Confidence intervals for ratio of means of delta-lognormal distributions based on left-censored data with application to rainfall data in Thailand

Generalized confidence interval approach

The generalized pivotal quantity (GPQ) is used to construct the GCI which is defined in Weerahandi (1993). According to Krishnamoorthy, Mallick & Mathew (2011), the GPQs for μ₁, σ₁, and θ₁ are defined by (16)${\displaystyle R_{{\mu }_1}={\hat{\mu }}_1-\frac{{\hat{\mu }}_1^{\ast }}{{\hat{\sigma }}_1^{\ast }}{\hat{\sigma }}_1}$ (17)${\displaystyle R_{{\sigma }_1}=\frac{{\hat{\sigma }}_1}{{\hat{\sigma }}_1^{\ast }}}$ and (18)${\displaystyle R_{{\theta }_1}=exp\left(R_{{\mu }_1}+\frac{1}{2}{\left(R_{{\sigma }_1}\right)}^2\right),}$ where ${\hat{\mu }}_1^{\ast }$ and ${\hat{\sigma }}_1^{\ast }$ are the maximum likelihood estimators based on a censored sample from standard normal distribution.

Similarly, the GPQs for μ₂, σ₂, and θ₂ are defined by (19)${\displaystyle R_{{\mu }_2}={\hat{\mu }}_2-\frac{{\hat{\mu }}_2^{\ast }}{{\hat{\sigma }}_2^{\ast }}{\hat{\sigma }}_2}$ (20)${\displaystyle R_{{\sigma }_2}=\frac{{\hat{\sigma }}_2}{{\hat{\sigma }}_2^{\ast }}}$ and (21)${\displaystyle R_{{\theta }_2}=exp\left(R_{{\mu }_2}+\frac{1}{2}{\left(R_{{\sigma }_2}\right)}^2\right),}$ where ${\hat{\mu }}_2^{\ast }$ and ${\hat{\sigma }}_2^{\ast }$ are the maximum likelihood estimators based on a censored sample from standard normal distribution.

The GPQ for the difference between means of delta-lognormal distributions based on left-censored data was used as previously described in Thangjai & Niwitpong (2023). In this article, the GPQ for the ratio of means of delta-lognormal distributions based on left-censored data is given by (22)${\displaystyle R_{\theta }=\frac{R_{{\theta }_1}}{R_{{\theta }_2}},}$ where R_θ1 and R_θ2 are defined in Eqs. (18) and (21), respectively.

Therefore, the 100(1 − α)% two-sided confidence interval for the ratio of means of delta-lognormal distributions based on left-censored data using the GCI approach is given by (23)${\displaystyle C{I}_{GCI}=\left[L_{GCI},U_{GCI}\right]=\left[R_{\theta }\left(\alpha /2\right),R_{\theta }\left(1-\alpha /2\right)\right],}$ where R_θ(α/2) and R_θ(1 − α/2) denote the 100(α/2)-th and 100(1 − α/2)-th percentiles of R_θ, respectively.

The Algorithm 1 is used to construct the GCI for the ratio of means of delta-lognormal distributions based on left-censored data.

Algorithm 1.

Step 1: Generate sample from the standard normal distribution and compute ${\hat{\mu }}_1^{\ast }$, ${\hat{\mu }}_2^{\ast }$, ${\hat{\sigma }}_1^{\ast }$, and ${\hat{\sigma }}_2^{\ast }$

Step 2: Compute R_μ1, R_σ1, and R_θ1 from Eqs. (16), (17), and (18), respectively.

Step 3: Compute R_μ2, R_σ2, and R_θ2 from Eqs. (19), (20), and (21), respectively.

Step 4: Compute R_θ from Eq. (22)

Step 5: Repeat step 1 - step 4, a total times and obtain an array of R_θ’s

Step 6: Compute L_GCI and U_GCI

Bayesian approach

The Bayesian approach offers a framework for updating beliefs and making predictions using new evidence or data. It is rooted in Bayes’ theorem, which combines prior probability and likelihood to calculate the posterior probability. The prior distribution represents uncertainty about parameters before observing data. In this article, we employed the Jeffreys Independence prior. According to Thangjai & Niwitpong (2023), the posterior distributions of ${\sigma }_1^2$, μ₁, and θ_1.BS are defined by (24)${\displaystyle {\sigma }_1^2|y_1\sim IG\left(\frac{n_{1\left(2\right)}-1}{2},\frac{\left(n_{1\left(2\right)}-1\right)s_1^2}{2}\right)}$ (25)${\displaystyle {\mu }_1|{\sigma }_1^2,y_1\sim N\left({\bar{y}}_1,\frac{{\sigma }_1^2}{n_{1\left(2\right)}}\right)}$

and (26)${\displaystyle {\theta }_{1.BS}=exp\left({\mu }_1+\frac{1}{2}{\sigma }_1^2\right),}$ where ${\bar{y}}_1$ is observed value of ${\bar{Y}}_1$ defined in Eq. (3) and $s_1^2$ is the observed value of $S_1^2$ defined in Eq. (4).

