TypeII censored samples from the TPLN for assessing the local and modified MLEs for the mean, variance and threshold parameters as well as the CIs for the threshold by using Monte Carlo simulations. Chen & Miao (2012) conducted the order statistics to construct the exact CIs and the exact upper CIs for the threshold parameter. Recently, Maneerat, Niwitpong & Nakjai (2022b) formulated the CIs for the median of the TPLN distribution based on bootstrapping, normal approximation (NA) and the generalized pivotal quantity. Maneerat, Nakjai & Niwitpong (2022a) also proposed Bayesian confidence intervals based on different noninformative priors for the deltalognormal mean.
However, the CIs for the parameters of a TPLN distribution, especially when zero values are included along with the nonzero values, have not yet been formulated. Therefore, we herein propose fiducial, NA, and Bayesian methods for constructing CIs for the ratio of the medians of TPLN distributions containing zero values. We conducted a Monte Carlo study to assess their performances in terms of their coverage percentages (CPs), lower and upper error probabilities (LEP and UEP, respectively), and expected lengths. These methods were applied to estimate hourly wind speeds during the transition from the winter to the summer season in northern Thailand. This information could be used to help design green corridors and implement other policies to reduce PM levels in northern Thailand in the future.
Threeparameter lognormal distributions containing zero values and the parameter of interest
Let X_{i} = (X_{i1}, X_{i2}, …, X_{ini}), i = 1, 2 be random samples drawn from a TPLN distribution containing zero values, denoted as $TPLNZ\left({\gamma}_{i},{\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}\right)$ where γ_{i} is the threshold parameter, μ_{Xi} is the scale parameter (the mean of X_{i}), ${\sigma}_{{X}_{i}}^{2}$ is the shape parameter (the variance of X_{i}) and ρ_{i} is the proportion of nonzero values. For X_{i} > 0, random variable Y_{i} = ln(X_{i} − γ_{i}) is normally distributed as $N\left({\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2}\right)$ if X_{i} is a lognormal random variable (Cohen & Whitten, 1980). Thus, the probability density function (pdf) of X_{i} is given by (1)$h\left({X}_{i};{\gamma}_{i},{\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}\right)=\left(1{\rho}_{i}\right)+\frac{{\rho}_{i}{\left({X}_{i}{\gamma}_{i}\right)}^{1}}{{\left(2\pi {\sigma}_{{X}_{i}}^{2}\right)}^{1/2}}exp\left\{\frac{1}{2{\sigma}_{{X}_{i}}^{2}}{\left[ln\left({X}_{i}{\gamma}_{i}\right){\mu}_{{X}_{i}}\right]}^{2}\right\},$
where γ_{i} < X_{i} < ∞, which are bounded as follows: γ_{i} ≥ 0, 0 < μ_{Xi} < ∞, ${\sigma}_{{X}_{i}}^{2}>0$ and 0 < ρ_{i} < 1. Otherwise, for X_{i} = 0, the pdf of W becomes $h\left({X}_{i};{\gamma}_{i},{\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}\right)=1{\rho}_{i}$. The likelihood and loglikelihood functions of $\omega =\left({\gamma}_{i},{\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}\right)$ are respectively given by
(2)$L\left(\omega X\right)={\left(1{\rho}_{i}\right)}^{{n}_{i0}}{\rho}_{i}^{{n}_{i1}}{\left(2\pi {\sigma}_{{X}_{i}}^{2}\right)}^{{n}_{i1}/2}\prod _{j=1}^{{n}_{i1}}exp\left\{ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{2{\sigma}_{{X}_{i}}^{2}}{\left[ln\left({X}_{ij}{\gamma}_{i}\right){\mu}_{{X}_{i}}\right]}^{2}\right\}$ (3)$l\left(\omega X\right)={n}_{i0}ln\left(1{\rho}_{i}\right)+{n}_{i1}ln{\rho}_{i}\frac{{n}_{i1}}{2}ln\left(2\pi {\sigma}_{{X}_{i}}^{2}\right)\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{2{\sigma}_{{X}_{i}}^{2}}\sum _{j:{x}_{ij}>0}{\left[ln\left({X}_{ij}{\gamma}_{i}\right){\mu}_{{X}_{i}}\right]}^{2}$
where n_{i1} is the number of nonzero values that are binomially distributed with sample size n_{i} = n_{i0} + n_{i1}; ρ_{i} is the proportion of nonzero values ; n_{i0} is the number of zero values. Eq. (3) is computed from the first derivative about $\left({\gamma}_{i},{\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}\right)$. The respective MLEs of $\left({\gamma}_{i},{\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}\right)$ are solved by setting their first derivative to zero as follows:
(4)$\frac{\partial l\left(\omega X\right)}{\partial {\gamma}_{i}}=\sum _{j:{x}_{ij}>0}\frac{1}{{X}_{ij}{\gamma}_{i}}+\frac{{n}_{i1}}{\sum _{j:{x}_{ij}>0}{\left[ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{{n}_{i1}}\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)\right]}^{2}}\sum _{j:{x}_{ij}>0}\left[\frac{1}{{X}_{ij}{\gamma}_{i}}\left\{ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{{n}_{i1}}\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)\right\}\right]=0$ (5)$\frac{\partial l\left(\omega X\right)}{\partial {\mu}_{{X}_{i}}}={\sigma}_{{X}_{i}}^{2}\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)=0$ (6)$\frac{\partial l\left(\omega X\right)}{\partial {\sigma}_{{X}_{i}}^{2}}={\left(2{\sigma}_{{X}_{i}}^{2}\right)}^{1}{n}_{i1}+{\left[2{\sigma}_{{X}_{i}}^{4}\right]}^{1}\sum _{j:{x}_{ij}>0}{\left[ln\left({X}_{ij}{\gamma}_{i}\right){\mu}_{{X}_{i}}\right]}^{2}=0$ (7)$\frac{\partial l\left(\omega X\right)}{\partial {\rho}_{i}}={n}_{i0}{\left(1\rho \right)}^{1}{n}_{i1}{\rho}^{1}=0.$
It is difficult to determine the explicit form of ${\stackrel{\u02c6}{\gamma}}_{i}$ from the Eq. (4), so the MLE of γ_{i} can be obtained by maximizing Eq. (4) using the Adam algorithm. It is a replacement optimization algorithm for stochastic gradient descent based on adaptive moment estimation that is mainly used in neural networks and other machine learning algorithms (Ruder, 2016). It provides a way of computing adaptive learning rates for specific parameters. Let β_{1} and β_{2} be the initial time step and decay rate, respectively. Adam performs well in practice when β_{1} = 0.9 and β_{2} = 0.999 (Singh, 2020). By optimizing γ_{i}, the decaying averages and the past squared gradients are the estimates of the first and second moments of the gradients as follows: (8)$\left({m}_{t},{v}_{t}\right)=\left({\beta}_{1}{m}_{t1}+\left[1{\beta}_{1}\right]{g}_{t},{\beta}_{2}{v}_{t1}+\left[1{\beta}_{2}\right]{g}_{t}^{2}\right)$
where g_{t} denotes the first derivative of the target function (4) at time t = 0. Note that (m_{t}, v_{t}) are initialized as vectors of 0’s, and the Adam algorithm operates until their gradients approach zero. The biascorrected first and secondmoment estimates are used to update parameter γ_{i} as follows: (9)$\left({\stackrel{\u02c6}{m}}_{t},{\stackrel{\u02c6}{v}}_{t}\right)=\left(\frac{{m}_{t}}{1{\beta}_{1}^{t}},\frac{{v}_{t}}{1{\beta}_{2}^{t}}\right).$
Finally, the parameter γ_{i} is updated by Adam update rule, which is defined as (10)$\gamma}_{i,t+1}={\gamma}_{i,t}\delta {\left(\sqrt{{\stackrel{\u02c6}{v}}_{t}}+\u03f5\right)}^{1}{\stackrel{\u02c6}{m}}_{t$
where δ and ϵ denote the step size and the learning rate, respectively. ϵ is fixed at 10^{−8} for a sufficient learning rate using the Adam optimization algorithm, as can be seen in Algorithm 1.
