Multiple comparisons of precipitation variations in different areas using simultaneous confidence intervals for all possible ratios of variances of several zero-inflated lognormal models

The Bayesian approach

The essential feature of Bayesian approach is to use the situation-specific prior distribution that reflects knowledge or subjective belief about the parameter of interest; this is modified in accordance with Baye’s Theorem to yield the posterior distribution. Thus, CIs based on the Bayesian approach are derived by using the posterior distribution. In Bayesian theory, the CI is referred to as the credible interval because it is not unique on the posterior distribution. The following methods are used to define suitable credible intervals: the narrowest interval for a univariate distribution (the highest posterior density interval) (Box & Tiao, 1973); the interval when the probability of being below is the same as being above, which is sometimes referred to as the equal-tailed interval (Gelman et al., 2014); or the interval with the mean as the central point (assuming that it exists). In the present study, the SCIs based on the Bayesian approach were constructed based on the equal-tailed interval. Motivated by Maneerat, Niwitpong & Niwitpong (2020), the probability-matching-beta (PMB) and reference-beta (RB) priors were our choice for parameter $\left(d_i^{\prime },{\mu }_i,{\sigma }_i^2\right)$ in this study. Thus, Bayesian SCIs for λ_ik were established as follows:

The PMB prior:

The probability-matching prior for $\left({\mu }_i,{\sigma }_i^2\right)$ is $\text{P}{\left({\mu }_i,{\sigma }_i^2\right)}_{pm}\propto {\sigma }_i^{-2}\sqrt{2+{\sigma }_i^{-2}}$ combined with the prior of $d_i^{\prime }$ as a beta distribution with a_i = b_i = 1/2. Thus, the PMB prior for $\left(d^{\prime },{\mu }_i,{\sigma }_i^2\right)$ can be defined as (8)${\displaystyle \text{P}{\left(d^{\prime },{\mu }_i,{\sigma }_i^2\right)}_{\text{pmb}}\propto \prod _{i=1}^h{\sigma }_i^{-2}\sqrt{\frac{2+{\sigma }_i^{-2}}{\left(1-d_i^{\prime }\right)d_i^{\prime }}}.}$

When updated with its likelihood, we obtain (9)${\displaystyle \text{P}\left(y|\lambda \right)\propto \prod _{i=1}^h{\left(1-d_i^{\prime }\right)}_i^{n_{i0}}{d}_i^{n_{i1}}{\left({\sigma }_i^2\right)}^{-n_{i1}/2}exp\left\{-\frac{1}{2{\sigma }_i^2}\sum _{j:x_{ij}>0}{\left(ln{y}_{ij}-{\mu }_i\right)}^2\right\}.}$

The respective marginal posterior distributions of $\left(d_i^{\prime },{\mu }_i,{\sigma }_i^2\right)$ are

(10)${\displaystyle \text{P}{\left(d_i^{\prime }|y_i\right)}_{\text{pmb}}\propto {\left(1-d_i^{\prime }\right)}_i^{n_{i0}+1/2}{d}_i^{n_{i1}+1/2}}$ (11)${\displaystyle \text{P}{\left({\mu }_i|y_i,{\sigma }_i^2\right)}_{\text{pmb}}\propto exp\left\{-\frac{1}{2{\sigma }_{i,\text{pmb}}^2}\sum _{j:x_{ij}>0}{\left(ln{y}_{ij}-{\mu }_i\right)}^2\right\}}$ (12)${\displaystyle \text{P}{\left({\sigma }_i^2|y_i\right)}_{\text{pmb}}\propto {\left({\sigma }_i^2\right)}^{-\frac{n_{i1}+1}{2}}\sqrt{2+{\sigma }_i^{-2}}exp\left\{-\frac{\left(n_{i1}-1\right){\hat{\sigma }}_i^2}{2{\sigma }_i^2}\right\}}$

