Estimating the average daily rainfall in Thailand using confidence intervals for the common mean of several deltalognormal distributions
 Published
 Accepted
 Received
 Academic Editor
 Graciela Raga
 Subject Areas
 Agricultural Science, Statistics, Computational Science, Natural Resource Management, Ecohydrology
 Keywords
 Agriculture, Bayesian approach, MOVER, Natural rainfall, Vague prior, Variance
 Copyright
 © 2021 Maneerat and Niwitpong
 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
 2021. Estimating the average daily rainfall in Thailand using confidence intervals for the common mean of several deltalognormal distributions. PeerJ 9:e10758 https://doi.org/10.7717/peerj.10758
Abstract
The daily average natural rainfall amounts in the five regions of Thailand can be estimated using the confidence intervals for the common mean of several deltalognormal distributions based on the fiducial generalized confidence interval (FGCI), large sample (LS), method of variance estimates recovery (MOVER), parametric bootstrap (PB), and highest posterior density intervals based on Jeffreys’ rule (HPDJR) and normalgammabeta (HPDNGB) priors. Monte Carlo simulation was conducted to assess the performance in terms of the coverage probability and average length of the proposed methods. The numerical results indicate that MOVER and PB provided better performances than the other methods in a variety of situations, even when the sample case was large. The efficacies of the proposed methods were illustrated by applying them to real rainfall datasets from the five regions of Thailand.
Introduction
Approximately 82.2% of Thailand’s cultivated land area depends on natural rainfall (Supasod, 2006), thereby indicating its importance for Thai agriculture. However, it is a natural phenomenon with a significant level of uncertainty that can cause natural disasters such as droughts, floods, and landslides. In many countries around the world, extreme rainfall events have been increasing in frequency and duration. On December 5, 2017, Storm Desmond led to heavy rainfall causing flooding in northern England, Southern Scotland, and Ireland (Otto & Oldenborgh, 2017). On July 6–7, 2018, extreme rainfall events such as floods and landslides affected over 5,000 houses, and approximately 1.9 million people in Japan were evacuated from the atrisk area (Oldenborgh, 2018). In midSeptember 2019, the amount of rainfall was extreme during Tropical Storm Imelda in Southeast Texas, USA, where over 1,000 people were affected by largescale flooding and there were five deaths (Oldenborgh et al., 2019). Thus, it is necessary to assess how rainfall varies in each region of a country on a daily basis. Due to the climate pattern and meteorological conditions, Thailand is commonly separated into five regions: northern, northeastern, central, eastern, and southern. The rainfall in each region varies widely due to both location and seasonality. Importantly, Thailand’s rainfall data include many zeros with probability δ > 0 and positive rightskewed data following a lognormal distribution for the remainder of the probability. Thus, applying a deltalognormal distribution (Aitchison, 1955) is appropriate.
The mean is a measure of the center of a set of observations (Casella & Berger, 2002) that can be used in statistical inference, while functions of the mean such as the ratio or difference between two means can also be used. These parameters have been applied in many research areas, such as medicine, fish stocks, pharmaceutics, and climatology. For example, they have been used for hypothesis testing of the effect of race on the average medical costs between African American and Caucasian patients with type I diabetes (Zhou, Gao & Hui, 1997), to estimate the mean charges for diagnostic tests on patients with unstable chronic medical conditions (Zhou & Tu, 2000; Tian, 2005; Tian & Wu, 2007; Li, Zhou & Tian, 2013), to estimate the maximum alcohol concentration in men in an alcohol interaction study (Tian & Wu, 2007; Krishnamoorthy & Oral, 2015), to estimate the mean red cod density around New Zealand as an indication of fish abundance (Fletcher, 2008; Wu & Hsieh, 2014), and to estimate the mean of the monthly rainfall totals to compare rainfall in Bloemfontein and Kimberley in South African (Harvey & van der Merwe, 2012).
In practice, the mean has been widely used in many fields, as mentioned before. When independent samples are recorded from several situations, then the common mean is of interest when studying more than one population. Many researchers have investigated methods for constructing confidence interval (CIs) for the common mean of several distributions. For example, Fairweather (1972) proposed a linear combination of Student’s t to construct CIs for the common mean of several normal distributions. Jordan & Krishnamoorthy (1996) solved the problem of CIs for the common mean under unknown and unequal variances based on Student’s t and independent F variables from several normal populations. Krishnamoorthy & Mathew (2003) presented the generalized CI (GCI) and compared it with the CIs constructed by Fairweather (1972), and Jordan & Krishnamoorthy (1996). Later, Lin & Lee (2005) developed a GCI for the common mean of several normal populations. Tian & Wu (2007) provided CIs for the common mean of several lognormal populations using the generalized variable approach, which was shown to be consistently better than the large sample (LS) approach. Lin & Wang (2013) studied the modification of the quadratic method to make inference via hypothesis testing and interval estimation for several lognormal means. Krishnamoorthy & Oral (2015) proposed the method of variance estimates recovery (MOVER) approach for the common mean of lognormal distributions.
As mentioned earlier, many researchers have developed CIs for the common mean of several normal and lognormal distributions. However, there has not yet been an investigation of statistical inference using the common mean of several deltalognormal distributions. Since the common mean is used to study more than one population, the average precipitation in the five regions in Thailand can be estimated using it as there is an important need to estimate the daily rainfall trends in these regions. Furthermore, the daily rainfall records from the five regions in Thailand satisfy the assumptions for a deltalognormal distribution. Herein, CIs for the common mean of several deltalognormal models based on the fiducial GCI (FGCI), LS, MOVER, parametric bootstrap (PB), and highest posterior density (HPD) intervals based on Jeffreys’ rule (HPDJR) and normalgammabeta (HPDNGB) priors are proposed. The outline of this article is as follows. The ideas behind the proposed methods are detailed in the Methods section. Numerical computations are reported in ‘Simulation Studies and Results’. In ‘An Empirical Application’, the daily natural rainfall records of the five regions in Thailand are used to illustrate the efficacy of the methods. Finally, the paper is ended with a discussion and conclusions.
Methods
Let W_{ij} = (W_{i1}, W_{i2}, …, W_{ini}) be random samples drawn from a deltalognormal distribution, for i = 1, 2, …, k and j = 1, 2, .., n_{i}. There are three parameters in this distribution: the mean μ_{i}, variance ${\sigma}_{i}^{2}$ and the probability of obtaining a zero observation δ_{i}. The distribution of W_{ij} is given by (1)$H\left({w}_{ij};{\mu}_{i},{\sigma}_{i}^{2},{\delta}_{i}\right)=\left\{\begin{array}{cc}{\delta}_{i}\phantom{\rule{10.00002pt}{0ex}}\hfill & ;{w}_{ij}=0\hfill \\ {\delta}_{i}+\left(1{\delta}_{i}\right)G\left({w}_{ij};{\mu}_{i},{\sigma}_{i}^{2}\right)\phantom{\rule{10.00002pt}{0ex}}\hfill & ;{w}_{ij}>0\hfill \end{array}\right.$ where $G\left({w}_{ij};{\mu}_{i},{\sigma}_{i}^{2}\right)$ is a lognormal distribution function, denoted as $LN\left({\mu}_{i},{\sigma}_{i}^{2}\right)$ such that $ln{W}_{ij}\sim N\left({\mu}_{i},{\sigma}_{i}^{2}\right)$. The number of zeros has a binomial distribution ${n}_{i\left(0\right)}=\#\left\{j:{w}_{ij}=0\right\}\sim B\left({n}_{i},{\delta}_{i}\right)$. The population mean of W_{ij} is given by (2)${\vartheta}_{i}=\left(1{\delta}_{i}\right)exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right)$
The unbiased estimates of ${\mu}_{i},{\sigma}_{i}^{2}$, and δ_{i} are ${\stackrel{\u02c6}{\mu}}_{i}={n}_{i\left(1\right)}^{1}{\sum}_{j:{w}_{ij}>0}ln{W}_{ij}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}={\left({n}_{i\left(1\right)}1\right)}^{1}{\sum}_{j:{w}_{ij}>0}{\left[ln{W}_{ij}{\stackrel{\u02c6}{\mu}}_{i}\right]}^{2}$, and ${\stackrel{\u02c6}{\delta}}_{i}={n}_{i\left(0\right)}\u2215{n}_{i}$, respectively, where n_{i} = n_{i(0)} + n_{i(1)}; ${n}_{i\left(1\right)}=\#\left\{j:{w}_{ij}>0\right\}$. Suppose that the deltalognormal mean in Eq. (2) for all k populations are the same, then according to Tian & Wu (2007) and Krishnamoorthy & Oral (2015), the common deltalognormal mean is defined as (3)$\vartheta =\left(1{\delta}_{i}\right)exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right).$
For the ith sample, the estimates of ϑ_{i} are ${\stackrel{\u02c6}{\vartheta}}_{i}^{\ast}=\left(1{\stackrel{\u02c6}{\delta}}_{i}\right)exp\left({\stackrel{\u02c6}{\mu}}_{i}+\frac{{\stackrel{\u02c6}{\sigma}}_{i}^{2}}{2}\right)$ which contains the unbiased estimates ${\stackrel{\u02c6}{\mu}}_{i}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ and ${\stackrel{\u02c6}{\delta}}_{i}$. According to Longford (2009), the expected value of ${\stackrel{\u02c6}{\vartheta}}_{i}^{\ast}$ is derived as
(4)$E\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\ast}\right]=\left[1E\left({\stackrel{\u02c6}{\delta}}_{i}\right)\right]E\left[exp\left\{{\stackrel{\u02c6}{\mu}}_{i}+\frac{{\stackrel{\u02c6}{\sigma}}_{i}^{2}}{2}\right\}\right]$ (5)$=\left(1{\delta}_{i}\right)exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{{n}_{i\left(1\right)}}\right){\left(\frac{{l}_{i}}{{l}_{i}{\sigma}_{i}^{2}}\right)}^{{l}_{i}\u22152}$ where ${\stackrel{\u02c6}{\delta}}_{i}\sim N\left({\delta}_{i},\frac{{\delta}_{i}\left(1{\delta}_{i}\right)}{{n}_{i}}\right)$ as n_{i} → ∞, $E\left[exp\left({\stackrel{\u02c6}{\mu}}_{i}\right)\right]=exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2{n}_{i\left(1\right)}}\right)$ and E[exp(c_{i}Y_{i})] = (1 − 2c_{i})^{−l∕2}; ${Y}_{i}={l}_{i}\frac{{\stackrel{\u02c6}{\sigma}}_{i}^{2}}{{\sigma}_{i}^{2}}\sim {\chi}_{{l}_{i}}^{2}$ and ${c}_{i}=\frac{{\sigma}_{i}^{2}}{2{l}_{i}}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}={\left({n}_{i\left(1\right)}1\right)}^{1}{\sum}_{j=1}^{{n}_{i\left(1\right)}}{\left[ln\left({W}_{ij}\right){\stackrel{\u02c6}{\mu}}_{i}\right]}^{2}$. If $\frac{{l}_{i}{\sigma}_{i}^{2}}{{l}_{i}}=exp\left[\frac{2{\sigma}_{i}^{2}}{{l}_{i}}\left(\frac{1}{2}\frac{1}{2{n}_{i\left(1\right)}}\right)\right]$, then we can obtain that