After obtaining ${\stackrel{\u02c6}{\gamma}}_{i}$ by using Adam, one can compute the MLEs of $\left({\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}\right)$ as follows: (11)$\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}},{\stackrel{\u02c6}{\sigma}}_{{X}_{i},mle}^{2},{\stackrel{\u02c6}{\rho}}_{i}\right)=\left({n}_{i1}^{1}\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\stackrel{\u02c6}{\gamma}}_{i}\right),{n}_{i1}^{1}\sum _{j:{x}_{ij}>0}{\left[ln\left({X}_{ij}{\stackrel{\u02c6}{\gamma}}_{i}\right){\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right]}^{2},\frac{{n}_{i1}}{{n}_{i}}.\right)$
Since η_{i} = ρ_{i}[γ_{i} + exp(μ_{Xi})] provides the medians of X_{i}, the ratio of the medians of X_{i} (the parameter of interest in the this study) is given by (12)$\omega ={\eta}_{1}/{\eta}_{2}.$
By substituting MLEs $\left({\stackrel{\u02c6}{\gamma}}_{i},{\stackrel{\u02c6}{\mu}}_{{X}_{i}},{\stackrel{\u02c6}{\sigma}}_{{X}_{i},mle}^{2},{\stackrel{\u02c6}{\rho}}_{i}\right)$, estimate ${\stackrel{\u02c6}{\eta}}_{i}={\stackrel{\u02c6}{\rho}}_{i}\left[{\stackrel{\u02c6}{\gamma}}_{i}+exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)\right]$ becomes the MLE of η_{i}. The concepts are elaborated in the methods for constructing CIs for ω in the next section.
Methods
Here, we present constructing the CIs for the ratio of the medians in the TPLN models containing zero observations based on the fiducial methods (the fiducial generalized pivotal quantity (fiducial GPQ)) and the method of variance estimates recoverybased fiducial generalized pivotal quantity (MOVERfiducial GPQ), NA (McKean & Schrader, 1984) and the Bayesian methods.
The fiducial method
Hannig, Iyer & Patterson (2006) described the fiducial method based on the fiducial GPQ, a GPQ subclass introduced by Weerahandi (1993). Fiducial techniques have been used in constructing CIs in several research studies (KharratiKopaei, Malekzadeh & SadooghiAlvandi, 2013; Li, Zhou & Tian, 2013; Maneerat, Niwitpong & Niwitpong, 2020). Let Y = (Y_{1}, Y_{2}, …, Y_{n}) be a random variable with probability density function f_{Y}(y; λ, θ), where λ and θ are vectors of the parameter of interest and the nuisance parameter, respectively. Moreover, let y = (y_{1}, y_{2}, …, y_{n}) be the observed values of Y, and assume that fiducial GPQ T(Y; y, λ, θ) is only a function of λ. This method is especially associated with the fiducial inference proposed by Fisher (1935). The fiducial GPQ CI depends on the fiducial GPQ defined by Hannig, Iyer & Patterson (2006) in Definition 1. Recently, Chankham, Niwitpong & Niwitpong (2022) recommended the fiducial GPQbased CI for estimating the coefficient of variation of an inverse gaussian distribution when the sample size was small. Similarly, the performance of a fiducial GPQ CI in terms of expected length was the shortest when used to estimate the common coefficient of variation of deltalognormal distributions by Yosboonruang, Niwitpong & Niwitpong (2022).
Definition 1.
A GPQ T(Y; y, λ, θ) for a parameter λ is called a fiducial generalized pivotal quantity (FGPQ) if it satisfies the following conditions: (FGPQ1) Given Y = y, the T(Y; y, λ, θ) distribution is free of all parameters. (FGPQ2) For every y ∈ R^{+}, the observed pivotal T(Y; y, λ, θ) = λ.
If the T(Y; y, λ, θ) satisfies the conditions (FGPQ1) and (FGPQ2), then it is possible to construct the 100(1 − φ)% fiducial GPQbased CI for λ is [T_{λ}(φ/2), T_{λ}(1 − φ/2)]; T_{λ}(φ) is the φ^{th} percentile of T(Y; y, λ, θ).
The fiducial GPQ CI
The fiducial GPQ CI can be constructed based on the fiducial GPQ conditions in Definition 1. Recall that (${\gamma}_{i},{\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2},{\rho}_{i}$) are the parameters controlling the behavior of a TPLN containing zero observations. Motivated by the possible value γ_{i} < X_{i(1)}, we proposed the fiducial GPQ of γ_{i} based on a continuously uniform distribution and the fiducial GPQ of ρ_{i} based on a beta distribution, respectively, as follows:
(13)${T}_{{\gamma}_{i}}\sim \text{Uniform}\left(\text{min}=0,\text{max}={X}_{i\left(1\right)}\right)$ (14)${T}_{1{\rho}_{i}}\sim \text{Beta}\left({n}_{i0}+1,{n}_{i1}+1\right)$
where X_{i(1)} denotes the minimum value of X_{i} and n_{i1} is the sample size of nonzero values n_{i} = n_{i1} + n_{i0}. Furthermore, the fiducial GPQ of μ_{Xi} can be obtained by using the concepts of Krishnamoorthy & Mathew (2003) as (15)$T}_{{\mu}_{{X}_{i}}}={\stackrel{\u02c6}{\mu}}_{{X}_{i}}{W}_{i}\sqrt{{T}_{{\sigma}_{{X}_{i}}^{2}}/{n}_{i1}$
where ${T}_{{\sigma}_{{X}_{i}}^{2}}=\left({n}_{i1}1\right){\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}/{V}_{i}$ is the fiducial GPQ of ${\sigma}_{{X}_{i}}^{2}$. Moreover, estimator ${\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}={\left({n}_{i1}1\right)}^{1}{\sum}_{j:{x}_{ij}>0}ln\left({X}_{ij}{\stackrel{\u02c6}{\gamma}}_{i}\right)$. Note that the random variables W_{i} and V_{i} are independently draw from a standard normal distribution and a chisquare distribution with n_{i1} − 1 degrees of freedom, respectively. Since the three fiducial pivots (T_{γi}, T_{ρi}, T_{μXi}) satisfy the fiducial GPQ properties and T_{ηi} = T_{1−ρi}[T_{γi} + exp(T_{μXi})] is the fiducial GPQ of η_{i}, then we can obtain the the 100(1 − φ)% fiducial GPQ CI for ω as follows: (16)$\left[{l}_{\omega}^{F},{u}_{\omega}^{F}\right]=\left[{T}_{\omega}\left(\phi /2\right),{T}_{\omega}\left(1\phi /2\right)\right]$
where T_{ω} = T_{η1}/T_{η2}, and T_{ω}(φ) denotes the φ^{th} percentile of T_{ω}. The steps for calculating the CP of fiducial GPQ CI (CP_{F}) for ω can be carried out by using Algorithm 2.