which are denoted as $d_{i,\text{pmb}}^{\left(post\right)}|y_i\sim \text{beta}\left(n_{i0}+1/2,n_{i1}+1/2\right)$, ${\mu }_{i,\text{pmb}}^{\left(post\right)}|y_i\sim \text{N}\left({\hat{\mu }}_{i,\text{pmb}},{\sigma }_{i,\text{pmb}}^{2\left(post\right)}\right)$, and ${\sigma }_{i,\text{pmb}}^{2\left(post\right)}\propto {\left({\sigma }_i^2\right)}^{-\frac{n_{i1}+1}{2}}\sqrt{2+{\sigma }_i^{-2}}exp\left\{-\frac{\left(n_{i1}-1\right){\hat{\sigma }}_i^2}{2{\sigma }_i^2}\right\}$, respectively. Thus, the posterior of λ becomes (13)${\displaystyle {\lambda }_{ik,\text{pmb}}^{\left(post\right)}=T_{i,\text{pmb}}^{\left(post\right)}-T_{k,\text{pmb}}^{\left(post\right)},}$

where $T_{i,\text{pmb}}^{\left(post\right)}\approx ln{d}_{i,\text{pmb}}^{\left(post\right)}+2\left({\mu }_i^{\left(post\right)}+{\sigma }_{i,\text{pmb}}^{2\left(post\right)}\right)$ and $T_{k,\text{pmb}}^{\left(post\right)}\approx ln{d}_{k,\text{pmb}}^{\left(post\right)}+2\left({\mu }_{k,\text{pmb}}^{\left(post\right)}+{\sigma }_{k,\text{pmb}}^{2\left(post\right)}\right)$. In agreement with Ganesh (2009), the 100(1 − α)% Bayesian-based SCI with PMB prior for λ_ik is (14)${\displaystyle {\left[L_{{\lambda }_{ik}},U_{{\lambda }_{ik}}\right]}_{\text{pmb}}=\left[{\lambda }_{ik,\text{pmb}}^{\left(post\right)}\mp v_{\alpha }^{\text{pmb}}\right],}$

where $v_{\alpha }^{\text{pmb}}$ stands for the (1 − α)^th percentile of the distribution of $V^{\text{pmb}}=\underset{h}{\text{max}}\left\{{\lambda }_{ik,\text{pmb}}^{\left(post\right)}\right\}-\underset{h}{\text{min}}\left\{{\lambda }_{ik,\text{pmb}}^{\left(post\right)}\right\}$.

The RB prior:

This is a non-informative prior derived from the Fisher information matrix (Maneerat, Niwitpong & Niwitpong, 2020). The RB prior of $\left(d^{\prime },{\mu }_i,{\sigma }_i^2\right)$ is defined as (15)${\displaystyle \text{P}{\left(d^{\prime },{\mu }_i,{\sigma }_i^2\right)}_{\text{rfb}}\propto \prod _{i=1}^h{\sigma }_i^{-1}\sqrt{\frac{1+{\left(2{\sigma }_i^2\right)}^{-1}}{\left(1-d_i^{\prime }\right)d_i^{\prime }}}}$

in which the prior of d′ is a beta distribution. When combined with its likelihood Eq. (9), the posterior of $\left({\mu }_i,{\sigma }_i^2\right)$ differs from the PMB prior as follows:

(16)${\displaystyle \text{P}{\left({\mu }_i|y_i,{\sigma }_i^2\right)}_{\text{rfb}}\propto exp\left\{-\frac{1}{2{\sigma }_{i,\text{rfb}}^2}\sum _{j:x_{ij}>0}{\left(ln{y}_{ij}-{\mu }_i\right)}^2\right\}}$ (17)${\displaystyle \text{P}{\left({\sigma }_i^2|y_i\right)}_{\text{rfb}}\propto {\left({\sigma }_i^2\right)}^{-\frac{n_{i1}}{2}}\sqrt{1+{\left(2{\sigma }_i^2\right)}^{-1}}exp\left\{-\frac{\left(n_{i1}-1\right){\hat{\sigma }}_i^2}{2{\sigma }_i^2}\right\}.}$