(6)$E\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\ast}\right]=\left(1{\delta}_{i}\right)exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2{n}_{i\left(1\right)}}\right){\left\{exp\left[\frac{2{\sigma}_{i}^{2}}{{l}_{i}}\left(\frac{1}{2}\frac{1}{2{n}_{i\left(1\right)}}\right)\right]\right\}}^{{l}_{i}\u22152}=\left(1{\delta}_{i}\right)exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right).$
According to Aitchison & Brown (1963), the Aitchison estimate of ϑ_{i} is expressed as (7)${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}=\left\{\begin{array}{cc}0\phantom{\rule{10.00002pt}{0ex}}\hfill & ;{n}_{i\left(1\right)}=0\hfill \\ {w}_{i1}\u2215{n}_{i}\phantom{\rule{10.00002pt}{0ex}}\hfill & ;{n}_{i\left(1\right)}=1\hfill \\ \left(1\stackrel{\u02c6}{{\delta}_{i}}\right)exp\left(\stackrel{\u02c6}{{\mu}_{i}}\right){\psi}_{{n}_{i\left(1\right)}}\left(\frac{{\stackrel{\u02c6}{\sigma}}_{i}^{2}}{2}\right)\phantom{\rule{10.00002pt}{0ex}}\hfill & ;{n}_{i\left(1\right)}>1\hfill \end{array}\right.$ where ψ_{a}(b) is a Bessel function defined as (8)${\psi}_{a}\left(b\right)=1+\frac{\left(a1\right)b}{a}+\frac{{\left(a1\right)}^{3}}{{a}^{2}2!}\frac{{b}^{2}}{a+1}+\frac{{\left(a1\right)}^{5}}{{a}^{3}3!}\frac{{b}^{3}}{\left(a+1\right)\left(a+3\right)}+...$
To investigate the unbiased estimate ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}$, the expected value is
$E\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]=\sum _{j=1}^{{n}_{i}}P\left({n}_{i\left(1\right)}=j\right)E\left[\stackrel{\u02c6}{{\vartheta}_{i}}{n}_{i\left(1\right)}=j\right]=0+P\left({n}_{i\left(1\right)}=1\right)E\left[{w}_{i1}\u2215{n}_{i}\right]+\sum _{j=2}^{{n}_{i}}P\left({n}_{i\left(1\right)}=j\right)E\left[\stackrel{\u02c6}{{\vartheta}_{i}}{n}_{i\left(1\right)}=j\right]=P\left({n}_{i\left(1\right)}=1\right)\frac{exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right)}{{n}_{i}}+\sum _{j=2}^{{n}_{i}}P\left({n}_{i\left(1\right)}=j\right)E\left[\frac{{n}_{i\left(1\right)}}{{n}_{i}}exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right){n}_{i\left(1\right)}=j\right]=\sum _{j=0}^{{n}_{i}}P\left({n}_{i\left(1\right)}=j\right)E\left[\frac{{n}_{i\left(1\right)}}{{n}_{i}}exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right){n}_{i\left(1\right)}=j\right]=E\left[\frac{{n}_{i\left(1\right)}}{{n}_{i}}exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right)\right]=\left(1{\delta}_{i}\right)exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right).$
According to Shimizu & Iwase (1981), the uniformly minimum variance unbiased (UMVU) estimate of ϑ_{i} is (9)${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Shi\right)}=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill ;{n}_{i\left(1\right)}<1\hfill \\ \hfill \frac{{n}_{i\left(1\right)}}{{n}_{i}}exp{\left({\stackrel{\u02c6}{\mu}}_{i}\right)}_{0}{F}_{1}\left(\frac{{n}_{i\left(1\right)}1}{2};\frac{{n}_{i\left(1\right)}1}{4{n}_{i\left(1\right)}}{S}_{i}^{2}\right)\hfill & \hfill ;{n}_{i\left(1\right)}\ge 1\hfill \end{array}\right.$ where ${S}_{i}^{2}={\sum}_{j=1}^{{n}_{i\left(1\right)}}{\left[ln\left({W}_{ij}\right){\stackrel{\u02c6}{\mu}}_{i}\right]}^{2}$ and ${}_{0}{F}_{1}\left(a;z\right)={\sum}_{m=0}^{\infty}\frac{{z}^{m}}{{\left(a\right)}_{m}m!}$; (10)${\left(a\right)}_{m}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill ;m=0\hfill \\ \hfill a\left(a+1\right)...\left(a+m1\right)\hfill & \hfill ;m\ge 1\hfill \end{array}\right.$
From Kunio (1983), $E\left[{\text{}}_{0}{F}_{1}\left(\frac{{n}_{i\left(1\right)}1}{2};\frac{a}{2}{S}_{i}^{2}\right)\right]=exp\left(a{\sigma}^{2}\right)$ is obtained, then (11)$E\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Shi\right)}\right]=E\left[\frac{{n}_{i\left(1\right)}}{n}exp{\left({\stackrel{\u02c6}{\mu}}_{i}\right)}_{0}{F}_{1}\left(\frac{{n}_{i\left(1\right)}1}{2},\frac{{n}_{i\left(1\right)}1}{4{n}_{i\left(1\right)}}{S}_{i}^{2}\right)\right]=\frac{{n}_{i}\left(1{\delta}_{i}\right)}{{n}_{i}}exp\left[{\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2{n}_{i\left(1\right)}}\right]exp\left[\frac{{n}_{i\left(1\right)}1}{2{n}_{i\left(1\right)}}{\sigma}_{i}^{2}\right]=\left(1{\delta}_{i}\right)exp\left({\mu}_{i}+\frac{{\sigma}_{i}^{2}}{2}\right)$
where E(n_{i(1)}) = n_{i}(1 − δ_{i}). The asymptotic variance of ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Shi\right)}$ is given by
$Var\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Shi\right)}\right]=exp\left(2{\mu}_{i}+{\sigma}_{i}^{2}\right)\left[\frac{1}{{n}_{i}^{2}}\sum _{j=1}^{{n}_{i}}\left(\genfrac{}{}{0.0pt}{}{{n}_{i}}{j}\right){\left(1{\delta}_{i}\right)}^{j}{\delta}^{{n}_{i}j}{j}^{2}exp\left(\frac{{\sigma}_{i}^{2}}{j}\right)\right.$ $\left.{\text{}}_{0}{F}_{1}\left(\frac{j1}{2};\frac{{\left(j1\right)}^{2}}{4{j}^{2}}{\sigma}_{i}^{4}\right){\left(1{\delta}_{i}\right)}^{2}\right]$ (12)$=\frac{exp\left(2{\mu}_{i}+{\sigma}_{i}^{2}\right)}{{n}_{i}}\left[{\delta}_{i}\left(1{\delta}_{i}\right)+\frac{1}{2}\left(1{\delta}_{i}\right)\left(2{\sigma}_{i}^{2}+{\sigma}_{i}^{4}\right)\right]+O\left({n}^{2}\right).$ Actually, ${\psi}_{{n}_{i\left(1\right)}}\left(\frac{{\stackrel{\u02c6}{\sigma}}_{i}^{2}}{2}\right){=}_{0}{F}_{1}\left(\frac{{n}_{i\left(1\right)}1}{2};\frac{{n}_{i\left(1\right)}1}{4{n}_{i\left(1\right)}}{S}_{i}^{2}\right)$ such that ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Shi\right)}$ and ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}$ are the unbiased estimates of ϑ_{i} under different ideas, although their variances are the same i.e., $Var\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Shi\right)}\right]=Var\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]$. Using $\stackrel{\u02c6}{{\mu}_{i}},{\stackrel{\u02c6}{\sigma}}_{i}^{2}$, and $\stackrel{\u02c6}{{\delta}_{i}}$ from the samples, the estimated deltalognormal mean ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}$ and variance of ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}$ are obtained. The following methods are the detailed construction of the CIs for the common deltalognormal mean.
Fiducial generalized confidence interval
Fiducial inference was introduced by Fisher (1930). Fisher’s fiducial argument was used to develop a generalized fiducial recipe that could be extended to the application of fiducial ideas (Hannig, 2009). The concept of the fiducial interval has been advanced by the idea of the generalized pivotal quantity (GPQ) such that it is directly used to apply for generalized inference. Later, Hannig, Iyer & Patterson (2006) argued that a subclass of GPQs, the fiducial GPQ (FGPQ), provides a framework that shows the connection between a distribution and a parameter. Recall that ${\stackrel{\u02c6}{\mu}}_{i}\sim N\left({\mu}_{i},{\sigma}_{i}^{2}\u2215{n}_{i\left(1\right)}\right)$ and $\left({n}_{i\left(1\right)}1\right){\stackrel{\u02c6}{\sigma}}_{i}^{2}\u2215{\sigma}_{i}^{2}\sim {\chi}_{{n}_{i\left(1\right)}1}^{2}$ are the independent random variables. The structure functions of ${\stackrel{\u02c6}{\mu}}_{i}$ and ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ are (13)$\stackrel{\u02c6}{\mu}}_{i}={\mu}_{i}+{V}_{i}\sqrt{\frac{{\sigma}_{i}^{2}}{{n}_{i\left(1\right)}}}\phantom{\rule{14.22636pt}{0ex}}\text{and}\phantom{\rule{14.22636pt}{0ex}}{\stackrel{\u02c6}{\sigma}}_{i}^{2}=\frac{{\sigma}_{i}^{2}{U}_{i}}{{n}_{i\left(1\right)}1$ which are the function of V_{i} and U_{i}, respectively, where V_{i} ∼ N(0, 1) and ${U}_{i}\sim {\chi}_{{n}_{i\left(1\right)}1}^{2}$. Given the observed values, the estimates ${\stackrel{\u02c6}{\mu}}_{i}$ and ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ can be obtained, and the unique solution of $\left({\stackrel{\u02c6}{\mu}}_{i},{\stackrel{\u02c6}{\sigma}}_{i}^{2}\right)=\left({\mu}_{i}+{V}_{i}\sqrt{\frac{{\sigma}_{i}^{2}}{{n}_{i\left(1\right)}}},\frac{{\sigma}_{i}^{2}{U}_{i}}{{n}_{i\left(1\right)}1}\right)$ becomes (14)${\mu}_{i}={\stackrel{\u02c6}{\mu}}_{i}{V}_{i}\frac{{\stackrel{\u02c6}{\sigma}}_{i}}{\sqrt{{n}_{i\left(1\right)}}}\sqrt{\frac{{n}_{i\left(1\right)}1}{{U}_{i}}},\phantom{\rule{28.45274pt}{0ex}}{\sigma}_{i}^{2}=\frac{\left({n}_{i\left(1\right)}1\right){\stackrel{\u02c6}{\sigma}}_{i}^{2}}{{U}_{i}}.$ The respective FGPQs of μ_{i} and ${\sigma}_{i}^{2}$ are (15)$G}_{{\mu}_{i}}={\stackrel{\u02c6}{\mu}}_{i}{V}_{i}^{\ast}\frac{{\stackrel{\u02c6}{\sigma}}_{i}}{\sqrt{{n}_{i\left(1\right)}}}\sqrt{\frac{{n}_{i\left(1\right)}1}{{U}_{i}^{\ast}}$ (16)$G}_{{\sigma}_{i}^{2}}=\frac{\left({n}_{i\left(1\right)}1\right){\stackrel{\u02c6}{\sigma}}_{i}^{2}}{{U}_{i}^{\ast}$ where ${V}_{i}^{\ast}$ and ${U}_{i}^{\ast}$ are independent copies of V_{i} and U_{i}, respectively. Hasan & Krishnamoorthy (2018) developed the FGPQ of δ_{i} using a beta distribution as ${G}_{{\delta}_{i}^{\prime}}\sim Beta\left({\alpha}_{i},{\beta}_{i}\right)$; α_{i} = n_{i(1)} + 0.5 and β_{i} = n_{i(0)} + 0.5. The FGPQ of ϑ based on k individual samples is (17)$G}_{\vartheta}=\frac{\sum _{i=1}^{k}{G}_{{w}_{i}}{G}_{{\vartheta}_{i}}}{\sum _{i=1}^{k}{G}_{{w}_{i}}$ where ${G}_{{\vartheta}_{i}}={G}_{{\delta}_{i}^{\prime}}exp\left({G}_{{\mu}_{i}}+{G}_{{\sigma}_{i}^{2}}\u22152\right)$, ${G}_{{w}_{i}}=1\u2215{G}_{Var\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]}$, and ${G}_{Var\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]}=exp\left(2{G}_{{\mu}_{i}}+{G}_{{\sigma}_{i}^{2}}\right)\left[{G}_{{\delta}_{i}^{\prime}}\left(1{G}_{{\delta}_{i}^{\prime}}\right)+\frac{1}{2}{G}_{{\delta}_{i}^{\prime}}\left(2{G}_{{\sigma}_{i}^{2}}+{G}_{{\sigma}_{i}^{4}}\right)\right]\u2215{n}_{i}$. Thus, the 100(1 − ζ)% FGCI for ϑ is (18)$C{I}_{\vartheta}^{\left(fgci\right)}=\left[{L}_{\vartheta}^{\left(fgci\right)},{U}_{\vartheta}^{\left(fgci\right)}\right]=\left[{G}_{\vartheta}\left(\zeta \u22152\right),{G}_{\vartheta}\left(1\zeta \u22152\right)\right]$ where G_{ϑ}(ζ) denotes the ζ^{th} percentiles of G_{ϑ}. Algorithm 1 shows the computational steps for obtaining the FGCI.
Algorithm 1: FGCI