The MOVERfiducial GPQ CI
MOVER is a wellknown method for estimating the CI of the parameter of interest (Donner & Zou, 2012; Harvey & Van der Merwe, 2012; Hasan & Krishnamoorthy, 2017; Maneerat & Niwitpong, 2020; Maneerat, Niwitpong & Niwitpong, 2021; Zhang et al., 2021; Maneerat, Nakjai & Niwitpong, 2022a). Moreover, it can produce an explicit form of the CI that is easy to compute. For these reasons, we derived the MOVERfiducial GPQ CI for ω as follows:
The fiducial GPQ CIs for (γ_{i}, μ_{Xi}, ρ_{i}) can respectively be written as
(17)$\left[{l}_{{\gamma}_{i}},{u}_{{\gamma}_{i}}\right]=\left[{T}_{{\gamma}_{i}}\left(\phi /2\right),{T}_{{\gamma}_{i}}\left(\phi /2\right)\right]$ (18)$\left[{l}_{{\mu}_{{X}_{i}}},{u}_{{\mu}_{{X}_{i}}}\right]=\left[{T}_{{\mu}_{{X}_{i}}}\left(\phi /2\right),{T}_{{\mu}_{{X}_{i}}}\left(\phi /2\right)\right]$ (19)$\left[{l}_{{\rho}_{i}},{u}_{{\rho}_{i}}\right]=\left[{T}_{{\rho}_{i}}\left(\phi /2\right),{T}_{{\rho}_{i}}\left(\phi /2\right)\right].$
These intervals can be formulated the MOVER CI by using the concept of Donner & Zou (2012) such that the 100(1 − φ)% MOVERfiducial CI for lnη_{i} becomes (20)$\left[{l}_{ln{\eta}_{i}},{u}_{ln{\eta}_{i}}\right]=\left[ln{\stackrel{\u02c6}{\eta}}_{i}\sqrt{{\left\{ln\left({\stackrel{\u02c6}{\rho}}_{i}\right){l}_{{\gamma}_{i}}\right\}}^{2}+{\left\{ln\left[{\stackrel{\u02c6}{\gamma}}_{i}+exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)\right]{l}_{ln\left[{\gamma}_{i}+exp\left({\mu}_{{X}_{i}}\right)\right]}\right\}}^{2}},ln{\stackrel{\u02c6}{\eta}}_{i}\right.\left.+\sqrt{{\left\{ln\left({\stackrel{\u02c6}{\rho}}_{i}\right){u}_{{\gamma}_{i}}\right\}}^{2}+{\left\{ln\left[{\stackrel{\u02c6}{\gamma}}_{i}+exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)\right]{u}_{ln\left[{\gamma}_{i}+exp\left({\mu}_{{X}_{i}}\right)\right]}\right\}}^{2}}\right]$
where
(21)$l}_{ln\left[{\gamma}_{i}+exp\left({\mu}_{{X}_{i}}\right)\right]}=ln\left\{{\stackrel{\u02c6}{\gamma}}_{i}+exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)\sqrt{{\left\{{\stackrel{\u02c6}{\gamma}}_{i}{l}_{{\gamma}_{i}}\right\}}^{2}+{\left\{exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)exp\left({l}_{{\gamma}_{i}}\right)\right\}}^{2}}\right\$ (22)${u}_{ln\left[{\gamma}_{i}+exp\left({\mu}_{{X}_{i}}\right)\right]}=ln\left\{{\stackrel{\u02c6}{\gamma}}_{i}+exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)+\sqrt{{\left\{{\stackrel{\u02c6}{\gamma}}_{i}{l}_{{\gamma}_{i}}\right\}}^{2}+{\left\{exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)exp\left({l}_{{\gamma}_{i}}\right)\right\}}^{2}}\right\},$
thereby providing the 100(1 − φ)% CIbased the MOVERfiducial GPQ CI for ω as (23)$\left[{l}_{\omega}^{\text{M}},{u}_{\omega}^{\text{M}}\right]=exp\left[\left(ln{\stackrel{\u02c6}{\eta}}_{1}ln{\stackrel{\u02c6}{\eta}}_{2}\right)\sqrt{{\left[ln{\stackrel{\u02c6}{\eta}}_{1}{l}_{ln{\stackrel{\u02c6}{\eta}}_{1}}\right]}^{2}+{\left[ln{\stackrel{\u02c6}{\eta}}_{2}{l}_{ln{\stackrel{\u02c6}{\eta}}_{2}}\right]}^{2}},\left(ln{\stackrel{\u02c6}{\eta}}_{1}ln{\stackrel{\u02c6}{\eta}}_{2}\right)\right.\left.+\sqrt{{\left[ln{\stackrel{\u02c6}{\eta}}_{1}{l}_{ln{\stackrel{\u02c6}{\eta}}_{1}}\right]}^{2}+{\left[ln{\stackrel{\u02c6}{\eta}}_{2}{l}_{ln{\stackrel{\u02c6}{\eta}}_{2}}\right]}^{2}}\right].$
Algorithm 3 presents the computational steps for calculating the CP of the MOVERfiducial GPQ CI (CP_{M}) for ω.
The NA method
According to probability theory, the concept behind this method is the assumption that the approximate distributions of all of the samples approach a normal distribution pattern if the sample size is sufficiently large. This idea is integrated with the central limit theorem, in which the distribution of a given sample mean is approximated as a normal pattern if the sample size is sufficiently large under the assumption that all of the samples are similar to each other regardless of the shape of the population distribution. Recently, Maneerat, Niwitpong & Nakjai (2022b) proposed an NAbased CI for the median of a TPLN distribution, which performed well for a large sample size. Thus, we also considered the NA method.
Given a set of observations, threshold γ_{i} can be estimated by using the Adam algorithm to find the MLEs of the mean and variance (${\stackrel{\u02c6}{\mu}}_{{X}_{i}},{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}$). Here, the medians of TPLN models with zero observations can be logtransformed to become (24)$ln{\eta}_{i}=ln{\rho}_{i}+ln\left[{\gamma}_{i}+exp\left({\mu}_{{X}_{i}}\right)\right]$
which can be approximated by using (${\stackrel{\u02c6}{\gamma}}_{i},{\stackrel{\u02c6}{\mu}}_{{X}_{i}},{\stackrel{\u02c6}{\rho}}_{i}$) to give $ln{\stackrel{\u02c6}{\eta}}_{i}=ln{\stackrel{\u02c6}{\rho}}_{i}+ln\left[{\stackrel{\u02c6}{\gamma}}_{i}+exp\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right)\right]$. Using the delta method, the variance of lnρ_{i} can be derived as (25)${V}_{ln{\stackrel{\u02c6}{\rho}}_{i}}=\frac{1{\rho}_{i}}{n{\rho}_{i}}.$
Likewise, McKean & Schrader (1984) estimated the variance of the median as a distributionfree estimate defined as follows: (26)$\text{V}}_{{\stackrel{\u02c6}{\eta}}_{i}}^{MS}={\left[\frac{{X}_{i\left({n}_{i1}c+1\right)}+{X}_{i\left(c\right)}}{2\left(1.96\right)}\right]}^{2$
where X_{i(r)} denotes the r^{th} order statistic in a random sample drawn from a TPLN model with zero observations of size n_{i1}, and $c=\left[\left({n}_{i1}+1\right)/2\right]1.96\left(\sqrt{{n}_{i1}/4}\right)$. Later, Hettmansperger & Sheather (1986) claimed that V^{MS} is a consistent estimator of the variance of the median. Similarly, the variance of $ln{\stackrel{\u02c6}{\eta}}_{i}$ can be derived by applying the delta method as follows: (27)$\text{V}}_{ln{\stackrel{\u02c6}{\eta}}_{i}}\cong \frac{{\text{V}}_{{\stackrel{\u02c6}{\eta}}_{i}}^{MS}}{{\left[{\text{E}}_{{\stackrel{\u02c6}{\eta}}_{i}}\right]}^{2}$
where ${\text{E}}_{{\stackrel{\u02c6}{\eta}}_{i}}={\stackrel{\u02c6}{\gamma}}_{i}+exp\left[{\stackrel{\u02c6}{\mu}}_{{X}_{i}}+{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}{\left(2{n}_{i1}\right)}^{1}\right]$ is the expectation of ${\stackrel{\u02c6}{\eta}}_{i}$. Estimated variance ${\widehat{\text{V}}}_{ln{\stackrel{\u02c6}{\eta}}_{i}}$ is obtainable by replacing $\left({\stackrel{\u02c6}{\gamma}}_{i},{\stackrel{\u02c6}{\mu}}_{{X}_{i}},{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2},{\stackrel{\u02c6}{\rho}}_{i}\right)$ from the sample. Thus, by applying the central limit theorem , the random variable W_{i} can be defined as (28)$W}_{i}=\frac{ln{\stackrel{\u02c6}{\eta}}_{i}ln{\eta}_{i}}{\sqrt{{\widehat{\text{V}}}_{ln{\stackrel{\u02c6}{\eta}}_{i}}}$
which approaches a standard normal distribution as n → ∞. Subsequently, the 100(1 − φ)% NAbased CI for ω can be written as (29)$\left[{l}_{\omega}^{\text{N}},{u}_{\omega}^{\text{N}}\right]=exp\left[\left(ln{\stackrel{\u02c6}{\eta}}_{1}ln{\stackrel{\u02c6}{\eta}}_{2}\right)\mp {W}_{1\phi /2}\sqrt{{\widehat{\text{V}}}_{ln{\stackrel{\u02c6}{\eta}}_{1}}+{\widehat{\text{V}}}_{ln{\stackrel{\u02c6}{\eta}}_{2}}}\right]$
where W_{φ} denotes the φ^{th} percentile of a standard normal distribution. Algorithm 4 was used to compute the CP of the NA (CP_{N}).