Moreover, it can be similarly denoted as $d_{i,\text{rfb}}^{\left(post\right)}|y_i\sim \text{beta}\left(n_{i0}+1/2,n_{i1}+1/2\right)$, ${\mu }_{i,\text{rfb}}^{\left(post\right)}|y_i\sim \text{N}\left({\hat{\mu }}_{i,\text{rfb}},{\sigma }_{i,\text{rfb}}^{2\left(post\right)}\right)$ and ${\sigma }_{i,\text{rfb}}^{2\left(post\right)}\propto {\left({\sigma }_i^2\right)}^{-\frac{n_{i1}}{2}}\sqrt{1+{\left(2{\sigma }_i^2\right)}^{-1}}exp\left\{-\frac{\left(n_{i1}-1\right){\hat{\sigma }}_i^2}{2{\sigma }_i^2}\right\}$, respectively. The posterior of λ_ik is ${\lambda }_{ik,\text{rfb}}^{\left(post\right)}=T_{i,\text{rfb}}^{\left(post\right)}-T_{k,\text{rfb}}^{\left(post\right)}$, where $T_{i,\text{rfb}}^{\left(post\right)}\approx ln{d}_{i,\text{rfb}}^{\left(post\right)}+2\left({\mu }_i^{\left(post\right)}+{\sigma }_{i,\text{rfb}}^{2\left(post\right)}\right)$ and $T_{k,\text{rfb}}^{\left(post\right)}\approx ln{d}_{k,\text{rfb}}^{\left(post\right)}+2\left({\mu }_{k,\text{rfb}}^{\left(post\right)}+{\sigma }_{k,\text{rfb}}^{2\left(post\right)}\right)$. According to Ganesh (2009), the 100(1 − α)% Bayesian-based SCI with the RB prior for λ_ik is (18)${\displaystyle {\left[L_{{\lambda }_{ik}},U_{{\lambda }_{ik}}\right]}_{\text{rfb}}=\left[{\lambda }_{ik,\text{rfb}}^{\left(post\right)}\mp v_{\alpha }^{\text{rfb}}\right],}$

where $v_{\alpha }^{\text{rfb}}$ stands for the (1 − α)^th percentile of the distribution of $V^{\text{rfb}}=\underset{h}{\text{max}}\left\{{\lambda }_{ik,\text{rfb}}^{\left(post\right)}\right\}-\underset{h}{\text{min}}\left\{{\lambda }_{ik,\text{rfb}}^{\left(post\right)}\right\}$.

The GPQ approach

Motivated by Wu & Hsieh (2014), the GPQ of d_i is formulated using the arcsin square-root transformation of the variance. Moreover, the GPQs for $\left({\mu }_i,{\sigma }_i^2\right)$ are also obtained from transformation of the normal approximation by using the central limit theorem (Tian, 2005; Hasan & Krishnamoorthy, 2017). The GPQ for T_i can be written as (19)${\displaystyle G_{T_i}=ln\left[1-{sin}^2\left\{{sin}^{-1}\sqrt{{\hat{d}}_i}-\frac{R_i}{2\sqrt{n_i}}\right\}\right]+2\left[{\hat{\mu }}_i-S_i\sqrt{\frac{{\text{G}}_{{\sigma }_i^2}}{n_{i1}}}+{\text{G}}_{{\sigma }_i^2}\right],}$

where ${\text{G}}_{{\sigma }_i^2}=\left(n_{i1}-1\right){\hat{\sigma }}_i^2/U_i$. The random variables $R_i=2\sqrt{n_i}\left({sin}^{-1}\sqrt{{\hat{d}}_i}-{sin}^{-1}\sqrt{d_i}\right)$, $S_i=\left({\hat{\mu }}_i-{\mu }_i\right)/\left({\sigma }_i^2/n_{i1}\right)$ and $U_i=\left(n_{i1}-1\right){\hat{\sigma }}_i^2/{\sigma }_i^2$ are independent from standard normal, normal and ${\chi }_{n_{i1-1}}^2$ distributions, respectively. Thus, the corresponding GPQ of λ_ik can be expressed as (20)${\displaystyle {\text{G}}_{{\lambda }_{ik}}={\text{G}}_{T_i}-{\text{G}}_{T_k}.}$