The posterior distributions of ${\sigma }_2^2$, μ₂, and θ_2.BS are defined by (27)${\displaystyle {\sigma }_2^2|y_2\sim IG\left(\frac{n_{2\left(2\right)}-1}{2},\frac{\left(n_{2\left(2\right)}-1\right)s_2^2}{2}\right)}$ (28)${\displaystyle {\mu }_2|{\sigma }_2^2,y_2\sim N\left({\bar{y}}_2,\frac{{\sigma }_2^2}{n_{2\left(2\right)}}\right)}$

and (29)${\displaystyle {\theta }_{2.BS}=exp\left({\mu }_2+\frac{1}{2}{\sigma }_2^2\right),}$ where ${\bar{y}}_2$ is observed value of ${\bar{Y}}_2$ defined in Eq. (9) and $s_2^2$ is the observed value of $S_2^2$ defined in Eq. (10).

Thangjai & Niwitpong (2023) proposed the posterior distribution of the difference between means of delta-lognormal distributions based on left-censored data. Therefore, the posterior distribution of the ratio of means of delta-lognormal distributions based on left-censored data is given by (30)${\displaystyle {\theta }_{BS}=\frac{{\theta }_{1.BS}}{{\theta }_{2.BS}},}$ where θ_1.BS and θ_2.BS are defined in Eqs. (26) and (29), respectively.

Therefore, the 100(1 − α)% two-sided credible interval for the ratio of means of delta-lognormal distributions based on left-censored data using the Bayesian approach is given by (31)${\displaystyle C{I}_{\theta .BS}=\left[L_{\theta .BS},U_{\theta .BS}\right],}$ where L_θ.BS and U_θ.BS denote the lower and upper limits of the shortest 100(1 − α)% highest posterior density interval of θ_BS, respectively.

The Algorithm 2 is used to construct the Bayesian credible interval for the ratio of means of delta-lognormal distributions based on left-censored data.

Algorithm 2.

Step 1: Compute ${\sigma }_1^2|y_1$, ${\mu }_1|{\sigma }_1^2,y_1$, and θ_1.BS from Eqs. (24), (25), and (26), respectively.

Step 2: Compute ${\sigma }_2^2|y_2$, ${\mu }_2|{\sigma }_2^2,y_2$, and θ_2.BS from Eqs. (27), (28), and (29), respectively.

Step 3: Compute θ_BS from Eq. (30)

Step 4: Repeat step 1 - step 3, a total m times and obtain an array of θ_BS’s

Step 5: Compute L_θ.BS and U_θ.BS

Parametric bootstrap approach

Let $Y_1^{\ast }=\left(Y_{11}^{\ast },Y_{12}^{\ast },\dots ,Y_{1n_{1\left(2\right)}}^{\ast }\right)$ be the observations above log(ξ₁). The mean and variance of $Y_1^{\ast }$ are given by (32)${\displaystyle {\bar{Y}}_1^{\ast }=\frac{1}{n_{1\left(2\right)}}\sum _{j=1}^{n_{1\left(2\right)}}{Y}_{1j}^{\ast }}$ and (33)${\displaystyle S_1^{2\ast }=\frac{1}{n_{1\left(2\right)}}\sum _{j=1}^{n_{1\left(2\right)}}{\left(Y_{1j}^{\ast }-{\bar{Y}}_1^{\ast }\right)}^2.}$

The estimator of the mean of delta-lognormal distribution based on left-censored data is (34)${\displaystyle {\hat{\theta }}_1^{\ast }=exp\left({\bar{Y}}_1^{\ast }+\frac{1}{2}{S}_1^{2\ast }\right),}$ where ${\bar{Y}}_1^{\ast }$ and $S_1^{2\ast }$ are defined in Eqs. (32) and (33), respectively.

