Estimating the average daily rainfall in Thailand using confidence intervals for the common mean of several delta-lognormal distributions

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 ${\hat{\mu }}_i\sim N\left({\mu }_i,{\sigma }_i^2∕n_{i\left(1\right)}\right)$ and $\left(n_{i\left(1\right)}-1\right){\hat{\sigma }}_i^2∕{\sigma }_i^2\sim {\chi }_{n_{i\left(1\right)}-1}^2$ are the independent random variables. The structure functions of ${\hat{\mu }}_i$ and ${\hat{\sigma }}_i^2$ are (13)${\displaystyle {\hat{\mu }}_i={\mu }_i+V_i\sqrt{\frac{{\sigma }_i^2}{n_{i\left(1\right)}}}\text{and}{\hat{\sigma }}_i^2=\frac{{\sigma }_i^2{U}_i}{n_{i\left(1\right)}-1}}$ 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 ${\hat{\mu }}_i$ and ${\hat{\sigma }}_i^2$ can be obtained, and the unique solution of $\left({\hat{\mu }}_i,{\hat{\sigma }}_i^2\right)=\left({\mu }_i+V_i\sqrt{\frac{{\sigma }_i^2}{n_{i\left(1\right)}}},\frac{{\sigma }_i^2{U}_i}{n_{i\left(1\right)}-1}\right)$ becomes (14)${\displaystyle {\mu }_i={\hat{\mu }}_i-V_i\frac{{\hat{\sigma }}_i}{\sqrt{n_{i\left(1\right)}}}\sqrt{\frac{n_{i\left(1\right)}-1}{U_i}},{\sigma }_i^2=\frac{\left(n_{i\left(1\right)}-1\right){\hat{\sigma }}_i^2}{U_i}.}$ The respective FGPQs of μ_i and ${\sigma }_i^2$ are (15)${\displaystyle G_{{\mu }_i}={\hat{\mu }}_i-V_i^{\ast }\frac{{\hat{\sigma }}_i}{\sqrt{n_{i\left(1\right)}}}\sqrt{\frac{n_{i\left(1\right)}-1}{U_i^{\ast }}}}$ (16)${\displaystyle G_{{\sigma }_i^2}=\frac{\left(n_{i\left(1\right)}-1\right){\hat{\sigma }}_i^2}{U_i^{\ast }}}$ 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)${\displaystyle G_{\vartheta }=\frac{\sum _{i=1}^k{G}_{w_i}{G}_{{\vartheta }_i}}{\sum _{i=1}^k{G}_{w_i}}}$ where $G_{{\vartheta }_i}=G_{{\delta }_i^{\prime }}exp\left(G_{{\mu }_i}+G_{{\sigma }_i^2}∕2\right)$, $G_{w_i}=1∕G_{Var\left[{\hat{\vartheta }}_i^{\left(Ait\right)}\right]}$, and $G_{Var\left[{\hat{\vartheta }}_i^{\left(Ait\right)}\right]}=exp\left(2G_{{\mu }_i}+G_{{\sigma }_i^2}\right)\left[G_{{\delta }_i^{\prime }}\left(1-G_{{\delta }_i^{\prime }}\right)+\frac{1}{2}{G}_{{\delta }_i^{\prime }}\left(2G_{{\sigma }_i^2}+G_{{\sigma }_i^4}\right)\right]∕n_i$. Thus, the 100(1 − ζ)% FGCI for ϑ is (18)${\displaystyle C{I}_{\vartheta }^{\left(fgci\right)}=\left[L_{\vartheta }^{\left(fgci\right)},U_{\vartheta }^{\left(fgci\right)}\right]=\left[G_{\vartheta }\left(\zeta ∕2\right),G_{\vartheta }\left(1-\zeta ∕2\right)\right]}$ where G_ϑ(ζ) denotes the ζ^th percentiles of G_ϑ. Algorithm 1 shows the computational steps for obtaining the FGCI.