The Bayesian method
Bayesian methods are based on treating probability as beliefs rather than frequencies. Given unknown parameter θ, a prior distribution p(θ) represents the subjective belief as a subjective distribution formulated before the data are seen. The posterior distribution is obtained from a prior that is updated with the likelihood function (or sample information) by using Bayes’ rule (Casella & Berger, 2002). Importantly, the posterior distribution is considered to be a random quantity and can be used to make a statement about θ, For example, the point and interval estimates of θ can be computed by using its posterior. Equaltailed Bayesian intervals based on the Jeffreys’ and uniform priors based on the posterior densities of the zero proportion and the variance have been shown to perform well in certain scenarios (Yosboonruang, Niwitpong & Niwitpong , 2022).
Here, the Bayesian CI for ω is formulated based on the Bayesian method. First, we define an informative prior for our objective assumption depending on the amount of information available in the data as (30)$P\left({\gamma}_{1i},{\mu}_{{X}_{i}},log{\sigma}_{{X}_{i}},{\rho}_{i}\right)=\text{constant}\ast {\left(1{\rho}_{i}\right)}^{{\alpha}_{i}1}{\rho}_{i}^{{\beta}_{i}1}\Gamma \left({\alpha}_{i}+{\beta}_{i}\right){\left[\Gamma \left({\alpha}_{i}\right)\Gamma \left({\beta}_{i}\right)\right]}^{1}.$
The posterior densities of (γ_{i}, ρ_{i}) are obtained by obtaining Eq. (30) with the likelihood function (2) as
(31)${f}_{{\gamma}_{i}}=P\left({\gamma}_{i}X,{\stackrel{\u02c6}{\mu}}_{{X}_{i}},{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}\right)=\left[\text{constant}\ast {\left(1{\rho}_{i}\right)}^{{\alpha}_{i}1}{\rho}_{i}^{{\beta}_{i}1}\Gamma \left({\alpha}_{i}+{\beta}_{i}\right){\left[\Gamma \left({\alpha}_{i}\right)\Gamma \left({\beta}_{i}\right)\right]}^{1}\right]\ast \left[{\left(1{\rho}_{i}\right)}^{{n}_{i0}}{\rho}_{i}^{{n}_{i1}}{\left(2\pi {\sigma}_{{X}_{i}}^{2}\right)}^{{n}_{i1}/2}exp\left\{\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{2{\sigma}_{{X}_{i}}^{2}}\sum _{j:{x}_{ij}>0}{\left[ln\left({X}_{ij}{\gamma}_{i}\right){\mu}_{{X}_{i}}\right]}^{2}\right\}\right]$ (32)$\propto {\left({\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}\right)}^{{n}_{i1}/2}exp\left\{\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{2{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}}\sum _{j:{x}_{ij}>0}{\left[ln\left({X}_{ij}{\gamma}_{i}\right){\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right]}^{2}\right\}$ (33)${f}_{{\rho}_{i}}=P\left({\rho}_{i}X\right)=\left[\text{constant}\ast {\left(1{\rho}_{i}\right)}^{{n}_{i0}+{\alpha}_{i}1}{\rho}_{i}^{{n}_{i1}{\beta}_{i}1}\Gamma \left({\alpha}_{i}+{\beta}_{i}\right){\left[\Gamma \left({\alpha}_{i}\right)\Gamma \left({\beta}_{i}\right)\right]}^{1}\right]\ast \left[{\left(2\pi {\sigma}_{{X}_{i}}^{2}\right)}^{{n}_{i1}/2}exp\left\{\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{2{\sigma}_{{X}_{i}}^{2}}\sum _{j:{x}_{ij}>0}{\left[ln\left({X}_{ij}{\gamma}_{i}\right){\mu}_{{X}_{i}}\right]}^{2}\right\}\right]$ (34)$\propto {\left(1{\rho}_{i}\right)}^{{n}_{i0}+{\alpha}_{i}1}{\rho}_{i}^{{n}_{i1}+{\beta}_{i}1}.$
Next, we apply NA to the posterior distribution of κ_{i} = (μ_{Xi}, logσ_{Xi}); the logarithm of the posterior density is approximated by using a quadratic function of κ_{i}. The second derivatives of the logposterior density are needed for constructing the approximation. From Eq. (3), the loglikelihood can be expressed as (35)$l\left({\kappa}_{i}X\right)={n}_{i0}ln\left(1{\rho}_{i}\right)+{n}_{i1}ln{\rho}_{i}\frac{{n}_{i1}}{2}ln\left(2\pi \right)+{n}_{i1}ln{\sigma}_{{X}_{i}}\sum _{j:{x}_{ij}>0}ln\left({X}_{ij}{\gamma}_{i}\right)\frac{1}{2{\sigma}_{{X}_{i}}^{2}}\left[\left({n}_{i1}1\right){\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}+{n}_{i1}{\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}{\mu}_{{X}_{i}}\right)}^{2}\right]\propto \text{constant.}+{n}_{i1}ln{\sigma}_{{X}_{i}}\frac{1}{2exp\left(2ln{\sigma}_{{X}_{i}}\right)}\left[\left({n}_{i1}1\right){\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}+{n}_{i1}{\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}{\mu}_{{X}_{i}}\right)}^{2}.\right]$
After that the first and second derivatives of (μ_{Xi}, logσ_{Xi}) respectively become
(36)$\left(\frac{\partial l\left({\kappa}_{i}X\right)}{\partial {\mu}_{{X}_{i}}},\frac{{\partial}^{2}l\left({\kappa}_{i}X\right)}{\partial {\mu}_{{X}_{i}}^{2}}\right)=\left(\frac{{n}_{i1}\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}{\mu}_{{X}_{i}}\right)}{{\sigma}_{{X}_{i}}^{2}},\frac{{n}_{i1}}{{\sigma}_{{X}_{i}}^{2}}\right)$ (37)$\left(\frac{\partial l\left({\kappa}_{i}X\right)}{\partial ln{\sigma}_{{X}_{i}}},\frac{{\partial}^{2}l\left({\kappa}_{i}X\right)}{\partial {\left[ln{\sigma}_{{X}_{i}}\right]}^{2}}\right)=\left({n}_{i1}+{\sigma}_{{X}_{i}}^{2}\left[\left({n}_{i1}1\right){\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}+{n}_{i1}{\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}{\mu}_{{X}_{i}}\right)}^{2}\right]\right.,\left.2{\sigma}_{{X}_{i}}^{1}\left[\left({n}_{i1}1\right){\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}+{n}_{i1}{\left({\stackrel{\u02c6}{\mu}}_{{X}_{i}}{\mu}_{{X}_{i}}\right)}^{2}\right]\right).$
The point estimates of (μ_{Xi}, logσ_{Xi}) are derived after setting their first derivatives to zero. Meanwhile, the variances of their estimates are respectively obtained by using an inverse Fisher information matrix as follows:
(38)$\left(\begin{array}{c}\hfill {\stackrel{\u02c6}{\mu}}_{{X}_{i}}\hfill \\ \hfill ln{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\stackrel{\u02c6}{\mu}}_{{X}_{i}}\hfill \\ \hfill ln\sqrt{\left({n}_{i1}1\right){n}_{i1}^{1}{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}}\hfill \end{array}\right)$ (39)$\left(\begin{array}{c}\hfill {\text{V}}_{{\stackrel{\u02c6}{\mu}}_{{X}_{i}}}\hfill \\ \hfill {V}_{ln{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}/{n}_{i1}\hfill \\ \hfill 2{n}_{i1}\hfill \end{array}\right).$