Generate V_{i} ∼ N(0, 1) and ${U}_{i}\sim {\chi}_{{n}_{i\left(1\right)}1}^{2}$ are independent.

Compute the FGPQs G_{μi}, ${G}_{{\sigma}_{i}^{2}}$ and ${G}_{{\delta}_{i}^{\prime}}$.

Compute G_{wi} and G_{ϑi} leading to obtain G_{ϑ}.

Repeat steps 13, a number of times, m = 2500, compute 95%FGCI for ϑ, as given in Eq. (18).
Large sample interval
Recall that the Aitchitson estimator is ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}=\left(1\stackrel{\u02c6}{{\delta}_{i}}\right)exp\left(\stackrel{\u02c6}{{\mu}_{i}}\right){\psi}_{{n}_{i\left(1\right)}}\left({\stackrel{\u02c6}{\sigma}}_{i}^{2}\u22152\right)$ and the variance of ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}$ is $Var\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]=exp\left(2{\mu}_{i}+{\sigma}_{i}^{2}\right)\left[{\delta}_{i}\left(1{\delta}_{i}\right)+\frac{1}{2}\left(1{\delta}_{i}\right)\left(2{\sigma}_{i}^{2}+{\sigma}_{i}^{4}\right)\right]\u2215{n}_{i}$. The approximated variance is obtained by replacing ${\stackrel{\u02c6}{\mu}}_{i}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ and ${\stackrel{\u02c6}{\delta}}_{i}$. The pooled estimate of ϑ_{i} is given by (19)$\stackrel{\u02c6}{\vartheta}=\frac{\sum _{i=1}^{k}{w}_{i}{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}}{\sum _{i=1}^{k}{w}_{i}}$ where ${w}_{i}=1\u2215\widehat{Var}\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]$. Hence, the 100(1 − ζ)% LS interval for ϑ is obtained as (20)$C{I}_{\vartheta}^{\left(ls\right)}=\left[{L}_{\vartheta}^{\left(ls\right)},{U}_{\vartheta}^{\left(ls\right)}\right]=\left[\stackrel{\u02c6}{\vartheta}{z}_{1\frac{\zeta}{2}}\sqrt{1\u2215\sum _{i=1}^{k}{w}_{i}},\stackrel{\u02c6}{\vartheta}+{z}_{1\frac{\zeta}{2}}\sqrt{1\u2215\sum _{i=1}^{k}{w}_{i}}\right]$ where z_{ζ} denotes the ζ^{th} percentiles of standard normal N(0, 1). The LS interval can be estimated easily via ‘Algorithm 2’.
Algorithm 2: LS

Compute ${\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}$ and $\widehat{Var}\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]$.

Compute $\stackrel{\u02c6}{\vartheta}$.

Compute 95%LS interval for ϑ, as given in Eq. (20).
Method of variance estimates recovery
This method produces a closedform CI that is easy to compute. For this reason, the MOVER CI for the common deltalognormal mean is considered for k individual random samples. The MOVER for a linear combination of ϑ_{i}; i=1 , 2, …, k is as follows. Let ${\stackrel{\u02c6}{\vartheta}}_{1},{\stackrel{\u02c6}{\vartheta}}_{2},\dots ,{\stackrel{\u02c6}{\vartheta}}_{k}$ be independent unbiased estimators of ϑ_{1}, ϑ_{2}, …, ϑ_{k}, respectively. In addition, let [l_{i}, u_{i}] stand for the 100(1 − ζ)%CI for ϑ_{i}. According to Krishnamoorthy & Oral (2015), the 100(1 − ζ)%MOVER for ${\sum}_{i=1}^{k}{c}_{i}{\vartheta}_{i}$ is given by (21)$C{I}_{\sum _{i=1}^{k}{c}_{i}{\vartheta}_{i}}=\left[{L}_{\sum _{i=1}^{k}{c}_{i}{\vartheta}_{i}},{U}_{\sum _{i=1}^{k}{c}_{i}{\vartheta}_{i}}\right]=\left[\sum _{i=1}^{k}{c}_{i}{\stackrel{\u02c6}{\vartheta}}_{i}\sqrt{\sum _{i=1}^{k}{c}_{i}^{2}{\left({\stackrel{\u02c6}{\vartheta}}_{i}{l}_{i}^{\ast}\right)}^{2}},\sum _{i=1}^{k}{c}_{i}{\stackrel{\u02c6}{\vartheta}}_{i}+\sqrt{\sum _{i=1}^{k}{c}_{i}^{2}{\left({\stackrel{\u02c6}{\vartheta}}_{i}{u}_{i}^{\ast}\right)}^{2}}\right]$ where ${l}_{i}^{\ast}=\left\{\begin{array}{cc}{l}_{i}\phantom{\rule{7.97224pt}{0ex}}\hfill & ;{c}_{i}>0\hfill \\ {u}_{i}\phantom{\rule{7.97224pt}{0ex}}\hfill & ;{c}_{i}<0\hfill \end{array}\right.$ and ${u}_{i}^{\ast}=\left\{\begin{array}{cc}{u}_{i}\phantom{\rule{7.97224pt}{0ex}}\hfill & ;{c}_{i}>0\hfill \\ {l}_{i}\phantom{\rule{7.97224pt}{0ex}}\hfill & ;{c}_{i}<0\hfill \end{array}\right.$. Next, the closedform CIs for ϑ_{i} are needed to construct MOVER for ϑ. Thus, ϑ_{i} is logtransformed as (22)$ln{\vartheta}_{i}=ln{\delta}_{i}^{\ast}+\left({\mu}_{i}+{\sigma}_{i}^{2}\right)$ where ${\delta}_{i}^{\ast}=1{\delta}_{i}$. Let ${\stackrel{\u02c6}{\mu}}_{i}$, and ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ and ${\stackrel{\u02c6}{\delta}}^{\ast}$ be the unbiased estimates of μ_{i}, ${\sigma}_{i}^{2}$, and δ_{i}, respectively. The MOVER for a single deltalognormal mean presented by Hasan & Krishnamoorthy (2018), the MOVER for ϑ_{i} is given by (23)$L}_{{\vartheta}_{i}}=exp\left\{ln{\stackrel{\u02c6}{\delta}}_{i}^{\ast}+\left({\stackrel{\u02c6}{\mu}}_{i}+{\stackrel{\u02c6}{\sigma}}_{i}^{2}\right)\sqrt{{\left(ln{\stackrel{\u02c6}{\delta}}_{i}^{\ast}{l}_{ln{\delta}_{i}^{\ast}}\right)}^{2}+{\left({\stackrel{\u02c6}{\mu}}_{i}+{\stackrel{\u02c6}{\sigma}}_{i}^{2}{l}_{{\mu}_{i}+{\sigma}_{i}^{2}}\right)}^{2}}\right\}{U}_{{\vartheta}_{i}}=exp\left\{ln{\stackrel{\u02c6}{\delta}}_{i}^{\ast}+\left({\stackrel{\u02c6}{\mu}}_{i}+{\stackrel{\u02c6}{\sigma}}_{i}^{2}\right)\sqrt{{\left(ln{\stackrel{\u02c6}{\delta}}_{i}^{\ast}{u}_{ln{\delta}_{i}^{\ast}}\right)}^{2}+{\left({\stackrel{\u02c6}{\mu}}_{i}+{\stackrel{\u02c6}{\sigma}}_{i}^{2}{u}_{{\mu}_{i}+{\sigma}_{i}^{2}}\right)}^{2}}\right\$ where $\left({l}_{ln{\delta}_{i}^{\ast}},{u}_{ln{\delta}_{i}^{\ast}}\right)=ln\left[\left({\stackrel{\u02c6}{\delta}}_{i}^{\ast}+\frac{{T}_{i,\zeta \u22152}^{2}}{2{n}_{i}}\mp {T}_{i,1\zeta \u22152}\sqrt{\frac{{\stackrel{\u02c6}{\delta}}_{i}^{\ast}\left(1{\stackrel{\u02c6}{\delta}}_{i}^{\ast}\right)}{{n}_{i}}+\frac{{T}_{i,\zeta \u22152}^{2}}{4{n}_{i}^{2}}}\right)\u2215\left(1+{T}_{i,\zeta \u22152}^{2}\u2215{n}_{i}\right)\right]$ (24)$\left({l}_{{\mu}_{i}+{\sigma}_{i}^{2}},{u}_{{\mu}_{i}+{\sigma}_{i}^{2}}\right)=\left[\left({\stackrel{\u02c6}{\mu}}_{i}+{\stackrel{\u02c6}{\sigma}}_{i}^{2}\u22152\right){\left\{{\left(\frac{{Z}_{i,\zeta \u22152}{\stackrel{\u02c6}{\sigma}}_{i}^{2}}{{n}_{i\left(1\right)}}\right)}^{2}+\frac{{\stackrel{\u02c6}{\sigma}}_{i}^{4}}{4}{\left(1\frac{{n}_{i\left(1\right)}1}{{\chi}_{i,1\zeta \u22152,{n}_{i\left(1\right)}1}^{2}}\right)}^{2}\right\}}^{1\u22152},\right.\left.\left({\stackrel{\u02c6}{\mu}}_{i}+{\stackrel{\u02c6}{\sigma}}_{i}^{2}\u22152\right)+{\left\{{\left(\frac{{Z}_{i,\zeta \u22152}{\stackrel{\u02c6}{\sigma}}_{i}^{2}}{{n}_{i\left(1\right)}}\right)}^{2}+\frac{{\stackrel{\u02c6}{\sigma}}_{i}^{4}}{4}{\left(\frac{{n}_{i\left(1\right)}1}{{\chi}_{i,\zeta \u22152,{n}_{i\left(1\right)}1}^{2}}1\right)}^{2}\right\}}^{1\u22152}\right].$ Note that both ${T}_{i}=\left({n}_{i\left(1\right)}{n}_{i}{\delta}^{\ast}\right)\u2215\sqrt{{n}_{i}{\delta}_{i}^{\ast}\left(1{\delta}_{i}^{\ast}\right)}\stackrel{d}{\sim}N\left(0,1\right)$, and ${Z}_{i}=\left({\stackrel{\u02c6}{\mu}}_{i}{\mu}_{i}\right)\u2215\sqrt{{\stackrel{\u02c6}{\sigma}}_{i}^{2}\u2215{n}_{i\left(1\right)}}\stackrel{d}{\sim}N\left(0,1\right)$ are independent random variables. According to Krishnamoorthy & Oral (2015), the 100(1 − ζ)% MOVER interval for ϑ is (25)$C{I}_{\vartheta}^{\left(mover\right)}\phantom{\rule{150.00023pt}{0ex}}=\left[{L}_{\vartheta},{U}_{\vartheta}\right]=\left[\frac{\sum _{i=1}^{k}{w}_{i}{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}}{\sum _{i=1}^{k}{w}_{i}}\sqrt{\frac{\sum _{i=1}^{k}{w}_{i}^{2}{\left({\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}{L}_{{\vartheta}_{i}}\right)}^{2}}{\sum _{i=1}^{k}{w}_{i}^{2}}},\right.\left.\frac{\sum _{i=1}^{k}{w}_{i}{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}}{\sum _{i=1}^{k}{w}_{i}}\sqrt{\frac{\sum _{i=1}^{k}{w}_{i}^{2}{\left({\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}{U}_{{\vartheta}_{i}}\right)}^{2}}{\sum _{i=1}^{k}{w}_{i}^{2}}}\right]$ where ${w}_{i}=1\u2215\widehat{Var}\left[{\stackrel{\u02c6}{\vartheta}}_{i}^{\left(Ait\right)}\right]$. ‘Algorithm 3’ describes the steps to construct the MOVER interval.
Algorithm 3: MOVER

Compute CIs for $ln\phantom{\rule{1em}{0ex}}{\delta}_{i}^{\ast}$ and ${\mu}_{i}+{\sigma}_{i}^{2}$ are $\left({l}_{ln{\delta}_{i}^{\ast}},{u}_{ln{\delta}_{i}^{\ast}}\right)$ and $\left({l}_{{\mu}_{i}+{\sigma}_{i}^{2}},{u}_{{\mu}_{i}+{\sigma}_{i}^{2}}\right)$, respectively.