Large sample interval

Recall that the Aitchitson estimator is ${\hat{\vartheta }}_i^{\left(Ait\right)}=\left(1-\widehat{{\delta }_i}\right)exp\left(\widehat{{\mu }_i}\right){\psi }_{n_{i\left(1\right)}}\left({\hat{\sigma }}_i^2∕2\right)$ and the variance of ${\hat{\vartheta }}_i^{\left(Ait\right)}$ is $Var\left[{\hat{\vartheta }}_i^{\left(Ait\right)}\right]=exp\left(2{\mu }_i+{\sigma }_i^2\right)\left[{\delta }_i\left(1-{\delta }_i\right)+\frac{1}{2}\left(1-{\delta }_i\right)\left(2{\sigma }_i^2+{\sigma }_i^4\right)\right]∕n_i$. The approximated variance is obtained by replacing ${\hat{\mu }}_i$, ${\hat{\sigma }}_i^2$ and ${\hat{\delta }}_i$. The pooled estimate of ϑ_i is given by (19)${\displaystyle \hat{\vartheta }=\frac{\sum _{i=1}^k{w}_i{\hat{\vartheta }}_i^{\left(Ait\right)}}{\sum _{i=1}^k{w}_i}}$ where $w_i=1∕\widehat{Var}\left[{\hat{\vartheta }}_i^{\left(Ait\right)}\right]$. Hence, the 100(1 − ζ)% LS interval for ϑ is obtained as (20)${\displaystyle C{I}_{\vartheta }^{\left(ls\right)}=\left[L_{\vartheta }^{\left(ls\right)},U_{\vartheta }^{\left(ls\right)}\right]=\left[\hat{\vartheta }-z_{1-\frac{\zeta }{2}}\sqrt{1∕\sum _{i=1}^k{w}_i},\hat{\vartheta }+z_{1-\frac{\zeta }{2}}\sqrt{1∕\sum _{i=1}^k{w}_i}\right]}$ where z_ζ denotes the ζ^th percentiles of standard normal N(0, 1). The LS interval can be estimated easily via ‘Algorithm 2’.

Method of variance estimates recovery

This method produces a closed-form CI that is easy to compute. For this reason, the MOVER CI for the common delta-lognormal 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 ${\hat{\vartheta }}_1,{\hat{\vartheta }}_2,\dots ,{\hat{\vartheta }}_k$ be independent unbiased estimators of ϑ₁, ϑ₂, …, 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)${\displaystyle C{I}_{\sum _{i=1}^k{c}_i{\vartheta }_i}=\left[L_{\sum _{i=1}^k{c}_i{\vartheta }_i},U_{\sum _{i=1}^k{c}_i{\vartheta }_i}\right]=\left[\sum _{i=1}^k{c}_i{\hat{\vartheta }}_i-\sqrt{\sum _{i=1}^k{c}_i^2{\left({\hat{\vartheta }}_i-l_i^{\ast }\right)}^2},\sum _{i=1}^k{c}_i{\hat{\vartheta }}_i+\sqrt{\sum _{i=1}^k{c}_i^2{\left({\hat{\vartheta }}_i-u_i^{\ast }\right)}^2}\right]}$ where $l_i^{\ast }=\left\{\begin{array}{cc}{l}_i\hfill & ;c_i>0\hfill \\ u_i\hfill & ;c_i<0\hfill \end{array}\right.$ and $u_i^{\ast }=\left\{\begin{array}{cc}{u}_i\hfill & ;c_i>0\hfill \\ l_i\hfill & ;c_i<0\hfill \end{array}\right.