After using the Jacobian to backtransform from $ln{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}$ to ${\sigma}_{{X}_{i}}^{2}$, the posterior densities of (${\mu}_{{X}_{i}},{\sigma}_{{X}_{i}}^{2}$) are respectively approximated as normal distribution as follows:
(40)$f}_{{\mu}_{{X}_{i}}}=p\left({\mu}_{{X}_{i}}X,{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2},{\stackrel{\u02c6}{\gamma}}_{i}\right)=\frac{1}{{\left(2\pi {\sigma}_{{X}_{i}}^{2}/{n}_{i1}\right)}^{1/2}}exp\left\{\frac{{n}_{i1}}{2{\sigma}_{{X}_{i}}^{2}}{\left[ln\left({X}_{ij}{\stackrel{\u02c6}{\gamma}}_{i}\right){\stackrel{\u02c6}{\mu}}_{{X}_{i}}\right]}^{2}\right\$ (41)$f}_{{\sigma}_{{X}_{i}}^{2}}=p\left({\sigma}_{{X}_{i}}^{2}X\right)={\left(\frac{{n}_{i1}+2}{4\pi {\stackrel{\u0303}{\sigma}}_{{X}_{i}}^{4}}\right)}^{1/2}exp\left\{\frac{{n}_{i1}+2}{4{\stackrel{\u0303}{\sigma}}_{{X}_{i}}^{4}}{\left[ln\left({X}_{ij}{\stackrel{\u02c6}{\gamma}}_{i}\right){\stackrel{\u0303}{\sigma}}_{{X}_{i}}^{2}\right]}^{2}\right\$
where ${\stackrel{\u0303}{\sigma}}_{{X}_{i}}^{2}={n}_{i1}{\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}/\left({n}_{i1}+2\right)$. Therefore, the posterior density of ω becomes (42)$f}_{\omega}=\frac{{f}_{{\rho}_{1}}\left[{f}_{{\gamma}_{1}}+exp\left({f}_{{\mu}_{{X}_{1}}}\right)\right]}{{f}_{{\rho}_{2}}\left[{f}_{{\gamma}_{2}}+exp\left({f}_{{\mu}_{{X}_{2}}}\right)\right]$
Finally, the 100(1 − φ)% Bayesianbased CI for ω is (43)$\left[{l}_{\omega}^{B},{u}_{\omega}^{B}\right]=\left[{f}_{\omega}\left(\phi /2\right),{f}_{\omega}\left(1\phi /2\right)\right]$
where f_{ω}(φ) denotes the φ^{th} percentile of f_{ω}. The CP of Bayesian CI (CP_{B}) can be computed by using Algorithm 5.
The Monte Carlo simulation study
The comparative performances of the proposed Bayesian, the fiducial GPQ and MOVERfiducial GPQ, and NAbased CI were evaluated via a Monte Carlo simulation study. The settings for the simulation parameters for the two simulation studies were as follows. For each specified parameter combination, the 95% CIs for the ratio of the medians (ω) of TPLN distributions containing zero values were constructed based on 5000 randomly generated samples. In addition, 2500 Monte Carlo sampling passes were used for each generated sample for the fiducial GPQbased method. To assess the performances of the methods, their CPs were calculated by using the proportion of 5000 simulated CIs covering ω. LEP and UEP are defined as the proportion of times that ω falls below and above the stimulated CIs, respectively. At the 95% nominal confidence level, the expected lengths (ELs) of the CIs is also needed for deciding which method performs the best. A good performance will produce CP = 95% and LEP = UEP = 2.5%. Likewise, the comparison between LEP and UEP can be expressed in terms of the relative bias, which is defined as (44)$\text{RB}=\frac{\text{UEP}\text{LEP}}{\text{UEP}+\text{LEP}}.$
Thus, a good balance between LEP and UEP will produce a relative bias close to zero. Last, the bestperforming method will provide the shortest EL.
In the first simulation study, we chose a small proportion of zeros and variance ((d_{1}, d_{2}) = (10%, 10%), (10%, 30%) and ${\sigma}_{{X}_{1}}^{2}={\sigma}_{{X}_{2}}^{2}=1.25$, respectively), the results of which provide insight into the sampling behavior of the CIs (Table 1 and Figs. 1, 2 and 3). It can be seen that although all of the methods generated CPs above or close to the nominal confidence level in almost all of the scenarios, the LEPs, UEPs, and ELs produced by the Bayesian method demonstrated its superiority. Thus, the Bayesian method performed the best in situations with a small proportion of zeros and sample variance except for with a large sample size, with the MOVERfiducial GPQ method performing the best in that case.
Scenarios  (n_{1}, n_{2})  (ρ_{1}, ρ_{2})%  (γ_{1}, γ_{2})  B  F  M  N  Relative bias  

LEP  CP  UEP  EL  LEP  CP  UEP  EL  LEP  CP  UEP  EL  LEP  CP  UEP  EL  B  F  M  N  
1  (25,25)  (10,10)  (1,1)  0.72  98.30  0.98  163.37  0.18  99.52  0.30  200.56  0.06  99.48  0.46  189.01  0.34  99.18  0.48  211.70  0.15  0.25  0.77  0.79 
2  (1,3)  0.66  98.44  0.90  156.22  0.28  99.48  0.24  198.36  0.00  99.58  0.42  186.41  0.28  99.26  0.46  202.52  0.15  −0.08  1.00  1.00  
3  (3,5)  0.74  98.44  0.82  142.20  0.24  99.68  0.08  195.00  0.00  99.80  0.20  182.46  0.36  99.28  0.36  182.52  0.05  −0.50  1.00  1.00  
4  (10,30)  (1,1)  0.96  98.30  0.74  180.80  0.36  99.42  0.22  222.67  0.18  99.58  0.24  208.90  0.48  99.14  0.38  240.36  0.13  −0.24  0.14  1.00  
5  (1,3)  0.90  97.98  1.12  171.38  0.54  99.22  0.24  218.83  0.04  99.46  0.50  204.42  0.44  99.00  0.56  225.81  0.11  −0.38  0.85  1.00  
6  (3,5)  0.90  98.02  1.08  157.74  0.28  99.54  0.18  215.19  0.08  99.52  0.40  200.41  0.44  99.08  0.48  206.96  0.09  −0.22  0.67  1.00  
7  (25,50)  (10,10)  (1,1)  0.38  96.92  2.70  136.08  0.12  98.84  1.04  167.07  0.12  99.00  0.88  158.35  0.14  98.98  0.88  177.91  0.75  0.79  0.76  0.56 
8  (1,3)  0.48  96.26  3.26  132.95  0.12  98.94  0.94  167.09  0.04  98.64  1.32  158.41  0.26  98.42  1.32  173.17  0.74  0.77  0.94  1.00  
9  (3,5)  0.28  96.96  2.76  121.35  0.02  99.36  0.62  165.16  0.02  98.96  1.02  156.11  0.14  98.86  1.00  156.03  0.82  0.94  0.96  1.00  
10  (10,30)  (1,1)  0.70  96.48  2.82  146.90  0.20  98.60  1.20  179.03  0.12  98.90  0.98  169.60  0.36  98.80  0.84  190.57  0.60  0.71  0.78  1.00  
11  (1,3)  0.44  96.86  2.70  142.87  0.18  98.86  0.96  178.49  0.06  98.70  1.24  168.96  0.36  98.70  0.94  183.63  0.72  0.68  0.91  1.00  
12  (3,5)  0.54  96.94  2.52  130.98  0.16  99.28  0.56  176.39  0.02  98.96  1.02  166.45  0.32  98.88  0.80  167.46  0.65  0.56  0.96  1.00  