Compute MOVER for ϑ_{i}, as given in Eq. (23).

Compute 95%MOVER for ϑ, given in Eq. (25).
Parametric Bootstrap
This is developed from the parametric bootstrap on the common mean of several heterogeneous lognormal distributions, proposed by Malekzadeh & KharratiKopaei (2019). The deltalognormal mean is transformed by taking the logarithm as (26)${\mu}_{i}=ln\left(\frac{\vartheta}{1{\delta}_{i}}\right)\frac{{\sigma}_{i}^{2}}{2}.$
The likelihood of $\left(\vartheta ,{\sigma}_{i}^{2},{\delta}_{i}\right)$ is (27)$L\left(\vartheta ,{\sigma}_{i}^{2},{\delta}_{i}{w}_{ij}\right)=\prod _{i=1}^{k}\left(\genfrac{}{}{0.0pt}{}{{n}_{i}}{{n}_{i\left(0\right)}}\right){\delta}_{i}\left(1{\delta}_{i}\right)\frac{1}{{\left(2\pi {\sigma}_{i}^{2}\right)}^{{n}_{i\left(1\right)}\u22152}}exp\left\{\frac{1}{2{\sigma}_{i}^{2}}\sum _{j=1}^{{n}_{i\left(1\right)}}{\left(ln{w}_{ij}ln\left(\frac{\vartheta}{1{\delta}_{i}}\right)+\frac{{\sigma}_{i}^{2}}{2}\right)}^{2}\right\}$ which enables obtaining the maximum likelihood estimates of lnϑ and ${\sigma}_{i}^{2}$ as (28)$ln{\stackrel{\u02c6}{\vartheta}}_{mle}=\frac{\sum _{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}\left[{\stackrel{\u02c6}{\mu}}_{i}+ln\left(1{\stackrel{\u02c6}{\delta}}_{i}\right)\right]+N\u22152}{\sum _{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}}{\stackrel{\u02c6}{\sigma}}_{mle,i}^{2}=2+2\sqrt{1+{\stackrel{\u02c6}{\sigma}}_{i}^{2}+{\left\{\stackrel{\u02c6}{\mu}ln\left[\stackrel{\u02c6}{\vartheta}\u2215\left(1{\stackrel{\u02c6}{\delta}}_{i}\right)\right]\right\}}^{2}}$ where ${\stackrel{\u02c6}{w}}_{mle,i}={n}_{i\left(1\right)}\u2215{\stackrel{\u02c6}{\sigma}}_{mle,i}^{2}$ and $ln{\stackrel{\u02c6}{\vartheta}}_{=}\frac{{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{i}\left[{\stackrel{\u02c6}{\mu}}_{i}+ln\left(1{\stackrel{\u02c6}{\delta}}_{i}\right)\right]+N\u22152}{{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{i}}$; ${\stackrel{\u02c6}{w}}_{i}={n}_{i\left(1\right)}\u2215{\stackrel{\u02c6}{\sigma}}_{i}^{2}$. If δ_{i} = 0, then it becomes the common lognormal mean (see Krishnamoorthy & Oral (2015) for a detailed explanation). By applying central limit theorem, we obtain $\left(ln{\stackrel{\u02c6}{\vartheta}}_{mle}ln\vartheta \right)\sqrt{{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}}\sim N\left(0,1\right)$ such that $T={\left(ln{\stackrel{\u02c6}{\vartheta}}_{mle}ln\vartheta \right)}^{2}{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}\sim {\chi}_{{n}_{i\left(1\right)}1}^{2}$. It is wellknown that ${\stackrel{\u02c6}{\mu}}_{i}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ and ${\stackrel{\u02c6}{\delta}}_{i}$ are independent random variables for which ${\stackrel{\u02c6}{\mu}}_{i}\sim N\left(ln\left(\frac{\vartheta}{1{\delta}_{i}}\right)\frac{{\sigma}_{i}^{2}}{2},{\sigma}_{i}^{2}\u2215{n}_{i\left(1\right)}\right)$, $\left({n}_{i\left(1\right)}1\right){\stackrel{\u02c6}{\sigma}}_{i}^{2}\u2215{\sigma}_{i}^{2}\sim {\chi}_{{n}_{i\left(1\right)}1}^{2}$ and ${\stackrel{\u02c6}{\delta}}_{i}\sim N\left(\delta ,\delta \left(1\delta \right)\u2215{n}_{i}\right)$ are obtained, respectively. Let $\eta ={\mu}_{i}+{\sigma}_{i}^{2}\u22152$ so that we can write $T=\frac{{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}\left[{\stackrel{\u02c6}{\mu}}_{i}+ln\left(1{\stackrel{\u02c6}{\delta}}_{i}\right)\eta ln\left(1{\delta}_{i}\right)\right]+N\u22152}{{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}}$. It can be seen that the distribution of T is complicated, possibly depending on nuisance parameters ${\sigma}_{i}^{2}$ and δ_{i}, but not on lnϑ. Thus, the exact distribution of T is unknown in practice, and so we propose the PB pivotal variable corresponding to T^{PB} as (29)$T}^{PB}={\left(ln{\stackrel{\u02c6}{\vartheta}}_{mle}^{PB}ln\stackrel{\u02c6}{\vartheta}\right)}^{2}\sum _{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}^{PB$ where $ln{\stackrel{\u02c6}{\vartheta}}_{mle}^{PB}=\frac{{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}^{PB}\left[{\stackrel{\u02c6}{\mu}}_{i}^{PB}+ln\left(1{\stackrel{\u02c6}{\delta}}_{i}^{PB}\right)\right]+N\u22152}{{\sum}_{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}^{PB}}$, ${\stackrel{\u02c6}{w}}_{i}^{PB}={n}_{i\left(1\right)}\u2215{\stackrel{\u02c6}{\sigma}}_{i}^{2B}$, ${\stackrel{\u02c6}{\mu}}_{i}^{PB}\sim N\left({\stackrel{\u02c6}{\mu}}_{i}^{B},{\stackrel{\u02c6}{\sigma}}_{i}^{2B}\u2215{n}_{i\left(1\right)}\right)$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2PB}\sim {\stackrel{\u02c6}{\sigma}}_{i}^{B2}{\chi}_{{n}_{i\left(1\right)}1}^{2}\u2215\left({n}_{i\left(1\right)}1\right)$ and ${\stackrel{\u02c6}{\delta}}^{PB}\sim beta\left({n}_{i\left(0\right)}^{B}+0.5,{n}_{i\left(1\right)}^{B}+0.5\right)$, ${n}_{i\left(0\right)}^{B}={n}_{i}{\stackrel{\u02c6}{\delta}}_{i}^{B}$, and ${n}_{i\left(1\right)}^{B}={n}_{i}{n}_{i\left(0\right)}^{B}$. Note that ${\stackrel{\u02c6}{\mu}}_{i}^{B}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2B}$, and ${\stackrel{\u02c6}{\delta}}_{i}^{B}$ are the observed values of ${\stackrel{\u02c6}{\mu}}_{i}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$, and ${\stackrel{\u02c6}{\delta}}_{i}$, respectively, from random sampling with replacement based on the bootstrap approach. Thus, the 100(1 − ζ)% PB interval for ϑ is given by (30)$C{I}_{\vartheta}^{\left(pb\right)}=exp\left[ln{\stackrel{\u02c6}{\vartheta}}_{mle}\mp \sqrt{{q}_{\zeta}^{PB}\u2215\sum _{i=1}^{k}{\stackrel{\u02c6}{w}}_{mle,i}}\right]$ where ${q}_{\zeta}^{PB}$ denotes the (1 − ζ)^{th} percentile of distribution of T^{PB}. The PB interval can be constructed as shown in ‘Algorithm 4’.
Algorithm 4: PB

Compute ${\stackrel{\u02c6}{\mu}}_{i}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ and $\stackrel{\u02c6}{\delta}$ leading to obtain $ln\stackrel{\u02c6}{\vartheta}$.

Compute $ln{\stackrel{\u02c6}{\vartheta}}_{mle}$ and ${\stackrel{\u02c6}{\sigma}}_{mle,i}^{2}$.

Generate ${\stackrel{\u02c6}{\mu}}_{i}^{PB}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2PB}$ and ${\stackrel{\u02c6}{\delta}}_{i}^{PB}$ leading to compute $ln{\stackrel{\u02c6}{\vartheta}}_{mle}^{PB}$.

Repeat steps 13, a number of time m = 2500, compute T^{PB} to obtain ${q}_{\zeta}^{PB}$.