$. Next, the closed-form CIs for ϑ_i are needed to construct MOVER for ϑ. Thus, ϑ_i is log-transformed as (22)${\displaystyle ln{\vartheta }_i=ln{\delta }_i^{\ast }+\left({\mu }_i+{\sigma }_i^2\right)}$ where ${\delta }_i^{\ast }=1-{\delta }_i$. Let ${\hat{\mu }}_i$, and ${\hat{\sigma }}_i^2$ and ${\hat{\delta }}^{\ast }$ be the unbiased estimates of μ_i, ${\sigma }_i^2$, and δ_i, respectively. The MOVER for a single delta-lognormal mean presented by Hasan & Krishnamoorthy (2018), the MOVER for ϑ_i is given by (23)${\displaystyle L_{{\vartheta }_i}=exp\left\{ln{\hat{\delta }}_i^{\ast }+\left({\hat{\mu }}_i+{\hat{\sigma }}_i^2\right)-\sqrt{{\left(ln{\hat{\delta }}_i^{\ast }-l_{ln{\delta }_i^{\ast }}\right)}^2+{\left({\hat{\mu }}_i+{\hat{\sigma }}_i^2-l_{{\mu }_i+{\sigma }_i^2}\right)}^2}\right\}{U}_{{\vartheta }_i}=exp\left\{ln{\hat{\delta }}_i^{\ast }+\left({\hat{\mu }}_i+{\hat{\sigma }}_i^2\right)-\sqrt{{\left(ln{\hat{\delta }}_i^{\ast }-u_{ln{\delta }_i^{\ast }}\right)}^2+{\left({\hat{\mu }}_i+{\hat{\sigma }}_i^2-u_{{\mu }_i+{\sigma }_i^2}\right)}^2}\right\}}$ where ${\displaystyle \left(l_{ln{\delta }_i^{\ast }},u_{ln{\delta }_i^{\ast }}\right)=ln\left[\left({\hat{\delta }}_i^{\ast }+\frac{T_{i,\zeta ∕2}^2}{2n_i}\mp T_{i,1-\zeta ∕2}\sqrt{\frac{{\hat{\delta }}_i^{\ast }\left(1-{\hat{\delta }}_i^{\ast }\right)}{n_i}+\frac{T_{i,\zeta ∕2}^2}{4n_i^2}}\right)∕\left(1+T_{i,\zeta ∕2}^2∕n_i\right)\right]}$ (24)${\displaystyle \left(l_{{\mu }_i+{\sigma }_i^2},u_{{\mu }_i+{\sigma }_i^2}\right)=\left[\left({\hat{\mu }}_i+{\hat{\sigma }}_i^2∕2\right)-{\left\{{\left(\frac{Z_{i,\zeta ∕2}{\hat{\sigma }}_i^2}{n_{i\left(1\right)}}\right)}^2+\frac{{\hat{\sigma }}_i^4}{4}{\left(1-\frac{n_{i\left(1\right)}-1}{{\chi }_{i,1-\zeta ∕2,n_{i\left(1\right)}-1}^2}\right)}^2\right\}}^{1∕2},\right.\left.\left({\hat{\mu }}_i+{\hat{\sigma }}_i^2∕2\right)+{\left\{{\left(\frac{Z_{i,\zeta ∕2}{\hat{\sigma }}_i^2}{n_{i\left(1\right)}}\right)}^2+\frac{{\hat{\sigma }}_i^4}{4}{\left(\frac{n_{i\left(1\right)}-1}{{\chi }_{i,\zeta ∕2,n_{i\left(1\right)}-1}^2}-1\right)}^2\right\}}^{1∕2}\right].