13  (50,50)  (10,10)  (1,1)  2.20  95.28  2.52  103.10  0.74  98.34  0.92  125.77  0.38  98.60  1.02  121.00  0.56  98.66  0.78  137.55  0.07  0.11  0.46  0.47 
14  (1,3)  2.30  95.34  2.36  99.18  1.12  98.40  0.48  125.60  0.18  98.80  1.02  120.90  0.46  98.90  0.64  131.36  0.01  −0.40  0.70  1.00  
15  (3,5)  2.20  95.40  2.40  91.07  0.70  99.10  0.20  126.29  0.00  98.96  1.04  121.84  0.48  99.02  0.50  119.67  0.04  −0.56  1.00  1.00  
16  (10,30)  (1,1)  2.32  96.16  1.52  116.99  0.90  98.52  0.58  140.93  0.46  98.80  0.74  135.19  0.64  99.04  0.32  153.10  0.21  −0.22  0.23  1.00  
17  (1,3)  2.44  95.88  1.68  111.47  1.14  98.36  0.50  140.01  0.18  98.92  0.90  134.02  0.70  98.72  0.58  145.15  0.18  −0.39  0.67  1.00  
18  (3,5)  2.52  95.76  1.72  102.94  0.88  98.86  0.26  139.91  0.04  99.06  0.90  134.30  0.66  98.74  0.60  132.03  0.19  −0.54  0.91  1.00  
19  (50,100)  (10,10)  (1,1)  1.28  92.60  6.12  86.81  0.36  97.48  2.16  106.31  0.28  97.58  2.14  102.76  0.42  98.28  1.30  116.03  0.65  0.71  0.77  0.07 
20  (1,3)  1.34  92.66  6.00  84.88  0.64  98.04  1.32  107.17  0.12  97.80  2.08  104.00  0.34  98.38  1.28  112.56  0.63  0.35  0.89  1.00  
21  (3,5)  1.18  92.76  6.06  77.90  0.28  98.80  0.92  108.69  0.00  97.64  2.36  105.99  0.18  98.52  1.30  102.12  0.67  0.53  1.00  1.00  
22  (10,30)  (1,1)  1.56  94.06  4.38  93.63  0.50  97.66  1.84  113.43  0.34  97.88  1.78  109.72  0.38  98.58  1.04  123.78  0.47  0.57  0.68  1.00  
23  (1,3)  1.56  94.08  4.36  91.27  0.80  97.98  1.22  114.13  0.18  97.98  1.84  110.65  0.34  98.66  1.00  119.67  0.47  0.21  0.82  1.00  
24  (3,5)  1.38  94.16  4.46  84.33  0.44  98.78  0.78  115.57  0.02  98.16  1.82  112.46  0.42  98.52  1.06  109.19  0.53  0.28  0.98  1.00  
25  (100,100)  (10,10)  (1,1)  3.94  92.68  3.38  66.72  1.40  97.36  1.24  82.27  0.66  98.10  1.24  80.42  0.96  98.42  0.62  90.22  0.08  −0.06  0.31  −0.05 
26  (1,3)  4.22  92.16  3.62  64.20  2.08  97.30  0.62  83.51  0.12  98.12  1.76  82.05  0.74  98.56  0.70  86.14  0.08  −0.54  0.87  1.00  
27  (3,5)  4.36  91.82  3.82  59.41  0.82  98.80  0.38  87.00  0.00  98.32  1.68  86.29  0.64  98.54  0.82  78.57  0.07  −0.37  1.00  1.00  
28  (10,30)  (1,1)  4.32  92.50  3.18  75.39  1.56  97.30  1.14  91.48  0.68  97.88  1.44  89.20  0.86  98.32  0.82  99.62  0.15  −0.16  0.36  1.00  
29  (1,3)  4.64  92.48  2.88  72.36  2.00  97.58  0.42  92.34  0.14  98.58  1.28  90.33  0.58  98.70  0.72  94.97  0.23  −0.65  0.80  1.00  
30  (3,5)  4.78  92.48  2.74  67.33  1.46  98.32  0.22  95.34  0.04  98.74  1.22  94.07  0.76  98.56  0.68  87.22  0.27  −0.74  0.94  1.00  
31  (100,200)  (10,10)  (1,1)  1.66  89.92  8.42  56.58  0.54  96.54  2.92  70.38  0.30  96.88  2.82  69.05  0.36  98.42  1.22  76.93  0.67  0.69  0.81  −0.44 
32  (1,3)  2.38  90.08  7.54  55.34  1.14  97.62  1.24  72.25  0.18  97.10  2.72  71.61  0.64  98.02  1.34  74.88  0.52  0.04  0.88  1.00  
33  (3,5)  2.14  90.02  7.84  51.27  0.62  98.86  0.52  76.67  0.00  97.22  2.78  76.83  0.48  97.98  1.54  68.25  0.57  −0.09  1.00  1.00  
34  (10,30)  (1,1)  2.42  91.16  6.42  61.27  0.90  97.04  2.06  75.23  0.34  97.48  2.18  73.82  0.58  98.46  0.96  81.88  0.45  0.39  0.73  1.00  
35  (1,3)  2.64  90.74  6.62  59.68  1.28  97.66  1.06  76.81  0.18  97.06  2.76  75.98  0.50  98.16  1.34  79.33  0.43  −0.09  0.88  1.00  
36  (3,5)  3.00  90.52  6.48  55.51  0.88  98.46  0.66  80.91  0.04  97.10  2.86  80.90  0.76  97.88  1.36  72.88  0.37  −0.14  0.97  1.00 
Notes:
 B

the Bayesian CI
 F

the fiducial GPQ CI
 M

the MOVERfiducial GPQ CI
 N

the NA CI
In the next simulation, we were interested in scenarios with a large proportion of zeros and variance (d_{1} = d_{2} = (20%, 40%), (40%, 40%) and ${\sigma}_{{X}_{1}}^{2}={\sigma}_{{X}_{2}}^{2}=3$, respectively) (Table 2 and Figs. 4, 5 and 6). Once again, the Bayesian method provided acceptable CPs, as well as better ELs and a better balance between LEP and UEP, than the other methods.
Application of the methods to compare hourly wind speed data from two areas in northern Thailand
Due to the rapid effects of climate change, agricultural growth, and the social economy, seasonal air pollution from the burning of agricultural waste in preparation for planting, forest fires, and waste disposal during the transition from the winter to the dry season are important factors that influence the environment in northern Thailand (IQAir, 2021). Wind can affect the movement of PM_{2.5} and PM_{10} when its speed is 7.2 km/hr or higher (Liu et al., 2020). When cold air mass moves from China to Thailand, the upper region of Thailand can potentially become very cold toward the end of winter, which reduces the northeast monsoon to a calm wind. For this reason, PM_{2.5} levels usually increase during the transition between the winter season to the dry season (Teerasuphaset & Culp, 2020). Phitsanulok is a city in lower northern Thailand about halfway between Chiang Mai and Bangkok where crop and forestland burning is extensive, resulting in extreme PM_{2.5} occurrences (IQAir, 2022),while Phayao is one of the three highestranking provinces for PM_{2.5} in the upper northern region (Group, 2021).
We used datasets of the hourly wind speed from Phitsanulok and Phayao (Table 3) recorded in January 2021 to illustrate the efficacies of our proposed methods for formulating CIs for the ratio of the medians of TPLN distributions containing zero values. The data were taken from the Thai Meteorological Department Automatic Weather System (Thai Meteorlogical Department Automatic Weather System, 2022). Since wind speed observations are always nonnegative, they are suitable for fitting to the following distributions: Cauchy, chisquared, exponential, lognormal, TPLN, logistic, normal, and tdistributions. The Akaike information criterion (AIC) can be used to determine the bestfitting distribution.