Similarly, ${\text{G}}_{T_k}=ln\left(1-{\text{G}}_{d_k}\right)+{\text{G}}_{2{\mu }_k}+{\text{G}}_{2{\sigma }_k^2}$ denotes the GPQ of T_k; ${\text{G}}_{d_k}={sin}^2\left\{{sin}^{-1}\sqrt{{\hat{d}}_k}-\left[R_k{\left(2\sqrt{n_k}\right)}^{-1}\right]\right\}$, ${\text{G}}_{2{\mu }_k}=2\left({\hat{\mu }}_k-S_k\sqrt{{\text{G}}_{{\sigma }_k^2}/n_{k1}}\right)$, and ${\text{G}}_{2{\sigma }_k^2}=2\left(n_{k1}-1\right){\hat{\sigma }}_k^2/U_k$. Therefore, the 100(1 − α)% SCI for λ_jk based on the GPQ approach is given by (21)${\displaystyle {\left[L_{{\lambda }_{ik}},U_{{\lambda }_{ik}}\right]}_{\text{gpq}}=\left[{\hat{\lambda }}_{ik}\mp q_{\alpha }^{\text{GPQ}}\sqrt{\widehat{Var}\left({\hat{\lambda }}_{ik}\right)}\right],}$

where $q_{\alpha }^{\text{GPQ}}$ denotes the (1 − α)^th percentile of the Q^GPQ distribution; the Q^GPQ is derived as (22)${\displaystyle Q^{\text{GPQ}}=\underset{j\ne l}{max}\left|\left\{{\hat{\lambda }}_{ik}-{\text{G}}_{{\lambda }_{ik}}\left(Y,Y^{\ast },d,\mu ,{\sigma }^2\right)\right\}/\sqrt{\widehat{Var}\left({\hat{\lambda }}_{ik}\right)}\right|.}$

In agreement with Hannig et al. (2006), Kharrati-Kopaei & Eftekhar (2017), the asymptotic coverage probability of the SCI for λ_ik based on the GPQ is slightly modified from that in Maneerat, Niwitpong & Niwitpong (2021b) (the proof of Theorem 1 in the Appendix).

for ∀i ≠ k and i, k =1 , …, h.

The PB approach

Here, we assume that the data come from a known distribution with unknown parameters that are estimated by using samples stimulated from the estimated distribution. In the present study, the PB approach is adjusted to suit our particular situation. Let ${\hat{d}}_i^{\ast }$, ${\hat{\mu }}_i^{\ast }$ and ${\hat{\sigma }}_i^{2\ast }$ be the observed values of ${\hat{d}}_i$, ${\hat{\mu }}_i$, and ${\hat{\sigma }}_i^2$ representing the estimated values of parameters d_i, μ_i, and ${\sigma }_i^2$, respectively. Thus, we can obtain the empirical distribution of T based on the PB approach. In accordance with Sadooghi-Alvandi & Malekzadeh (2014), the respective sampling distributions of (${\hat{d}}_i^{\ast }$, ${\hat{\mu }}_i^{\ast }$, ${\hat{\sigma }}_i^{2\ast }$) are