Compute 95%PB interval for ϑ, as given in Eq. (30).
Highest posterior density intervals
The HPD interval is constructed from the posterior distribution, as defined by Box & Tiao (1973). Note that the prior of ϑ_{i} is updated with its likelihood function thereby obtaining the posterior distribution based on the Bayesian approach. Recall that ${W}_{ij}\sim \Delta \left({\mu}_{i},{\sigma}_{i}^{2},{\delta}_{i}\right)$, then the likelihood is given by (31)$P\left({w}_{ij}{\mu}_{i},{\sigma}_{i}^{2},{\delta}_{i}\right)\propto \prod _{i=1}^{k}{\delta}_{i}^{{n}_{i\left(0\right)}}{\left(1{\delta}_{i}\right)}^{{n}_{i\left(1\right)}}{\left({\sigma}_{i}^{2}\right)}^{{n}_{i\left(1\right)}\u22152}exp\left\{\frac{1}{2{\sigma}_{i}^{2}}\sum _{j=1}^{{n}_{i\left(1\right)}}{\left(ln{w}_{ij}{\mu}_{i}\right)}^{2}\right\}.$
For k individual samples, Miroshnikov, Wei & Conlon (2015) described the pooled independent subposterior samples toward the joint posterior distributions ϑ are combined using weighted averages as follows: (32)$\vartheta}^{post}=\sum _{i=1}^{k}{w}_{i}{\vartheta}_{i}^{post}{\left(\sum _{i=1}^{k}{w}_{i}\right)}^{1$ where ${\vartheta}_{i}^{post}$ are the posterior samples of ϑ_{i}, for i = 1, 2, …, k. The inverse of the sample variance is used to weight the posterior based on the ith samples is denoted as ${w}_{i}=Va{r}^{1}\left({\stackrel{\u02c6}{\vartheta}}_{i}{w}_{ij}\right)$. Different priors have been developed for estimating the common deltalognormal mean, two of which are derived in the following subsections.
Jeffreys’ rule prior
Harvey & van der Merwe (2012) defined this prior as (33)$P{\left(\vartheta \right)}_{JR}\propto \prod _{i=1}^{k}{\sigma}_{i}^{3}{\delta}_{i}^{1\u22152}{\left(1{\delta}_{i}\right)}^{1\u22152}$ which is combined with the likelihood Eq. (34) to obtain the posterior of ϑ as (34)$P\left({w}_{ij}\vartheta \right)\propto \prod _{i=1}^{k}{\delta}_{i}^{{n}_{i\left(0\right)}1\u22152}{\left(1{\delta}_{i}\right)}^{{n}_{i\left(1\right)}+1\u22152}{\left({\sigma}_{i}^{2}\right)}^{\left({n}_{i\left(1\right)}+3\right)\u22152}exp\left\{\frac{1}{2{\sigma}_{i}^{2}}\sum _{j=1}^{{n}_{i\left(1\right)}}{\left(ln{w}_{ij}{\mu}_{i}\right)}^{2}\right\}\propto \prod _{i=1}^{k}{\delta}_{i}^{\left({n}_{i\left(0\right)}+1\u22152\right)1}{\left(1{\delta}_{i}\right)}^{\left({n}_{i\left(1\right)}+3\u22152\right)1}{\left({\sigma}_{i}^{2}\right)}^{\frac{\left({n}_{i\left(1\right)}+1\right)}{2}1}exp\left\{\frac{1}{2{\sigma}_{i}^{2}}\left[\left({n}_{i\left(1\right)}1\right){\stackrel{\u02c6}{\sigma}}_{i}^{2}+{n}_{i\left(1\right)}{\left({\stackrel{\u02c6}{\mu}}_{i}{\mu}_{i}\right)}^{2}\right]\right\}.$ This leads to obtaining the marginal posterior distributions of μ_{i}, ${\sigma}_{i}^{2}$ and δ_{i} as (35)${\mu}_{i}^{\left(JR\right)}{\sigma}_{i,JR}^{2},{w}_{ij}\sim N\left({\stackrel{\u02c6}{\mu}}_{i},{\sigma}_{i}^{2\left(JR\right)}\u2215{n}_{i\left(1\right)}\right){\sigma}_{i}^{2\left(JR\right)}{w}_{ij}\sim IG\left(\left({n}_{i\left(1\right)}+1\right)\u22152,\left({n}_{i\left(1\right)}+1\right){\stackrel{\u02c6}{\sigma}}_{i}^{2}\u22152\right){\delta}_{i}^{\left(JR\right)}{w}_{ij}\sim beta\left({n}_{i\left(0\right)}+1\u22152,{n}_{i\left(1\right)}+3\u22152\right).$
The pooled posterior of ϑ is weighted by its inversely estimated variance as follows: (36)$\vartheta}^{post}=\sum _{i=1}^{k}{w}_{i}^{\left(JR\right)}{\vartheta}_{i}^{{\left(JR\right)}_{p}}{\left(\sum _{i=1}^{k}{w}_{i}^{\left(JR\right)}\right)}^{1$ where
${\vartheta}_{i}^{{\left(JR\right)}_{p}}=\left(1{\delta}_{i}^{\left(JR\right)}\right)exp\left({\mu}_{i}^{\left(JR\right)}+{\sigma}_{i}^{2\left(JR\right)}\u22152\right)$
${w}_{i}^{\left(JR\right)}={\left\{{n}_{i}^{1}exp\left(2{\mu}_{i}^{\left(JR\right)}+{\sigma}_{i}^{2\left(JR\right)}\right)\left[{\delta}_{i}^{\left(JR\right)}\left(1{\delta}_{i}^{\left(JR\right)}\right)+\frac{1}{2}\left(1{\delta}_{i}^{\left(JR\right)}\right)\left(2{\sigma}_{i}^{2\left(JR\right)}+{\sigma}_{i}^{4\left(JR\right)}\right)\right]\right\}}^{1}$.
From Eq. (36), the 100(1 − ζ)%HPDbased Jeffreys’ rule prior (HPDJR) for ϑ is constructed as follows:
Normalgammabeta prior
Maneerat, Niwitpong & Niwitpong (2020) proposed a HPD based on the normalgamma prior for the ratio of deltalognormal variances that worked better than the HPDJR of Harvey & van der Merwe (2012). Suppose that Y = lnW be a random variable of normal distribution with mean μ = (μ_{1}, μ_{2}, …, μ_{k}) and precision λ = (λ_{1}, λ_{2}, …, λ_{k}) where W ∼ LN(μ, λ) and ${\lambda}_{i}={\sigma}_{i}^{2}$. The HPDbased normalgammabeta prior (HPDNGB) of ϑ = (μ_{i}, λ_{i}, δ_{i})′ is defined as (37)$P\left(\vartheta \right)\propto \prod _{i=1}^{k}{\lambda}_{i}^{1}{\left[{\delta}_{i}\left(1{\delta}_{i}\right)\right]}^{1\u22152}$ where (μ_{i}, λ_{i}) follows a normalgamma distribution, and δ_{i} follows a beta distribution, denoted as (μ_{i}, λ_{i}) ∼ NG(μ_{i}, λ_{i}μ, k_{i(0)} = 0, α_{i(0)} = − 1∕2, β_{i(0)} = 0) and δ_{i} ∼ beta(1∕2, 1∕2), respectively. When the the prior Eq. (37) is combined with the likelihood Eq. (34), then the posterior density of ϑ becomes (38)$P\left(\vartheta {w}_{ij}\right)\propto \prod _{i=1}^{k}{\delta}_{i}^{{n}_{i\left(0\right)}1\u22152}{\left(1{\delta}_{i}\right)}^{{n}_{i\left(1\right)}1\u22152}{\lambda}_{i}^{\frac{{n}_{i\left(1\right)}1}{2}1}exp\left\{\frac{{\lambda}_{i}}{2}\sum _{j=1}^{{n}_{i\left(1\right)}}{\left(ln{w}_{ij}{\stackrel{\u02c6}{\mu}}_{i}\right)}^{2}\right\}{\lambda}_{i}^{1\u22152}exp\left\{\frac{{n}_{i\left(1\right)}{\lambda}_{i}}{2}{\left({\mu}_{i}{\mu}_{i}^{\ast}\right)}^{2}\right\}$ which can be integrated out to obtain the marginal posterior distributions of μ_{i}, λ_{i} and δ_{i} as follows: (39)${\mu}_{i}^{\left(NGB\right)}{w}_{ij}\sim {t}_{df}\left({\mu}_{i}{\stackrel{\u02c6}{\mu}}_{i},\sum _{j=1}^{{n}_{i\left(1\right)}}{\left(ln{w}_{ij}{\stackrel{\u02c6}{\mu}}_{i}\right)}^{2}\u2215\left[{n}_{i\left(1\right)}\left({n}_{i\left(1\right)}1\right)\right]\right){\lambda}_{i}^{\left(NGB\right)}{w}_{ij}\sim G\left({\lambda}_{i}\left({n}_{i\left(1\right)}1\right)\u22152,\sum _{j=1}^{{n}_{i\left(1\right)}}{\left(ln{w}_{ij}{\stackrel{\u02c6}{\mu}}_{i}\right)}^{2}\u22152\right){\delta}_{i}^{\left(NGB\right)}{w}_{ij}\sim beta\left({n}_{i\left(0\right)}+1\u22152,{n}_{i\left(1\right)}+1\u22152\right)$ where df = 2 (n_{i(1)} − 1) and ${\sigma}_{i}^{2\left(NGB\right)}{w}_{ij}\sim IG\left({\sigma}_{i}^{2}\left({n}_{i\left(1\right)}1\right)\u22152,{\sum}_{j=1}^{{n}_{i\left(1\right)}}{\left(ln{w}_{ij}{\stackrel{\u02c6}{\mu}}_{i}\right)}^{2}\u22152\right)$. Similarly, the pooled posterior of ϑ is given by (40)$\vartheta}^{post}=\sum _{i=1}^{k}{w}_{i}^{\left(NGB\right)}{\vartheta}_{i}^{{\left(NGB\right)}_{p}}{\left(\sum _{i=1}^{k}{w}_{i}^{\left(NGB\right)}\right)}^{1$ where ${\vartheta}_{i}^{{\left(NGB\right)}_{p}}=\left(1{\delta}_{i}^{\left(NGB\right)}\right)exp\left({\mu}_{i}^{\left(NGB\right)}+{\sigma}_{i}^{2\left(NGB\right)}\u22152\right){w}_{i}^{\left(NGB\right)}$ $={\left\{{n}_{i}^{1}exp\left(2{\mu}_{i}^{\left(NGB\right)}+{\sigma}_{i}^{2\left(NGB\right)}\right)\left[{\delta}_{i}^{\left(NGB\right)}\left(1{\delta}_{i}^{\left(NGB\right)}\right)\frac{1}{2}\left(1{\delta}_{i}^{\left(NGB\right)}\right)\left(2{\sigma}_{i}^{2\left(NGB\right)}+{\sigma}_{i}^{4\left(NGB\right)}\right)\right]\right\}}^{1}$.
Hence, the 100(1 − ζ)%HPDHGB for ϑ is constructed in Eq. (40). ‘Algorithm 5’ details the steps to construct the HPDJR and HPDNGB.
Algorithm 5: HPDJR and HPDNGB

Compute ${\stackrel{\u02c6}{\mu}}_{i}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ and $\stackrel{\u02c6}{\delta}$.

Generate the posterior densities of μ_{i}, ${\sigma}_{i}^{2}$ and δ_{i} basedJeffreys’ rule (JR) and normalgammabeta (NGB) priors, as given in Eq. (35) and Eq. (39), respectively.

Compute the pooled posterior of ϑ based on JR and NGB priors, as given in Eq. (36) and Eq. (40), respectively.

Compute 95%HPDJR and HPDNGB for ϑ, defined by Box & Tiao (1973).
Simulation Studies and Results
The performances of the CIs were assessed by comparing their coverage probabilities (CPs) and average length (ALs) using Monte Carlo simulation. The bestperforming CI is the one where the CP is closest to or greater than the nominal confidence level 1 − ζ while also having an AL with the narrowest width. The CIs for the common deltalognormal mean constructed using FGCI, LS, MOVER, PB, HPDJR, and HPDNGB were assessed in the study, the parameter settings for which are provided in Table 1. The number of generated random samples was fixed at M = 5000. For FGCI, the number of FGPQs was Q = 2500 for each set of 5,000 random samples. ‘Algorithm 6’ shows the computational steps to estimate the CP and AL performances of all of the methods.
Scenarios  (n_{1}, …, n_{k})  (δ_{1}, …, δ_{k})  $\left({\sigma}_{1}^{2},\dots ,{\sigma}_{k}^{2}\right)$ 