Let $Y_2^{\ast }=\left(Y_{21}^{\ast },Y_{22}^{\ast },\dots ,Y_{2n_{2\left(2\right)}}^{\ast }\right)$ be the observations above log(ξ₂). The mean and variance of $Y_2^{\ast }$ are given by (35)${\displaystyle {\bar{Y}}_2^{\ast }=\frac{1}{n_{2\left(2\right)}}\sum _{j=1}^{n_{2\left(2\right)}}{Y}_{2j}^{\ast }}$ and (36)${\displaystyle S_2^{2\ast }=\frac{1}{n_{2\left(2\right)}}\sum _{j=1}^{n_{2\left(2\right)}}{\left(Y_{2j}^{\ast }-{\bar{Y}}_2^{\ast }\right)}^2.}$

The estimator of the mean of delta-lognormal distribution based on left-censored data is (37)${\displaystyle {\hat{\theta }}_2^{\ast }=exp\left({\bar{Y}}_2^{\ast }+\frac{1}{2}{S}_2^{2\ast }\right),}$ where ${\bar{Y}}_2^{\ast }$ and $S_2^{2\ast }$ are defined in Eqs. (35) and (36), respectively.

The estimator of the difference between means of delta-lognormal distributions based on left-censored data was proposed as previously described in Thangjai & Niwitpong (2023). According to Thangjai & Niwitpong (2023), the estimator of the ratio of means of delta-lognormal distributions based on left-censored data is given by (38)${\displaystyle {\hat{\theta }}^{\ast }=\frac{{\hat{\theta }}_1^{\ast }}{{\hat{\theta }}_2^{\ast }},}$ where ${\hat{\theta }}_1^{\ast }$ and ${\hat{\theta }}_2^{\ast }$ are defined in Eqs. (34) and (37), respectively.

The lower and upper limits of the confidence interval for the ratio of means of delta-lognormal distributions based on left-censored data are given by (39)${\displaystyle L_{PB}=\overline{{\hat{\theta }}^{\ast }}-z_{1-\alpha /2}sd\left({\hat{\theta }}^{\ast }\right)}$ and (40)${\displaystyle U_{PB}=\overline{{\hat{\theta }}^{\ast }}+z_{1-\alpha /2}sd\left({\hat{\theta }}^{\ast }\right),}$ where $\overline{{\hat{\theta }}^{\ast }}$ is the mean of ${\hat{\theta }}^{\ast }$, $sd\left({\hat{\theta }}^{\ast }\right)$ is the standard deviation of ${\hat{\theta }}^{\ast }$, and z_1−α/2 is the 100(1 − α/2)-th percentile of the standard normal distribution.

Therefore, the 100(1 − α)% two-sided confidence interval for the ratio of means of delta-lognormal distributions based on left-censored data using the parametric bootstrap approach is given by (41)${\displaystyle C{I}_{PB}=\left[L_{PB},U_{PB}\right],}$ where L_PB and U_PB are defined in Eqs. (39) and (40), respectively.

The Algorithm 3 is used to construct the parametric bootstrap confidence interval for the ratio of means of delta-lognormal distributions based on left-censored data.

Algorithm 3.

Step 1: Parametric bootstrapping assumes that the data comes from a known distribution with unknown parameters. We estimate these parameters from the data and then use the estimated distributions to simulate the samples. Generate $y_1^{\ast }=\left(y_{11}^{\ast },y_{12}^{\ast },\dots ,y_{1n_{1\left(2\right)}}^{\ast }\right)$ from normal distribution with ${\hat{\mu }}_1$ and ${\hat{\sigma }}_1^2$ and generate $y_2^{\ast }=\left(y_{21}^{\ast },y_{22}^{\ast },\dots ,y_{2n_{2\left(2\right)}}^{\ast }\right)$ from normal distribution with ${\hat{\mu }}_2$ and ${\hat{\sigma }}_2^2$

Step 2: Compute ${\bar{y}}_1^{\ast }$ from Eq. (32), $s_1^{2\ast }$ from Eq. (33), and ${\hat{\theta }}_1^{\ast }$ from Eq. (34)

Step 3: Compute ${\bar{y}}_2^{\ast }$ from Eq. (35), $s_2^{2\ast }$ from Eq. (36), and ${\hat{\theta }}_2^{\ast }$ from Eq. (37)

Step 4: Compute ${\hat{\theta }}^{\ast }$ from Eq. (38)

Step 5: Repeat step 1 - step 4, a total m times and obtain an array of ${\hat{\theta }}^{\ast }$’s

Step 6: Compute L_PB and U_PB from Eqs. (39) and (40)