where $D_i=\left[{\hat{\mu }}_i^{\left(pboot\right)}-{\hat{\mu }}_i^{\ast }\right]/\sqrt{{\hat{\sigma }}_j^{2\ast }/n_{i1}^{\ast }}\sim N\left(0,1\right)$ and $U_j=\left[n_{i1}^{\ast }-1\right]{\hat{\sigma }}_i^{2\left(pboot\right)}/{\hat{\sigma }}_i^{2\ast }$ $\sim {\chi }_{n_{i1}^{\ast }-1}^2$ are independent random variables with standard normal and Chi-square distributions, respectively. The PB variable-based pivotal quantity is expressed as (27)${\displaystyle M^{\text{PB}}=\left|\left({\hat{\lambda }}_{ik}^{pboot}-{\hat{\lambda }}_{ik}^{\ast }\right)/\sqrt{\widehat{Var}\left({\hat{\lambda }}_{ik}^{\ast }\right)}\right|,}$

where ${\hat{\lambda }}_{ik}^{\left(pboot\right)}={\hat{T}}_i^{\left(pboot\right)}-{\hat{T}}_k^{\left(pboot\right)}$ and ${\hat{\lambda }}_{ik}^{\ast }={\hat{T}}_i^{\ast }-{\hat{T}}_k^{\ast }$. By replacing observed values $\left({\hat{d}}_i^{\ast }{\hat{\mu }}_i^{\ast },{\hat{\sigma }}_i^{2\ast }\right)$ from the samples, we respectively obtain

(28)${\displaystyle {\hat{\lambda }}_{ik}^{\ast }=ln\left(\frac{{\hat{d}}_i^{\ast }}{{\hat{d}}_k^{\ast }}\right)+2\left[\left({\hat{\mu }}_i^{\ast }-{\hat{\mu }}_k^{\ast }\right)+\left({\hat{\sigma }}_i^{2\ast }-{\hat{\sigma }}_k^{2\ast }\right)\right]}$ (29)${\displaystyle {\hat{\lambda }}_{ik}^{\left(pboot\right)}=ln\left(\frac{{\hat{d}}_i^{\left(pboot\right)}}{{\hat{d}}_k^{\left(pboot\right)}}\right)+2\left[\left({\hat{\mu }}_i^{\left(pboot\right)}-{\hat{\mu }}_k^{\left(pboot\right)}\right)+\left({\hat{\sigma }}_i^{2\left(pboot\right)}-{\hat{\sigma }}_k^{2\left(pboot\right)}\right)\right]}$ (30)${\displaystyle \widehat{Var}\left({\hat{\lambda }}_{ik}^{\ast }\right)=\frac{{\hat{d}}_i^{\ast }}{n_i{\hat{d}}_i^{\ast }}+\frac{{\hat{d}}_i^{\ast }}{n_k{\hat{d}}_k^{\ast }}+4\left(\frac{{\hat{\sigma }}_i^{2\ast }}{n_{i1}^{\ast }}+\frac{2{\hat{\sigma }}_i^{2\ast }}{n_{i1}^{\ast }-1}+\frac{{\hat{\sigma }}_k^{2\ast }}{n_{k1}^{\ast }}+\frac{2{\hat{\sigma }}_k^{2\ast }}{n_{k1}^{\ast }-1}\right),}$

where ${\hat{d}}_i^{\ast }=1-{\hat{d}}_i^{\ast }$ and $n_{i1}^{\ast }=n_i{\hat{d}}_i^{\ast }$. Hence, the 100(1 − α)% SCI for λ_ik based on the PB approach is (31)${\displaystyle {\left[L_{{\lambda }_{ik}},U_{{\lambda }_{ik}}\right]}_{\left(PB\right)}=\left[{\hat{\lambda }}_{ik}\mp M_{\alpha }^{\text{PB}}\sqrt{\widehat{Var}\left({\hat{\lambda }}_{ik}\right)}\right],}$

where $m_{\alpha }^{\text{PB}}$ is the (1 − α)^th percentile of the distribution of M^PB. Theorem 2 shows the asymptotic coverage probability of the 100(1 − α)% SCI for λ_ik based on the PB approach (see the proof in the Appendix ).