}$ Note that both $T_i=\left(n_{i\left(1\right)}-n_i{\delta }^{\ast }\right)∕\sqrt{n_i{\delta }_i^{\ast }\left(1-{\delta }_i^{\ast }\right)}\stackrel{d}{\sim }N\left(0,1\right)$, and $Z_i=\left({\hat{\mu }}_i-{\mu }_i\right)∕\sqrt{{\hat{\sigma }}_i^2∕n_{i\left(1\right)}}\stackrel{d}{\sim }N\left(0,1\right)$ are independent random variables. According to Krishnamoorthy & Oral (2015), the 100(1 − ζ)% MOVER interval for ϑ is (25)${\displaystyle C{I}_{\vartheta }^{\left(mover\right)}=\left[L_{\vartheta },U_{\vartheta }\right]=\left[\frac{\sum _{i=1}^k{w}_i{\hat{\vartheta }}_i^{\left(Ait\right)}}{\sum _{i=1}^k{w}_i}-\sqrt{\frac{\sum _{i=1}^k{w}_i^2{\left({\hat{\vartheta }}_i^{\left(Ait\right)}-L_{{\vartheta }_i}\right)}^2}{\sum _{i=1}^k{w}_i^2}},\right.\left.\frac{\sum _{i=1}^k{w}_i{\hat{\vartheta }}_i^{\left(Ait\right)}}{\sum _{i=1}^k{w}_i}-\sqrt{\frac{\sum _{i=1}^k{w}_i^2{\left({\hat{\vartheta }}_i^{\left(Ait\right)}-U_{{\vartheta }_i}\right)}^2}{\sum _{i=1}^k{w}_i^2}}\right]}$ where $w_i=1∕\widehat{Var}\left[{\hat{\vartheta }}_i^{\left(Ait\right)}\right]$. ‘Algorithm 3’ describes the steps to construct the MOVER interval.

Parametric Bootstrap

This is developed from the parametric bootstrap on the common mean of several heterogeneous log-normal distributions, proposed by Malekzadeh & Kharrati-Kopaei (2019). The delta-lognormal mean is transformed by taking the logarithm as (26)${\displaystyle {\mu }_i=ln\left(\frac{\vartheta }{1-{\delta }_i}\right)-\frac{{\sigma }_i^2}{2}.}$

The likelihood of $\left(\vartheta ,{\sigma }_i^2,{\delta }_i\right)$ is (27)${\displaystyle L\left(\vartheta ,{\sigma }_i^2,{\delta }_i|w_{ij}\right)=\prod _{i=1}^k\left(\genfrac{}{}{0.0pt}{}{n_i}{n_{i\left(0\right)}}\right){\delta }_i\left(1-{\delta }_i\right)\frac{1}{{\left(2\pi {\sigma }_i^2\right)}^{n_{i\left(1\right)}∕2}}exp\left\{-\frac{1}{2{\sigma }_i^2}\sum _{j=1}^{n_{i\left(1\right)}}{\left(ln{w}_{ij}-ln\left(\frac{\vartheta }{1-{\delta }_i}\right)+\frac{{\sigma }_i^2}{2}\right)}^2\right\}}$ which enables obtaining the maximum likelihood estimates of lnϑ and ${\sigma }_i^2$ as (28)${\displaystyle ln{\hat{\vartheta }}_{mle}=\frac{\sum _{i=1}^k{\hat{w}}_{mle,i}\left[{\hat{\mu }}_i+ln\left(1-{\hat{\delta }}_i\right)\right]+N∕2}{\sum _{i=1}^k{\hat{w}}_{mle,i}}{\hat{\sigma }}_{mle,i}^2=-2+2\sqrt{1+{\hat{\sigma }}_i^2+{\left\{\hat{\mu }-ln\left[\hat{\vartheta }∕\left(1-{\hat{\delta }}_i\right)\right]\right\}}^2}}$ where ${\hat{w}}_{mle,i}=n_{i\left(1\right)}∕{\hat{\sigma }}_{mle,i}^2$ and $ln{\hat{\vartheta }}_{=}\frac{{\sum }_{i=1}^k{\hat{w}}_i\left[{\hat{\mu }}_i+ln\left(1-{\hat{\delta }}_i\right)\right]+N∕2}{{\sum }_{i=1}^k{\hat{w}}_i}$; ${\hat{w}}_i=n_{i\left(1\right)}∕{\hat{\sigma }}_i^2$. 