Method of variance estimates recovery approach

According to Maneerat, Niwitpong & Niwitpong (2020), the lower and upper limits of the confidence interval for μ₁ are defined by (42)${\displaystyle l_{{\mu }_1}={\hat{\mu }}_1-z_{1-\alpha /2}\sqrt{\frac{\left(n_{1\left(2\right)}-1\right){\hat{\sigma }}_1^2}{n_{1\left(2\right)}{\chi }_{n_{1\left(2\right)}-1}^2}}}$ and (43)${\displaystyle u_{{\mu }_1}={\hat{\mu }}_1+z_{1-\alpha /2}\sqrt{\frac{\left(n_{1\left(2\right)}-1\right){\hat{\sigma }}_1^2}{n_{1\left(2\right)}{\chi }_{n_{1\left(2\right)}-1}^2}},}$ where z_1−α/2 is the 100(1 − α/2)-th percentile of the standard normal distribution, ${\chi }_{n_{1\left(2\right)}-1}^2$ is the chi-squared distribution with n₁₍₂₎ − 1 degrees of freedom, and ${\hat{\mu }}_1$ and ${\hat{\sigma }}_1^2$ are defined in Eqs. (5) and (6), respectively.

Similarly, the lower and upper limits of the confidence interval for μ₂ are given by (44)${\displaystyle l_{{\mu }_2}={\hat{\mu }}_2-z_{1-\alpha /2}\sqrt{\frac{\left(n_{2\left(2\right)}-1\right){\hat{\sigma }}_2^2}{n_{2\left(2\right)}{\chi }_{n_{2\left(2\right)}-1}^2}}}$ and (45)${\displaystyle u_{{\mu }_2}={\hat{\mu }}_2+z_{1-\alpha /2}\sqrt{\frac{\left(n_{2\left(2\right)}-1\right){\hat{\sigma }}_2^2}{n_{2\left(2\right)}{\chi }_{n_{2\left(2\right)}-1}^2}},}$ where z_1−α/2 is the 100(1 − α/2)-th percentile of the standard normal distribution, ${\chi }_{n_{2\left(2\right)}-1}^2$ is the chi-squared distribution with n₂₍₂₎ − 1 degrees of freedom, and ${\hat{\mu }}_2$ and ${\hat{\sigma }}_2^2$ are defined in Eqs. (11) and (12), respectively.

Following Maneerat, Niwitpong & Niwitpong (2020), the lower and upper limits of the confidence interval for ${\sigma }_1^2$ are defined by (46)${\displaystyle l_{{\sigma }_1^2}=\frac{\left(n_{1\left(2\right)}-1\right){\hat{\sigma }}_1^2}{{\chi }_{1-\alpha /2,n_{1\left(2\right)}-1}^2}}$ and (47)${\displaystyle u_{{\sigma }_1^2}=\frac{\left(n_{1\left(2\right)}-1\right){\hat{\sigma }}_1^2}{{\chi }_{\alpha /2,n_{1\left(2\right)}-1}^2},}$ where ${\chi }_{1-\alpha /2,n_{1\left(2\right)}-1}^2$ and ${\chi }_{\alpha /2,n_{1\left(2\right)}-1}^2$ are the 100(1 − α/2)-th and 100(α/2)-th percentiles of the chi-squared distribution with n₁₍₂₎ − 1 degrees of freedom, and ${\hat{\sigma }}_1^2$ is defined in Eq. (6).

The lower and upper limits of the confidence interval for ${\sigma }_2^2$ are defined by (48)${\displaystyle l_{{\sigma }_2^2}=\frac{\left(n_{2\left(2\right)}-1\right){\hat{\sigma }}_2^2}{{\chi }_{1-\alpha /2,n_{2\left(2\right)}-1}^2}}$ and (49)${\displaystyle u_{{\sigma }_2^2}=\frac{\left(n_{2\left(2\right)}-1\right){\hat{\sigma }}_2^2}{{\chi }_{\alpha /2,n_{2\left(2\right)}-1}^2},}$ where ${\chi }_{1-\alpha /2,n_{2\left(2\right)}-1}^2$ and ${\chi }_{\alpha /2,n_{2\left(2\right)}-1}^2$ are the 100(1 − α/2)-th and 100(α/2)-th percentiles of the chi-squared distribution with n₂₍₂₎ − 1 degrees of freedom, and ${\hat{\sigma }}_2^2$ is defined in Eq (12).