k = 2  
1–9  (30_{2})  (0.1,0.2), (0.2,0.5), (0.3,0.7)  (1,2), (2,4), (3,5) 
10–18  (30,50)  (0.1,0.2), (0.2,0.5), (0.3,0.7)  (1,2), (2,4), (3,5) 
19–27  (50_{2})  (0.1,0.2), (0.2,0.5), (0.3,0.7)  (1,2), (2,4), (3,5) 
28–36  (50,100)  (0.1,0.2), (0.2,0.5), (0.3,0.7)  (1,2), (2,4), (3,5) 
37–45  (100_{2})  (0.1,0.2), (0.2,0.5), (0.3,0.7)  (1,2), (2,4), (3,5) 
k = 5  
46–54  (30_{5})  (0.05, 0.1_{2}, 0.2_{2}), (0.2_{2}, 0.4_{3}), (0.5_{2}, 0.7_{3})  (1_{2}, 2_{3}), (2_{2}, 3_{3}), (3_{2}, 5_{3}) 
55–63  (30_{2}, 50_{3})  (0.05, 0.1_{2}, 0.2_{2}), (0.2_{2}, 0.4_{3}), (0.5_{2}, 0.7_{3})  (1_{2}, 2_{3}), (2_{2}, 3_{3}), (3_{2}, 5_{3}) 
64–72  (30_{2}, 50_{2}, 100)  (0.05, 0.1_{2}, 0.2_{2}), (0.2_{2}, 0.4_{3}), (0.5_{2}, 0.7_{3})  (1_{2}, 2_{3}), (2_{2}, 3_{3}), (3_{2}, 5_{3}) 
73–81  (30, 50_{2}, 100_{2})  (0.05, 0.1_{2}, 0.2_{2}), (0.2_{2}, 0.4_{3}), (0.5_{2}, 0.7_{3})  (1_{2}, 2_{3}), (2_{2}, 3_{3}), (3_{2}, 5_{3}) 
82–90  (50_{5})  (0.05, 0.1_{2}, 0.2_{2}), (0.2_{2}, 0.4_{3}), (0.5_{2}, 0.7_{3})  (1_{2}, 2_{3}), (2_{2}, 3_{3}), (3_{2}, 5_{3}) 
91–99  (50_{2}, 100_{3})  (0.05, 0.1_{2}, 0.2_{2}), (0.2_{2}, 0.4_{3}), (0.5_{2}, 0.7_{3})  (1_{2}, 2_{3}), (2_{2}, 3_{3}), (3_{2}, 5_{3}) 
100–108  (100_{5})  (0.05, 0.1_{2}, 0.2_{2}), (0.2_{2}, 0.4_{3}), (0.5_{2}, 0.7_{3})  (1_{2}, 2_{3}), (2_{2}, 3_{3}), (3_{2}, 5_{3}) 
k = 10  
109–114  (30_{5}, 50_{5})  (0.1_{5}, 0.2_{5}), (0.2_{5}, 0.5_{5})  (1_{5}, 2_{5}), (2_{5}, 4_{5}), (3_{5}, 5_{5}) 
115–120  (30_{3}, 50_{3}, 100_{4})  (0.1_{5}, 0.2_{5}), (0.2_{5}, 0.5_{5})  (1_{5}, 2_{5}), (2_{5}, 4_{5}), (3_{5}, 5_{5}) 
121–126  (50_{5}, 100_{5})  (0.1_{5}, 0.2_{5}), (0.2_{5}, 0.5_{5})  (1_{5}, 2_{5}), (2_{5}, 4_{5}), (3_{5}, 5_{5}) 
Notes:
Note: (30_{5}) stands for (30, 30, 30, 30, 30).
Scenarios  CP  AL  

FG  LS  MO  PB  HJ  HN  FG  LS  MO  PB  HJ  HN  
k = 2  
1  0.959  0.897  0.967  0.994  0.916  0.941  1.556  1.296  2.005  2.324  1.353  1.436 
2  0.958  0.857  0.947  0.996  0.924  0.941  5.169  3.770  7.287  8.631  4.186  4.335 
3  0.963  0.821  0.959  0.996  0.919  0.932  13.088  8.675  23.312  22.883  9.905  10.220 
4  0.962  0.886  0.978  0.995  0.917  0.939  1.487  1.211  2.181  2.155  1.247  1.386 
5  0.953  0.832  0.962  0.995  0.913  0.922  4.875  3.487  9.881  7.818  3.811  4.066 
6  0.951  0.793  0.971  0.991  0.901  0.912  12.311  7.740  37.615  21.129  8.875  9.378 
7  0.961  0.829  0.972  0.982  0.920  0.940  1.511  1.095  3.968  2.173  1.224  1.406 
8  0.950  0.778  0.974  0.995  0.900  0.911  4.821  3.123  293.620  7.649  3.566  3.916 
9  0.939  0.725  0.973  0.988  0.866  0.887  13.159  7.067  8.0e4  23.632  8.680  9.419 
10  0.960  0.900  0.965  0.992  0.915  0.941  1.503  1.249  1.936  2.225  1.362  1.395 
11  0.961  0.848  0.941  0.992  0.924  0.940  5.128  3.712  6.765  8.667  4.298  4.368 
12  0.965  0.819  0.952  0.998  0.919  0.931  12.297  8.382  20.057  21.597  9.819  9.894 
13  0.960  0.896  0.977  0.992  0.917  0.942  1.366  1.147  1.909  2.004  1.203  1.271 
14  0.961  0.851  0.964  0.996  0.916  0.931  4.593  3.422  7.236  7.458  3.761  3.889 
15  0.949  0.790  0.958  0.994  0.894  0.905  11.116  7.517  22.293  19.310  8.507  8.718 
16  0.963  0.860  0.972  0.974  0.928  0.943  1.354  1.033  2.141  1.928  1.155  1.257 
17  0.952  0.803  0.976  0.992  0.900  0.917  4.397  3.048  10.772  6.889  3.418  3.630 
18  0.940  0.737  0.968  0.989  0.872  0.889  11.065  6.663  43.755  19.011  7.903  8.247 
19  0.961  0.914  0.966  0.992  0.921  0.946  1.153  1.009  1.382  1.696  1.043  1.076 
20  0.965  0.895  0.946  0.991  0.938  0.949  3.668  2.924  4.309  5.981  3.178  3.229 
21  0.962  0.863  0.952  0.996  0.930  0.940  8.747  6.665  11.805  14.651  7.272  7.395 
22  0.958  0.910  0.978  0.985  0.919  0.944  1.091  0.945  1.414  1.555  0.945  1.031 
23  0.965  0.883  0.969  0.996  0.926  0.937  3.336  2.695  4.578  5.204  2.811  2.950 
24  0.961  0.840  0.972  0.995  0.921  0.928  7.887  5.987  13.164  12.757  6.338  6.605 
25  0.969  0.868  0.980  0.958  0.930  0.953  1.120  0.866  1.610  1.503  0.937  1.070 
26  0.954  0.839  0.970  0.997  0.916  0.926  3.208  2.433  6.544  4.735  2.621  2.830 
27  0.946  0.773  0.970  0.992  0.893  0.903  7.803  5.443  26.105  12.382  6.011  6.376 
28  0.958  0.912  0.972  0.979  0.916  0.947  1.119  0.952  1.397  1.615  1.054  1.051 
29  0.956  0.872  0.921  0.958  0.927  0.943  3.745  2.836  4.238  6.098  3.338  3.330 
30  0.961  0.846  0.937  0.987  0.925  0.936  8.488  6.274  10.833  13.991  7.332  7.320 
31  0.962  0.927  0.985  0.978  0.919  0.949  0.984  0.876  1.322  1.433  0.908  0.929 
32  0.960  0.880  0.958  0.992  0.925  0.940  3.214  2.618  4.169  5.150  2.818  2.860 
33  0.958  0.838  0.960  0.994  0.910  0.925  7.360  5.744  10.824  12.105  6.256  6.279 
34  0.963  0.888  0.977  0.922  0.938  0.954  0.975  0.785  1.322  1.352  0.876  0.922 
35  0.958  0.860  0.971  0.995  0.917  0.929  2.915  2.343  4.321  4.424  2.486  2.586 
36  0.951  0.820  0.973  0.995  0.901  0.916  6.726  5.103  11.951  10.823  5.511  5.626 
37  0.957  0.935  0.960  0.970  0.927  0.948  0.802  0.722  0.923  1.168  0.743  0.753 
38  0.955  0.916  0.926  0.953  0.942  0.948  2.442  2.044  2.541  3.935  2.220  2.219 
39  0.957  0.888  0.939  0.981  0.937  0.945  5.608  4.594  6.295  9.049  4.984  4.998 
40  0.961  0.942  0.975  0.957  0.924  0.954  0.740  0.679  0.911  1.062  0.659  0.702 
41  0.961  0.920  0.960  0.988  0.933  0.950  2.199  1.925  2.558  3.401  1.958  2.012 
42  0.955  0.875  0.960  0.994  0.925  0.931  4.976  4.209  6.298  7.813  4.318  4.439 
43  0.967  0.909  0.980  0.863  0.937  0.960  0.773  0.625  0.972  1.012  0.659  0.743 
44  0.960  0.896  0.970  0.993  0.928  0.939  2.076  1.750  2.684  3.013  1.788  1.921 
45  0.952  0.835  0.970  0.996  0.908  0.914  4.683  3.786  7.007  7.008  3.952  4.182 
Notes:
 FG

fiducial generalized confidence interval
 MO

method of variance estimates
 HJ

HPDbased Jeffreys’ rule prior
 HPDJR

HN, HPDbased normalgammabeta prior
Bold denoted as the bestperforming method each case.
Algorithm 6: Comparison of CPs and ALs for all CIs

For g = 1 to M. Generate ${w}_{ij}\sim \Delta \left({\mu}_{i},{\sigma}_{i}^{2},{\delta}_{i}\right)$.

Compute the unbiased estimates ${\stackrel{\u02c6}{\mu}}_{i}$, ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$ and $\stackrel{\u02c6}{\delta}$.

Compute the 95%CIs for ϑ based on FGCI, LS, MOVER, PB and the HPDs via Algorithm 1, 2, 3, 4 and 5, respectively.

Let A_{g} = 1 if ϑ falls within the intervals of FGCI, LS, MOVER, PB or the HPDs, else A_{g} = 0.

The CP and AL for each method are obtained by $\text{CP}=\left(1\u2215M\right){\sum}_{g=1}^{M}{A}_{g}$ and AL = (U − L)∕M, respectively, where U and L are the upper and lower confidence limits, respectively. (end g loop)
The numerical results for the CI performances are presented in terms of CP and AL for various sample cases. For k = 2 (Table 2 and Fig. 1), FGCI performed well for smalltomoderate sample sizes, as well as for large ${\sigma}_{i}^{2}$ and a moderatetolarge sample size. HPDNGB attained stable and the best CP and AL values for small ${\sigma}_{i}^{2}$ and a moderatetolarge sample size. MOVER and PB attained correct CPs but wider ALs than the other methods whereas LS and HPDJR had lower CPs and narrower ALs. For k = 5 (Table 3 and Fig. 2), there were only two methods producing better CPs than the other methods in the various situations: MOVER (small δ_{i} and ${\sigma}_{i}^{2}$) and PB (large δ_{i} and ${\sigma}_{i}^{2}$). Moreover, the results were similar for k = 10 (Table 4 and Fig. 3).
Scenarios  CP  AL  