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 $\left(ln{\hat{\vartheta }}_{mle}-ln\vartheta \right)\sqrt{{\sum }_{i=1}^k{\hat{w}}_{mle,i}}\sim N\left(0,1\right)$ such that $T={\left(ln{\hat{\vartheta }}_{mle}-ln\vartheta \right)}^2{\sum }_{i=1}^k{\hat{w}}_{mle,i}\sim {\chi }_{n_{i\left(1\right)}-1}^2$. It is well-known that ${\hat{\mu }}_i$, ${\hat{\sigma }}_i^2$ and ${\hat{\delta }}_i$ are independent random variables for which ${\hat{\mu }}_i\sim N\left(ln\left(\frac{\vartheta }{1-{\delta }_i}\right)-\frac{{\sigma }_i^2}{2},{\sigma }_i^2∕n_{i\left(1\right)}\right)$, $\left(n_{i\left(1\right)}-1\right){\hat{\sigma }}_i^2∕{\sigma }_i^2\sim {\chi }_{n_{i\left(1\right)}-1}^2$ and ${\hat{\delta }}_i\sim N\left(\delta ,\delta \left(1-\delta \right)∕n_i\right)$ are obtained, respectively. Let $\eta ={\mu }_i+{\sigma }_i^2∕2$ so that we can write $T=\frac{{\sum }_{i=1}^k{\hat{w}}_{mle,i}\left[{\hat{\mu }}_i+ln\left(1-{\hat{\delta }}_i\right)-\eta -ln\left(1-{\delta }_i\right)\right]+N∕2}{{\sum }_{i=1}^k{\hat{w}}_{mle,i}}$. 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)${\displaystyle T^{PB}={\left(ln{\hat{\vartheta }}_{mle}^{PB}-ln\hat{\vartheta }\right)}^2\sum _{i=1}^k{\hat{w}}_{mle,i}^{PB}}$ where $ln{\hat{\vartheta }}_{mle}^{PB}=\frac{{\sum }_{i=1}^k{\hat{w}}_{mle,i}^{PB}\left[{\hat{\mu }}_i^{PB}+ln\left(1-{\hat{\delta }}_i^{PB}\right)\right]+N∕2}{{\sum }_{i=1}^k{\hat{w}}_{mle,i}^{PB}}$, ${\hat{w}}_i^{PB}=n_{i\left(1\right)}∕{\hat{\sigma }}_i^{2B}$, ${\hat{\mu }}_i^{PB}\sim N\left({\hat{\mu }}_i^B,{\hat{\sigma }}_i^{2B}∕n_{i\left(1\right)}\right)$, ${\hat{\sigma }}_i^{2PB}\sim {\hat{\sigma }}_i^{B2}{\chi }_{n_{i\left(1\right)}-1}^2∕\left(n_{i\left(1\right)}-1\right)$ and ${\hat{\delta }}^{PB}\sim beta\left(n_{i\left(0\right)}^B+0.5,n_{i\left(1\right)}^B+0.5\right)$, $n_{i\left(0\right)}^B=n_i{\hat{\delta }}_i^B$, and $n_{i\left(1\right)}^B=n_i-n_{i\left(0\right)}^B$. Note that ${\hat{\mu }}_i^B$, ${\hat{\sigma }}_i^{2B}$, and ${\hat{\delta }}_i^B$ are the observed values of ${\hat{\mu }}_i$, ${\hat{\sigma }}_i^2$, and ${\hat{\delta }}_i$, respectively, from random sampling with replacement based on the bootstrap approach. Thus, the 100(1 − ζ)% PB interval for ϑ is given by (30)${\displaystyle C{I}_{\vartheta }^{\left(pb\right)}=exp\left[ln{\hat{\vartheta }}_{mle}\mp \sqrt{q_{\zeta }^{PB}∕\sum _{i=1}^k{\hat{w}}_{mle,i}}\right]}$ 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’.