Scenarios  (n_{1}, n_{2})  (ρ_{1}, ρ_{2})%  (γ_{1}, γ_{2})  B  F  M  N  Relative bias  

LEP  CP  UEP  EL  LEP  CP  UEP  EL  LEP  CP  UEP  EL  LEP  CP  UEP  EL  B  F  M  N  
37  (25,25)  (20,40)  (1,1)  1.70  97.30  1.00  259.44  1.28  98.10  0.62  293.83  1.52  97.88  0.60  280.54  0.40  99.30  0.30  444.30  −0.26  −0.35  −0.43  0.89 
38  (1,3)  1.36  97.60  1.04  245.93  1.08  98.52  0.40  286.48  0.92  98.70  0.38  269.61  0.46  99.30  0.24  417.03  −0.13  −0.46  −0.42  1.00  
39  (3,5)  1.44  97.24  1.32  224.79  1.04  98.40  0.56  276.06  0.46  98.92  0.62  255.06  0.34  99.10  0.56  380.39  −0.04  −0.30  0.15  1.00  
40  (40,40)  (1,1)  1.66  97.02  1.32  283.20  1.00  98.28  0.72  323.39  1.38  97.84  0.78  307.55  0.24  99.36  0.40  516.55  −0.11  −0.16  −0.28  1.00  
41  (1,3)  0.78  98.06  1.16  270.82  0.42  99.00  0.58  317.18  0.50  98.72  0.78  297.65  0.20  99.52  0.28  486.72  0.20  0.16  0.22  1.00  
42  (3,5)  1.48  97.20  1.32  248.95  0.82  98.58  0.60  306.52  0.46  98.88  0.66  282.32  0.36  99.40  0.24  445.82  −0.06  −0.15  0.18  1.00  
43  (25,50)  (20,40)  (1,1)  1.22  96.54  2.24  213.10  0.92  97.48  1.60  235.89  1.06  97.38  1.56  227.43  0.44  99.02  0.54  320.58  0.29  0.27  0.19  0.73 
44  (1,3)  1.36  95.90  2.74  205.82  1.16  97.12  1.72  232.74  0.92  97.18  1.90  222.39  0.64  98.62  0.74  309.26  0.34  0.19  0.35  1.00  
45  (3,5)  1.14  96.56  2.30  188.47  0.82  97.82  1.36  225.40  0.42  98.10  1.48  211.28  0.26  99.02  0.72  283.15  0.34  0.25  0.56  1.00  
46  (40,40)  (1,1)  1.00  96.68  2.32  242.04  0.70  97.66  1.64  272.84  0.82  97.60  1.58  260.74  0.40  98.94  0.66  404.46  0.40  0.40  0.32  1.00  
47  (1,3)  0.70  96.42  2.88  235.54  0.56  97.68  1.76  270.25  0.48  97.60  1.92  256.41  0.34  98.80  0.86  395.13  0.61  0.52  0.60  1.00  
48  (3,5)  1.04  96.58  2.38  214.38  0.72  97.78  1.50  259.15  0.48  97.84  1.68  241.05  0.38  98.86  0.76  360.16  0.39  0.35  0.56  1.00  
49  (50,50)  (20,40)  (1,1)  1.96  96.32  1.72  177.64  1.52  97.26  1.22  193.23  1.60  97.12  1.28  187.25  0.90  98.70  0.40  249.14  −0.07  −0.11  −0.11  0.67 
50  (1,3)  2.02  96.36  1.62  168.55  2.06  96.98  0.96  189.16  1.28  97.54  1.18  180.85  0.76  98.72  0.52  237.29  −0.11  −0.36  −0.04  1.00  
51  (3,5)  1.88  95.98  2.14  155.15  1.64  97.46  0.90  184.51  0.74  98.02  1.24  173.93  0.76  98.64  0.60  218.59  0.06  −0.29  0.25  1.00  
52  (40,40)  (1,1)  1.50  97.02  1.48  192.68  1.20  97.58  1.22  209.88  1.20  97.64  1.16  203.27  0.54  98.90  0.56  273.71  −0.01  0.01  −0.02  1.00  
53  (1,3)  1.96  96.44  1.60  184.84  2.00  97.12  0.88  206.78  1.26  97.58  1.16  198.06  0.96  98.42  0.62  261.51  −0.10  −0.39  −0.04  1.00  
54  (3,5)  1.48  96.80  1.72  169.90  1.28  97.74  0.98  200.86  0.58  98.20  1.22  189.24  0.52  98.74  0.74  241.20  0.08  −0.13  0.36  1.00  
55  (50,100)  (20,40)  (1,1)  1.56  96.02  2.42  147.79  1.12  97.06  1.82  159.74  1.24  97.00  1.76  155.22  0.52  98.76  0.72  202.75  0.22  0.24  0.17  0.34 
56  (1,3)  1.30  96.16  2.54  142.96  1.26  97.20  1.54  158.18  0.74  97.24  2.02  152.63  0.48  98.98  0.54  196.49  0.32  0.10  0.46  1.00  
57  (3,5)  1.36  96.22  2.42  131.20  1.24  97.54  1.22  154.66  0.46  97.88  1.66  147.32  0.42  98.82  0.76  180.39  0.28  −0.01  0.57  1.00  
58  (40,40)  (1,1)  1.38  96.32  2.30  165.56  1.06  96.80  2.14  179.74  1.10  96.86  2.04  174.31  0.30  98.60  1.10  230.58  0.25  0.34  0.30  1.00  
59  (1,3)  1.16  96.36  2.48  160.90  1.16  97.36  1.48  178.08  0.70  97.36  1.94  171.83  0.48  98.48  1.04  223.81  0.36  0.12  0.47  1.00  
60  (3,5)  1.32  96.10  2.58  148.06  1.14  97.42  1.44  173.56  0.50  97.54  1.96  164.89  0.60  98.46  0.94  206.05  0.32  0.12  0.59  1.00  
61  (100,100)  (20,40)  (1,1)  2.40  95.50  2.10  122.51  1.98  96.50  1.52  131.58  1.68  96.78  1.54  128.30  0.70  98.56  0.74  165.11  −0.07  −0.13  −0.04  0.25 
62  (1,3)  2.28  95.80  1.92  116.67  2.30  96.74  0.96  129.62  1.14  97.30  1.56  125.17  0.86  98.54  0.60  156.54  −0.09  −0.41  0.16  1.00  
63  (3,5)  2.30  95.52  2.18  107.65  2.20  96.92  0.88  128.12  0.48  98.06  1.46  122.77  0.86  98.46  0.68  143.65  −0.03  −0.43  0.51  1.00  
64  (40,40)  (1,1)  2.20  95.52  2.28  133.62  1.78  96.48  1.74  143.42  1.64  96.46  1.90  139.78  0.72  98.46  0.82  180.55  0.02  −0.01  0.07  1.00  
65  (1,3)  1.88  95.86  2.26  128.21  2.00  96.68  1.32  141.60  0.94  97.30  1.76  136.82  0.68  98.54  0.78  172.85  0.09  −0.20  0.30  1.00  
66  (3,5)  1.58  95.90  2.52  118.25  1.56  97.06  1.38  139.27  0.40  97.62  1.98  133.33  0.44  98.66  0.90  158.17  0.23  −0.06  0.66  1.00  
67  (100,200)  (20,40)  (1,1)  1.58  95.16  3.26  101.46  1.34  96.36  2.30  108.80  1.24  96.34  2.42  106.37  0.64  98.36  1.00  134.21  0.35  0.26  0.32  −0.20 
68  (1,3)  1.46  95.54  3.00  98.30  1.70  96.74  1.56  108.44  0.88  96.68  2.44  105.68  0.40  98.50  1.10  130.20  0.35  −0.04  0.47  1.00  
69  (3,5)  1.68  95.32  3.00  90.42  1.54  97.32  1.14  107.71  0.36  97.30  2.34  104.56  0.56  98.46  0.98  119.13  0.28  −0.15  0.73  1.00  
70  (40,40)  (1,1)  1.40  95.62  2.98  114.31  1.04  96.50  2.46  122.60  1.02  96.54  2.44  119.66  0.38  98.56  1.06  152.21  0.36  0.41  0.41  1.00  
71  (1,3)  1.28  95.74  2.98  111.45  1.54  96.78  1.68  122.19  0.68  96.70  2.62  118.93  0.36  98.48  1.16  148.24  0.40  0.04  0.59  1.00  
72  (3,5)  1.58  95.00  3.42  102.93  1.24  97.16  1.60  121.02  0.48  96.70  2.82  116.90  0.52  98.26  1.22  135.95  0.37  0.13  0.71  1.00 
Notes:
 B

the Bayesian CI
 F

the fiducial GPQ CI
 M

the MOVERfiducial GPQ CI
 N

the NA CI
Phisanulok  Phayao  

19.9  0  4.3  13.7  9.4  0  26.7  15 
20.6  0  21.6  21.9  16.1  2.2  20.9  13.3 
26  11.6  12.6  19.1  17.3  7.5  19.1  9.7 