FG  LS  MO  PB  HJ  HN  FG  LS  MO  PB  HJ  HN  
k = 5  
46  0.885  0.790  0.988  0.989  0.757  0.846  0.963  0.819  1.794  1.532  0.848  0.956 
47  0.789  0.627  0.973  0.996  0.674  0.715  2.240  1.908  4.982  3.897  1.991  2.176 
48  0.840  0.613  0.953  0.997  0.723  0.746  5.325  4.529  13.769  12.250  4.744  4.870 
49  0.894  0.800  0.993  0.978  0.779  0.864  0.900  0.765  1.825  1.439  0.773  0.905 
50  0.783  0.623  0.972  0.998  0.680  0.711  2.008  1.711  5.203  3.608  1.750  1.955 
51  0.797  0.580  0.959  0.996  0.680  0.701  4.700  4.066  16.626  11.353  4.118  4.287 
52  0.893  0.735  0.989  0.896  0.816  0.853  0.753  0.589  2.849  1.433  0.636  0.764 
53  0.768  0.517  0.977  0.997  0.666  0.676  1.474  1.168  19.967  3.364  1.282  1.406 
54  0.742  0.467  0.983  0.996  0.624  0.629  3.250  2.654  1.5e4  11.238  2.817  2.855 
55  0.884  0.779  0.988  0.979  0.743  0.846  0.940  0.777  1.739  1.434  0.857  0.930 
56  0.806  0.645  0.973  0.995  0.681  0.740  2.204  1.822  4.586  3.561  2.045  2.141 
57  0.858  0.622  0.949  0.986  0.725  0.771  5.620  4.542  12.575  12.073  5.122  5.162 
58  0.901  0.827  0.995  0.962  0.770  0.870  0.845  0.728  1.699  1.326  0.771  0.841 
59  0.793  0.644  0.978  0.997  0.675  0.726  1.904  1.629  4.351  3.262  1.750  1.850 
60  0.825  0.605  0.952  0.997  0.710  0.734  4.753  4.058  12.745  10.793  4.373  4.353 
61  0.905  0.785  0.992  0.822  0.809  0.865  0.685  0.564  1.632  1.219  0.620  0.686 
62  0.786  0.578  0.969  0.993  0.683  0.704  1.368  1.142  4.477  2.775  1.260  1.309 
63  0.755  0.496  0.963  0.998  0.639  0.637  3.177  2.714  18.995  8.911  2.884  2.822 
64  0.892  0.787  0.991  0.970  0.737  0.858  0.928  0.751  1.740  1.364  0.872  0.919 
65  0.822  0.647  0.975  0.996  0.673  0.763  2.168  1.738  4.371  3.326  2.047  2.114 
66  0.852  0.593  0.943  0.981  0.715  0.767  5.710  4.413  12.195  11.422  5.267  5.278 
67  0.905  0.827  0.996  0.949  0.768  0.873  0.816  0.697  1.637  1.256  0.770  0.811 
68  0.801  0.654  0.979  0.995  0.683  0.737  1.839  1.549  4.069  3.016  1.753  1.797 
69  0.821  0.595  0.947  0.994  0.693  0.733  4.806  3.976  12.174  10.326  4.431  4.432 
70  0.917  0.803  0.994  0.775  0.817  0.886  0.650  0.539  1.499  1.133  0.616  0.650 
71  0.804  0.612  0.973  0.992  0.692  0.730  1.310  1.094  3.962  2.543  1.236  1.262 
72  0.756  0.502  0.958  0.997  0.631  0.646  3.158  2.695  16.604  8.356  2.888  2.835 
73  0.924  0.832  0.994  0.942  0.772  0.893  0.822  0.673  1.505  1.186  0.856  0.808 
74  0.853  0.699  0.985  0.990  0.696  0.798  1.971  1.589  3.823  2.899  2.000  1.923 
75  0.883  0.652  0.952  0.945  0.755  0.817  5.330  4.072  9.997  9.911  5.224  4.974 
76  0.924  0.857  0.997  0.913  0.771  0.901  0.723  0.626  1.418  1.088  0.746  0.715 
77  0.826  0.695  0.986  0.989  0.689  0.767  1.670  1.406  3.476  2.610  1.692  1.632 
78  0.854  0.638  0.955  0.984  0.718  0.771  4.456  3.628  9.715  8.788  4.311  4.160 
79  0.930  0.846  0.998  0.683  0.811  0.900  0.581  0.486  1.253  0.964  0.586  0.580 
80  0.830  0.658  0.981  0.980  0.705  0.762  1.215  1.019  3.179  2.168  1.225  1.181 
81  0.788  0.555  0.967  0.997  0.675  0.689  2.992  2.554  11.873  7.026  2.927  2.738 
82  0.915  0.844  0.993  0.964  0.788  0.889  0.769  0.662  1.337  1.158  0.692  0.753 
83  0.858  0.735  0.982  0.993  0.741  0.804  1.882  1.599  3.605  2.920  1.698  1.825 
84  0.886  0.705  0.969  0.981  0.782  0.827  4.650  3.895  8.767  9.068  4.208  4.335 
85  0.925  0.865  0.998  0.939  0.803  0.897  0.707  0.618  1.315  1.068  0.618  0.700 
86  0.834  0.705  0.987  0.994  0.735  0.775  1.683  1.439  3.493  2.683  1.482  1.642 
87  0.855  0.684  0.968  0.994  0.751  0.783  4.027  3.489  8.924  8.068  3.613  3.766 
88  0.929  0.824  0.994  0.677  0.835  0.903  0.611  0.495  1.322  0.993  0.515  0.616 
89  0.823  0.627  0.981  0.985  0.729  0.749  1.284  1.045  3.692  2.296  1.121  1.250 
90  0.799  0.578  0.972  0.997  0.699  0.705  2.875  2.453  13.603  6.644  2.519  2.641 
91  0.927  0.831  0.997  0.906  0.777  0.898  0.753  0.614  1.389  1.064  0.703  0.735 
92  0.871  0.731  0.988  0.986  0.720  0.820  1.821  1.466  3.459  2.601  1.721  1.769 
93  0.905  0.693  0.957  0.897  0.791  0.852  5.015  3.768  8.461  8.829  4.621  4.690 
94  0.931  0.879  0.999  0.873  0.781  0.909  0.651  0.571  1.279  0.972  0.608  0.639 
95  0.847  0.738  0.991  0.986  0.719  0.797  1.541  1.313  3.117  2.351  1.447  1.499 
96  0.875  0.679  0.966  0.969  0.760  0.806  4.125  3.374  8.002  7.707  3.808  3.865 
97  0.935  0.866  0.998  0.541  0.832  0.911  0.529  0.450  1.097  0.856  0.493  0.523 
98  0.848  0.697  0.986  0.971  0.735  0.782  1.126  0.956  2.572  1.916  1.060  1.091 
99  0.817  0.613  0.963  0.994  0.698  0.725  2.784  2.418  7.510  6.042  2.565  2.571 
100  0.941  0.888  0.998  0.863  0.813  0.920  0.557  0.484  0.954  0.806  0.510  0.536 
101  0.906  0.827  0.995  0.973  0.799  0.875  1.413  1.201  2.515  2.029  1.288  1.361 
102  0.929  0.790  0.975  0.861  0.845  0.889  3.639  2.946  5.529  6.174  3.365  3.428 
103  0.948  0.923  1.000  0.801  0.816  0.931  0.501  0.456  0.909  0.741  0.452  0.487 
104  0.888  0.816  0.996  0.978  0.784  0.853  1.253  1.095  2.373  1.852  1.121  1.216 
105  0.905  0.775  0.981  0.953  0.822  0.859  3.147  2.678  5.326  5.441  2.893  2.975 
106  0.955  0.907  0.999  0.289  0.852  0.943  0.438  0.372  0.838  0.668  0.373  0.433 
107  0.881  0.761  0.994  0.939  0.781  0.833  0.992  0.823  2.044  1.536  0.863  0.972 
108  0.868  0.722  0.984  0.987  0.781  0.805  2.331  2.005  5.072  4.308  2.088  2.208 
Notes:
 FG

fiducial generalized confidence interval
 MO

method of variance estimates
 HJ

HPDbased Jeffreys’ rule prior
 HPDJR

HN, HPDbased normalgammabeta prior
Bold denoted as the bestperforming method each case.
CP  AL  

Scenarios  FG  LS  MO  PB  HJ  HN  FG  LS  MO  PB  HJ  HN 
k = 10  
109  0.728  0.675  0.998  0.927  0.566  0.692  0.612  0.501  1.554  0.932  0.545  0.623 
110  0.661  0.500  0.979  0.891  0.570  0.588  1.644  1.291  3.867  3.278  1.500  1.637 
111  0.504  0.352  0.950  0.978  0.481  0.404  3.159  2.561  8.645  7.286  2.996  3.076 
112  0.720  0.692  0.999  0.904  0.587  0.690  0.557  0.459  1.519  0.832  0.483  0.574 
113  0.532  0.452  0.976  0.985  0.512  0.462  1.393  1.159  3.853  2.682  1.260  1.404 
114  0.361  0.290  0.955  0.998  0.403  0.274  2.556  2.218  8.570  5.943  2.411  2.505 
115  0.789  0.723  0.999  0.808  0.561  0.762  0.554  0.440  1.416  0.789  0.546  0.560 
116  0.716  0.524  0.985  0.578  0.590  0.653  1.635  1.180  3.478  2.915  1.559  1.624 
117  0.593  0.406  0.964  0.872  0.519  0.507  3.289  2.406  7.754  6.380  3.189  3.214 
118  0.782  0.773  1.000  0.780  0.586  0.758  0.477  0.404  1.317  0.696  0.474  0.483 
119  0.626  0.514  0.988  0.947  0.535  0.561  1.337  1.076  3.348  2.360  1.284  1.341 
120  0.447  0.347  0.965  0.992  0.450  0.355  2.570  2.108  7.290  5.180  2.506  2.531 
121  0.826  0.773  1.000  0.736  0.592  0.796  0.488  0.399  1.266  0.695  0.444  0.486 
122  0.774  0.620  0.994  0.438  0.647  0.720  1.460  1.086  3.072  2.512  1.328  1.438 
123  0.659  0.460  0.977  0.798  0.553  0.577  3.002  2.236  6.597  5.502  2.775  2.921 
124  0.828  0.826  1.000  0.708  0.606  0.802  0.426  0.368  1.187  0.615  0.387  0.427 
125  0.688  0.595  0.995  0.912  0.591  0.627  1.205  0.992  2.912  2.039  1.094  1.197 
126  0.520  0.426  0.979  0.984  0.486  0.439  2.390  1.989  6.222  4.479  2.224  2.344 
Notes:
 FG

fiducial generalized confidence interval
 MO

method of variance estimates recovery
 HJ

HPDbased Jeffreys’ rule prior
 HPDJR

HN, HPDbased normalgammabeta prior
Bold denoted as the bestperforming method each case.
As previously mentioned, our findings show that FGCI works well for small sample case because the FGPQ of ${\sigma}_{i}^{2}$ might contain some weak points that affect the FGPQ of μ_{i} as the sample case increases. For large sample sizes, MOVER was the best method for small σ^{2}, which is possibly caused by the CI for ${\mu}_{i}+{\sigma}_{i}^{2}$. Meanwhile, the next best one was PB, which has the strong point of using a resampling technique to collect information about several populations even when the variance σ^{2} is large.
Northern  Northeastern  Central  Eastern  Southern  