Applying the concept of Donner & Zou (1993), the lower and upper limits of confidence interval for ${\theta }_1=exp\left({\mu }_1+\frac{1}{2}{\sigma }_1^2\right)$ are given by (50)${\displaystyle l_{{\theta }_1}=exp\left[\left({\hat{\mu }}_1+\frac{1}{2}{\hat{\sigma }}_1^2\right)-\sqrt{{\left({\hat{\mu }}_1-l_{{\mu }_1}\right)}^2+{\left(\frac{1}{2}{\hat{\sigma }}_1^2-\frac{1}{2}{l}_{{\sigma }_1^2}\right)}^2}\right]}$ and (51)${\displaystyle u_{{\theta }_1}=exp\left[\left({\hat{\mu }}_1+\frac{1}{2}{\hat{\sigma }}_1^2\right)+\sqrt{{\left(u_{{\mu }_1}-{\hat{\mu }}_1\right)}^2+{\left(\frac{1}{2}{u}_{{\sigma }_1^2}-\frac{1}{2}{\hat{\sigma }}_1^2\right)}^2}\right],}$ where ${\hat{\mu }}_1$, ${\hat{\sigma }}_1^2$, l_μ1, u_μ1, $l_{{\sigma }_1^2}$, and $u_{{\sigma }_1^2}$ are defined in Eqs. (5), (6), (42), (43), (46), and (47), respectively.

Similarly, the lower and upper limits of confidence interval for ${\theta }_2=exp\left({\mu }_2+\frac{1}{2}{\sigma }_2^2\right)$ are given by (52)${\displaystyle l_{{\theta }_2}=exp\left[\left({\hat{\mu }}_2+\frac{1}{2}{\hat{\sigma }}_2^2\right)-\sqrt{{\left({\hat{\mu }}_2-l_{{\mu }_2}\right)}^2+{\left(\frac{1}{2}{\hat{\sigma }}_2^2-\frac{1}{2}{l}_{{\sigma }_2^2}\right)}^2}\right]}$ and (53)${\displaystyle u_{{\theta }_2}=exp\left[\left({\hat{\mu }}_2+\frac{1}{2}{\hat{\sigma }}_2^2\right)+\sqrt{{\left(u_{{\mu }_2}-{\hat{\mu }}_2\right)}^2+{\left(\frac{1}{2}{u}_{{\sigma }_2^2}-\frac{1}{2}{\hat{\sigma }}_2^2\right)}^2}\right],}$ where ${\hat{\mu }}_2$, ${\hat{\sigma }}_2^2$, l_μ2, u_μ2, $l_{{\sigma }_2^2}$, and $u_{{\sigma }_2^2}$ are defined in Eqs. (11), (12), (44), (45), (48), (49), respectively.

Using the concept of Donner & Zou (2012) and Thangjai & Niwitpong (2023), the lower and upper limits of confidence interval for $\theta =\frac{{\theta }_1}{{\theta }_2}$ are given by (54)${\displaystyle L_{MOVER}=\frac{{\hat{\theta }}_1{\hat{\theta }}_2-\sqrt{{\left({\hat{\theta }}_1{\hat{\theta }}_2\right)}^2-l_{{\theta }_1}{u}_{{\theta }_2}\left(2{\hat{\theta }}_1-l_{{\theta }_1}\right)\left(2{\hat{\theta }}_2-u_{{\theta }_2}\right)}}{u_{{\theta }_2}\left(2{\hat{\theta }}_2-u_{{\theta }_2}\right)}}$ and (55)${\displaystyle U_{MOVER}=\frac{{\hat{\theta }}_1{\hat{\theta }}_2+\sqrt{{\left({\hat{\theta }}_1{\hat{\theta }}_2\right)}^2-u_{{\theta }_1}{l}_{{\theta }_2}\left(2{\hat{\theta }}_1-u_{{\theta }_1}\right)\left(2{\hat{\theta }}_2-l_{{\theta }_2}\right)}}{l_{{\theta }_2}\left(2{\hat{\theta }}_2-l_{{\theta }_2}\right)},}$ where ${\hat{\theta }}_1$, ${\hat{\theta }}_2$, l_θ1, u_θ1, l_θ2, and u_θ2 are defined in Eqs. (8), (14), (50), (51), (52), and (53), respectively.

Therefore, the 100(1 − α)% two-sided confidence interval for the ratio of means of delta-lognormal distributions based on left-censored data using the MOVER approach is given by (56)${\displaystyle C{I}_{MOVER}=\left[L_{MOVER},U_{MOVER}\right],}$ where L_MOVER and U_MOVER are defined in Eqs. (54) and (55), respectively.