9.4  22.7  18.4  21.2  8.6  15.8  20.5  14 
9.8  16.5  16.6  20.1  2.9  14.5  21.7  11.9 
12.6  6.1  23.3  20.2  1.8  4.3  29.9  11.1 
11.6  0  31  9  0  2.9  33.1  14.3 
0  6.8  6.1  22  1.4  4.3  23  12.3 
20.9  3.6  1.8  20.9  9.7  2.2  22.6  17.2 
14.8  1.5  5  12.6  23.4  0  29.9  6.1 
27.3  9  1.8  0  19.8  5.4  24.1  4.7 
13  10.4  0  0  3.9  12.6  16.2  3.7 
0  18.3  14.4  4.3  1.1  15.4  32  6.1 
5.4  8.3  14.5  6.9  0  6.2  29.4  4.6 
3.2  0  24.4  16.6  0  3.2  14  11.1 
8.7  0.4  6.9  15.9  1.8  2.9  25.2  23.4 
8.7  0  2.5  11.6  4.3  0.4  30.2  21.2 
13  0  4.3  6.9  22.3  0.4  23.4  4.7 
25.9  8.7  2.2  0  21.2  10.8  31.6  1.8 
8  25.9  2.5  0  7.6  12.2  29.5  4.7 
5  21.7  15.8  0  0.4  21.6  20.1  1.4 
3.6  8.3  15.2  2.1  1.1  14.8  30.6  0 
5.7  4.7  14.4  4.3  0  13  18.4  7.2 
6.2  4.7  7.2  14.4  0.7  17.6  21.2  17.3 
7.6  2.2  15.8  19.5  7.2  16.9  14.1  16.6 
13  10.1  26.3  5  15.5  2.1  14.8  3.9 
25.6  14  32  0  16.2  8.2  19.1  3.2 
15.1  20.5  32.7  4.7  5  21.9  25.3  1.8 
2.2  17.6  32.4  0  0  21.3  24.5  0.7 
3.6  5.8  20.2  4  0  23.5  19.5  0 
Notes:
Source: Thai Meteorological Department Automatic Weather System.
We found that the TPLN model was suitable for the wind speed data, as evidenced by its smallest AIC value in Table 4. The basic statistics for the datasets are reported in Table 5. By way of comparison, the ratio of the medians of the TPLN distributions of the hourly wind speed data from Phayao and Phitsanulok is $\stackrel{\u02c6}{\omega}=1.0217$ for medians ${\stackrel{\u02c6}{\eta}}_{phayao}=7.4091$ and ${\stackrel{\u02c6}{\eta}}_{phitsanulok}=7.2514$. The 95% CIs and corresponding lengths based on the fiducial, NA, and Bayesian methods for the ratio of the medians of TPLN distributions containing zero values are reported in Table 6.
Provinces  Cauchy  Chisuqare  Exponential  Lognormal  TPLN  Logistic  Normal  T 

Phitsanulok  787.8140  818.3827  739.9249  733.9261  725.9633  742.5044  736.4505  736.7834 
Phayao  865.3148  1008.1470  793.3140  816.9264  800.6639  812.1632  803.3119  803.8294 
Notes:
Bold font denotes the bestfitting model for the data.
Basic statistics  Provinces  

Phitsanulok  Phayao  
Groups (i)  1  2 
sample sizes (n_{i})  120  120 
Mean (${\stackrel{\u02c6}{\mu}}_{i}$)  2.1581  2.0196 
Variances (${\stackrel{\u02c6}{\sigma}}_{{X}_{i}}^{2}$)  1.4911  2.3389 
Zero proportions (${\stackrel{\u02c6}{\rho}}_{i}:\text{\%}$)  13.33  8.33 
Threshold (${\stackrel{\u02c6}{\gamma}}_{i}$)  0.399  0.389 
Median wind speed (${\stackrel{\u02c6}{\eta}}_{i}$)  8.0248  6.8204 
The ratio of the medians ($\stackrel{\u02c6}{\omega}={\stackrel{\u02c6}{\eta}}_{1}/{\stackrel{\u02c6}{\eta}}_{2}$)  $\stackrel{\u02c6}{\omega}=1.1765$ 
Methods  95% CIs  Lengths 

Fiducial GPQ  (0.7825, 1.7937)  1.0112 
MOVERfiducial GPQ  (0.7823, 1.7203)  0.9379 
NA  (0.6922, 1.9997)  1.3074 
Bayesian  (0.8047, 1.8238)  1.0190 
It can be interpreted that there is no difference between the wind speeds in Phayao and Phitsanulok. The majority of the population in both areas are agriculturists, and so agricultural burning is often carried out in preparation for planting and after harvesting. Furthermore, the empirical example results are in agreement with the Monte Carlo simulation results in the previous section; the EL of the MOVERfiducial GPQ CI was the smallest with a suitable CP for a small variance and a large sample size. Overall, the Bayesianbased method is the most suitable for formulating CIs for the ratio of the medians of TPLN distributions containing zero values when taking the checking criteria results from scenarios 1–12 and 37–72 into account.
Discussion
We applied fiducial, NA, and Bayesianbased method to formulate CIs for the ratio of the medians of TPLN distributions containing zero values. From the results of the simulation study, the fiducial methods based on fiducial GPQ and MOVERfiducial GPQ always provided CPs greater than the nominal 95% confidence level because the fiducial GPQ of (γ_{i}, ρ_{i}, μ_{Xi}) have strong points (Hannig, Iyer & Patterson, 2006), as revealed by the conditions for FGPQ2 in Definition 1. However, the MOVERfiducial GPQ method produced shorter interval than the fiducial GPQ and worked well for a small variance and a large sample size. The NA method provided CPs greater than the nominal 95% confidence level, thereby making its ELs longer than the other methods, which could have been caused by the variance in the estimated median (McKean & Schrader, 1984).
Meanwhile, the Bayesian method provided suitable CPs with the shortest interval length, except for a small variance and a large sample size for which the MOVERfiducial GPQ method performed the best. This could be because of using the uniform prior to gain information about the parameters from the data to obtain their posterior density. As such, the constructed CIs for the ratio of the medians of TPLN distributions containing zero values based on the Bayesian method with a uniform prior performed well.
Conclusions
CIs for the ratio of the medians of TPLN distributions containing zero values were formulated by using fiducial, NA, and Bayesianbased methods. Since a theoretical comparison was not possible, a Monte Carlo simulation and empirical application with two real datasets of wind speed observations were used to evaluate their performances in terms of their CPs and ELs. The results of the simulation study led us to recommend the Bayesian method for constructing the CIs for the ratio of the medians of TPLN distributions containing zero values because it attained CPs close to the nominal 95% confidence level and the shortest EL in most cases, except for a small variance and a large sample size for which the MOVERfiducial GPQ method should be used.