3  0  3  0  0  49.5  0  0  0  2.9  3.2  0  4.1  0  0  2.7 
2.6  5  0  40  1.5  10.5  0  0  0  0.2  0  3.2  0  0  0  0 
1  23.8  0  3.5  18.5  60.4  4  0  11  0.3  0  10.4  11.5  3.5  0  0 
3.6  16  0  0  42  12.7  0  0  0  2.5  4.7  1.1  2.5  13.6  0  0 
0  11.5  0  12  9.1  6.8  0  20.3  0  0.4  19.3  0.2  9.7  0  0.2  0 
13.2  1.2  0  15  6  69.3  0  0  0  0.4  3.1  4.3  10.4  0  0  0 
22.4  10.3  0  0  7.5  36.5  0  2.4  0.3  1.1  2.9  0  9.6  0  0  0 
1.4  1.7  0  1.5  0  8.6  0  0  1  0  5.7  0 19  0  0  0  
18.3  5.5  0  0.7  6.3  0  0  0  0  1.3  0.9  0  8.3  0  0  0 
0  7.3  0  0  0  0  0  0  0  0.1  0  0  0  4.8  0  6.2 
15.5  24.3  1.7  3  0.4  0  0  0  0  2.9  0  0.2  0  0  0  0 
0  27.2  2.3  0  0  3.8  0  0  0  0  2.6  0.1  0  0  0  0 
0  12.6  0.5  0  0  0  0  3.2  0  1  17  62.8  0  0  0  6.1 
0  22.7  3.9  0  0  0  0  0  0  4.7  0  36.7  17.8  0  0  0 
9.8  0  6.9  29.4  1.8  0  0  0  0  0.5  3.5  15.6  12.3  0  0  0 
24.3  2.6  2.2  48  0  0  0  0  0  5  0  50  2.5  0  0  0 
24.6  0  3.2  0  0  0  6  0  0  2.5  0  35.5  0  0  0  0.3 
8.8  3.2  5.3  70.8  14.3  0  0  0  0  0  0  35  0.9  0  0  0 
0  2.6  11  3.5  0  0  0  0  0  0  5.1  5.9  0  0  0  0 
19.8  2  0.6  14.2  0  0  0  4.8  0  0  60.4  0  2.6  0  0  0 
5  8  0  7  0  0  2.3  0  0  0  6.9  0  0  0  0  0 
12.3  1.9  1  0  0  21.5  0  0  0  6.6  3  3  0  0  0  0 
8.1  0.8  2.4  0  0  2.5  1  0  0  0  15.1  60.4  2  0  0  0 
4.8  2.2  13.2  0  0  0  0  0  0  9.5  6  60  0  0  0  0 
5.8  6.5  0.4  0  0  13  0  0  5.1  13.4  76  0  0  0  0  
17  0  0  10.8  0  26.2  0  0  12.5  6.2  79.7  0  0  0  0  
25.1  2.2  1.3  0  10.1  2.2  4.6  5.4  0  65.7  3.5  0  0  
8.3  0  10  6.3  0  3  0  0  0  108  0  0  36.1  
22.9  4.3  2.5  0  4.8  10.5  10  0  3.2  10.5  0  0  41.8  
26.9  0.2  4.6  4  0  0  0  12  0  0  0  30  
0  0  0  19.3  0  0  9.5  0  2.2  0  0  0 
Notes:
Source: Thai Meteorological Department: https://www.tmd.go.th/services/weekly report.php.
Northern  Northeastern  Central  Eastern  Southern  

9.5  0  25.3  20  6.6  8.4  0  67  0  39.6  0  0  27.9  4.1  0.4  114.6 
4.9  10  25.5  14.5  16.9  0.8  2.9  65.4  0  25  0  0  0  9  3.8  0 
0  21.6  24  3  10  20.2  0  21  0  0  0  26.5  3.4  27.3  0.6  0 
4.7  15  8  28  48.2  0  14.3  6.4  7.2  0  0  36.4  0  6.5  0  0 
0  15.5  0  27  6.5  0.5  0  0  3.5  29.7  0.1  0  0.8  3.5  10.8  0 
63.2  14  20  50  4.8  5.3  6  52  0  0  0.3  4.5  37.9  0  5  18.2 
9.6  8.5  0  24  25  16.7  0  45  40.5  3.1  0.5  0  32.4  0  12.2  40.4 
10.7  11.5  0  30  0  45.2  28  41.4  25.8  8.2  31.5  0.5  33.8  0  3.6  0 
13  17.4  0  22  0  0  0  14.3  30.4  3.2  8.2  0.7  15.8  0  0  0 
0  15.6  0  16  3.2  0.6  0  45  0  7.1  0  12.3  0  3.6  8.8  10.8 
0  31.6  33.8  0  44  0  0  27  0  0  0  0.5  0  3  0  0 
0  20.6  33.7  0  0  3.1  27.6  0.2  0  3.2  0  1.9  0  1  0  0 
0  31.1  15.1  0  9.3  33.3  33  30  0  4.2  0  66.4  0  3.7  6.2  35 
0  16.3  18.5  0  0  6  0  0  0  5.7  0  93.6  11.5  15.6  0  0 
2.8  0  44.8  39.7  20  0  0  0  8.3  30  0  68.7  1.7  11.2  3.8  33.5 
11.3  33.1  37.5  9.3  0  13.2  0  0  0  4  0  40  1.2  24  0  57 
0.6  29.2  0  0  4.8  0  0  0  0  0  0  65  21.2  0  0  10.5 
36.1  11.2  47  2.1  0  21  0  0  0  0  0  63.7  0  0  0  0 
0  14.4  20  0  0  0  0  1  36.1  0  0  9.2  30  10.2  0.2  0 
2.6  60  30.8  46.7  0  8.4  15  0  0  0  1.2  0  5.1  0  0  0 
5  42.3  30  10.5  0  0  0  0  12.5  0  0  0  2.5  0  0  30.8 
13.4  9.5  1  0  56.5  0  0  2.5  0  14.7  0.1  11  2.4  0  0.4  10.7 
12.3  34.5  1.2  41  39.2  0.5  0  0  0  0  1  69.6  5  0  0  0 
25.8  36.5  56.3  10.3  0  4.5  25.7  9.5  0  0  3  89.6  1.7  0  0  15.9 
30.2  9.7  0  1.2  6.4  16.2  41.4  0  0  0.5  160  0  0  2.2  0  
16.4  0  6  23.9  5.3  0  41.6  0  1.6  34.3  0  0  0  
6  0  0  22.2  0  3.5  53.8  0  0  0  2.1  0  0.6  
33.1  7.6  5.3  24.1  9.8  20  48.5  0  0  25  0  0  76.6  
16.4  9.6  7.2  38  0  0  78.5  2.1  0  19.5  10.5  7  121.6  
19.8  9.3  24.6  9  9.7  0  12.7  0  0  15.3  10.6  60  
0  0  30  9.2  4.5  1.2  80.9  0  0  0  3.6  0 
Notes:
Source: Thai Meteorological Department: https://www.tmd.go.th/services/weekly_report.php.
An empirical application
Daily rainfall data obtained from the Thai Meteorological Department (TMD) were divided into the northern, northeastern, central, and eastern regions, while the southern region was a combination of the data from the southeastern and southwestern shores. Due to the differences in the climate patterns and meteorological conditions in the five regions, we focused was on estimating the daily rainfall data in these regions by treating them as separate sets of observations rather than using the average rainfall for the whole of Thailand by pooling them and treating them as a single population. The daily rainfall amounts were recorded on August 5 and 9, 2019, which is in the middle of the rainy season (midMay to midOctober) when rice farming is conducted in Thailand. Entries with rainfall of less than 0.1 mm were considered as zero records.
Tables 5–6 contain the daily rainfall records for the five regions, while Figs. 4–5 show histogram plots of rainfall observations, and Figs. 6–7 exhibit normal QQ plots of the logpositive rainfall data on August 5 and 9, 2019, respectively. It can be seen that the data for all of the regions contained zero observations. After that, the fitted distribution of the positive observations was checked using the Akaike information criterion (AIC), as reported in Table 7. It can be concluded that the rainfall data in all of the regions on August 5 and 9, 2019 follow a deltalognormal distribution. All data sets and R code are available in the Supplemental Information. The summary statistics are reported in Table 8. In the approximation of the daily rainfall amounts in the five regions, the estimated common means were 4.4506 and 13.2621 mm/day on August 5 and 9, 2019, respectively. The computed 95% CIs of the common rainfall mean are reported in Table 9. Under the rain criteria issued by the TMD (Department, 2018), it can be interpreted that the daily rainfall in Thailand on August 5, 2019, was light (0.1–10.0 mm), while it was moderate (10.1–35.0 mm) on August 9, 2019. These results confirm the simulation results for k = 5 in the previous section.
Regions  AIC  

Cauchy  Logistic  Lognormal  Normal  Tdistribution  
On August 5, 2019  
Northern  373.1958  357.3122  336.8724  353.7757  354.3055 
Northeastern  600.9473  642.1779  543.9619  667.2334  664.6152 
Central  240.0227  266.4162  220.8503  293.9151  283.2302 
Eastern  229.8995  220.2523  202.8394  218.7240  219.1471 
Southern  194.9368  197.5586  178.5587  201.1654  200.1388 
On August 9, 2019  
Northern  389.6257  387.3072  375.7994  391.1802  390.2479 
Northeastern  1123.7491  1080.8694  1052.8953  1080.1467  1079.9365 
Central  178.8516  189.5353  155.0261  190.6855  190.5103 
Eastern  233.5236  227.1725  215.9306  228.0501  227.4559 
Southern  541.0477  569.2615  487.4667  592.2242  588.2377 
Regions  Estimated parameters  

n_{i}  ${\stackrel{\u02c6}{\mu}}_{i}$  ${\stackrel{\u02c6}{\sigma}}_{i}^{2}$  ${\stackrel{\u02c6}{\delta}}_{i}$  ${\stackrel{\u02c6}{\vartheta}}_{i}$  
August 5, 2020  
Northern  62  1.866  1.277  0.210  9.472 
Northeastern  210  1.734  1.578  0.619  4.668 
Central  57  1.085  1.784  0.316  4.741 
Eastern  29  2.366  4.545  0.241  59.391 
Southern  119  1.684  1.730  0.782  2.639 
August 9, 2020  
Northern  62  2.621  0.732  0.226  15.187 
Northeastern  210  2.577  1.502  0.405  16.429 
Central  57  1.190  3.054  0.579  5.542 
Eastern  29  2.860  3.070  0.241  52.813 
Southern  119  2.007  2.051  0.462  10.811 
Methods  95%CIs for ϑ  Lengths  

Lower  Upper  
On August 5, 2020  
FGCI  2.5545  6.3342  3.7798 
LS  3.2166  5.6846  2.4681 
MOVER  2.7216  9.0296  6.3080 
PB  5.8876  11.4965  5.6089 
HPDJR  3.5216  7.8533  4.3317 
HPDNGB  2.4969  6.0904  3.5935 
On August 9, 2020  
FGCI  7.1127  16.8809  9.7682 
LS  10.4880  16.0363  5.5483 
MOVER  7.5814  23.3171  15.7357 
PB  14.5229  23.5821  9.0591 
HPDJR  12.8404  20.4349  7.5945 
HPDNGB  7.2928  17.1265  9.8337 
Discussion
It can be seen that for MOVER and PB developed from the studies of Krishnamoorthy & Oral (2015) and Malekzadeh & KharratiKopaei (2019), respectively, the simulation results are similar to both of these studies provided that the zero observations are omitted. CIs for the common mean have been investigated in both normal and lognormal distributions (Fairweather, 1972; Jordan & Krishnamoorthy, 1996; Krishnamoorthy & Mathew, 2003; Lin & Lee, 2005; Tian & Wu, 2007; Krishnamoorthy & Oral, 2015). However, the common mean of deltalognormal populations is especially of interest because it can be used to fit the data from realworld situations such as investigating medical costs (Zou, Taleban & Huo, 2009; Tierney et al., 2003; Tian, 2005), analyzing airborne contaminants (Owen & DeRouen, 1980; Tian, 2005) and measuring fish abundance (Fletcher, 2008; Wu & Hsieh, 2014). Furthermore, it is possible that some extreme rainfall data also fulfill the assumptions of a deltalognormal distribution. Note that such natural disasters as floods and landslides have been caused by the extreme rainfall events, as evidenced in many country around the world: Europe (e.g., Northern England, Southern Scotland and Ireland Otto & Oldenborgh, 2017), Asia (e.g., Japan Oldenborgh, 2018) and North America (e.g., Southeast Texas Oldenborgh et al., 2019). Our findings show that some of the methods studied had CPs that were too low or too high for large sample cases, a shortcoming that should be addressed in future work.
Conclusions
The objective of this study was to propose CIs for the common mean of several deltalognormal distributions using FGCI, LS, MOVER, PB, HPDJR, and HPDNGB. The CP and AL as performance measures of the methods were assessed via Monte Carlo simulation. The findings confirm that for small sample case ()k=2 (), FGCI and HPDNGB are the recommended methods in different situations: FGCI (a smalltomoderate sample size and a large ${\sigma}_{i}^{2}$ with a moderatetolarge sample size) and HPDNGB (small ${\sigma}_{i}^{2}$ with a moderatetolarge sample size). For large sample cases (k = 5, 10), MOVER small δ_{i} and ${\sigma}_{i}^{2}$) and PB (large δ_{i} and ${\sigma}_{i}^{2}$) performed the best.