Estimation in generalized bivariate Birnbaum–Saunders models

Abstract

In this paper, we propose two moment-type estimation methods for the parameters of the generalized bivariate Birnbaum–Saunders distribution by taking advantage of some properties of the distribution. The proposed moment-type estimators are easy to compute and always exist uniquely. We derive the asymptotic distributions of these estimators and carry out a simulation study to evaluate the performance of all these estimators. The probability coverages of confidence intervals are also discussed. Finally, two examples are used to illustrate the proposed methods.

Appendices

Appendix 1: Asymptotic distribution of the MMEs

Let ${\varvec{T}}=(T_{1},T_{2})^{\top }$ follow a ${\mathrm{GBBS}}({\varvec{\alpha }},{\varvec{\beta }},\rho ,\nu )$ distribution, then

$$\begin{aligned} {\mathrm{E}}\left[ T_{k}\right]= & {} \beta _{k}\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) , \quad \sigma _{k}^{11}={\mathrm{Var}}\left[ T_{k}\right] = \beta _{k}^2\alpha _{k}^2\left( u_{k1} +\frac{2u_{k2}-u_{k1}^2}{4}\alpha _{k}^2\right) ,\\ \sigma _{k}^{22}= & {} {\mathrm{Var}}\left[ T_{k}^{-1}\right] = \beta _{k}^{-2}\alpha _{k}^2\left( u_{k1} +\frac{2u_{k2}-u_{k1}^2}{4}\alpha _{k}^2\right) \quad \sigma _{k}^{12}=\sigma _{k}^{22}={\mathrm{Cov}}\left[ T_{k}\right] \\= & {} 1-\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^2, \quad k=1,2, \end{aligned}$$

where $u_{kr}$ is as in (14).

Now, let $\{(t_{1i},t_{2i}),i=1,\ldots ,n\}$ be a bivariate random sample from the ${\mathrm{GBBS}}({\varvec{\alpha }},{\varvec{\beta }},\rho )$ distribution. The sample arithmetic and harmonic means are defined by

$$\begin{aligned} s_{k}=\frac{1}{n}\sum \limits _{i=1}^n t_{ki} \quad \text{ and } \quad r_{k}^*=r_{ki}^{-1}=\frac{1}{n}\sum \limits _{i=1}^n t_{ki}^{-1}, \quad k=1,2, \end{aligned}$$

and the MMEs are given by

$$\begin{aligned} \tilde{\alpha }_{k}=\left\{ \frac{2}{u_{k1}} \left[ \left( s_{k}r_{k}^*\right) ^\frac{1}{2}-1\right] \right\} ^\frac{1}{2} \quad \text{ and } \quad \tilde{\beta }_{k}=\left( s_{k}/r_{k}^*\right) ^{\frac{1}{2}}, \quad k=1,2. \end{aligned}$$

Consider $S_{k}=\frac{1}{n}\sum _{i=1}^{n}X_{kj}$ and $R_{k}^{*}=R_{k}^{-1}=\sum _{i=1}^{n}\frac{1}{X_{ki}}$, with $k=1,2$, which implies that the vector $(S_{k},R_{k}^{-1})^{\top }$ is bivariate normal distributed, that is,

$$\begin{aligned} \sqrt{n} \left( \begin{array}{c} S_{k}-E\left[ T_{k}\right] \\ R_{k}^*-E\left[ T_{k}^{-1}\right] \end{array} \right) \sim N \left[ \left( \begin{array}{c} 0\\ 0 \end{array} \right) , \left( \begin{array}{cc} {\mathrm{Var}}\left[ T_{k}\right] ,1-{\mathrm{E}}\left[ T_{k}\right] {\mathrm{E}}\left[ T_{k}^{-1}\right] \\ 1-{\mathrm{E}}\left[ T_{k}\right] {\mathrm{E}}\left[ T_{k}^{-1}\right] ,{\mathrm{Var}}\left[ T_{k}\right] \end{array} \right) \right] . \end{aligned}$$

We need to find the asymptotic joint distribution of $(\tilde{\alpha }_{k},\tilde{\beta }_{k})^{\top }$. Note that

$$\begin{aligned} \frac{\partial \tilde{\alpha }_{k}}{\partial s_{k}}= & {} \frac{1}{2u_{k1}}\left\{ \frac{2}{u_{k1}} \left[ \left( s_{k}r_{k}^*\right) ^\frac{1}{2}-1\right] \right\} ^{-\frac{1}{2}} \left( r_{k}^*/s_{k}\right) ^\frac{1}{2},\\ \frac{\partial \tilde{\alpha }_{k}}{\partial r_{k}^*}= & {} \frac{1}{2u_{k1}}\left\{ \frac{2}{u_{k1}} \left[ \left( s_{k}r_{k}^*\right) ^\frac{1}{2}-1\right] \right\} ^{-\frac{1}{2}} \left( s_{k}/r_{k}^*\right) ^\frac{1}{2},\\ \frac{\partial \tilde{\beta }_{k}}{\partial s_{k}}= & {} \frac{1}{2}\left( s_{k}r_{k}^*\right) ^{-\frac{1}{2}},\\ \frac{\partial \tilde{\beta }_{k}}{\partial s_{k}}= & {} -\frac{1}{2}\left( s_{k}/r_{k}^*\right) ^{\frac{1}{2}}(r_{k}^*)^{-1}, \end{aligned}$$

and

$$\begin{aligned} a_{k}=\frac{\partial \tilde{\alpha }_{k}}{\partial s_{k}}\big |_{s_{k}=E\left[ S_{k}\right] , r_{k}^*=E\left[ r_{k}^*\right] }= & {} \frac{1}{2\alpha _{k}\beta u_{k1}},\\ b_{k}=\frac{\partial \tilde{\alpha }_{k}}{\partial r_{k}^*} \big |_{s_{k}=E\left[ S_{k}\right] , r_{k}^*=E\left[ r_{k}^*\right] }= & {} \frac{\beta _{k} }{2\alpha _{k} u_{k1}},\\ c_{k}=\frac{\partial \tilde{\beta }_{k}}{\partial s_{k}}\big |_{s_{k} =E\left[ S_{k}\right] , r_{k}^*=E\left[ r_{k}^*\right] }= & {} \frac{1}{2\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) },\\ d_{k}=\frac{\partial \tilde{\beta }_{k}}{\partial s_{k}}\big |_{s_{k} =E\left[ S_{k}\right] , r_{k}^*=E\left[ r_{k}^*\right] }= & {} -\frac{\beta ^2}{2\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) }. \end{aligned}$$

By using the Taylor series expansion, we readily have

$$\begin{aligned} \sqrt{n}\left( \begin{array}{c} \tilde{\alpha }_{k}-\alpha _{k}\\ \tilde{\beta }_{k}-\beta _{k} \end{array} \right) \sim N \left[ \left( \begin{array}{c} 0\\ 0 \end{array} \right) , {\varvec{\varSigma }}_{k} \right] , \quad k=1,2, \end{aligned}$$

where

$$\begin{aligned} {\varvec{\varSigma }}_{k} = \left( \begin{array}{cc} a_{k} &{} b_{k}\\ c_{k} &{} d_{k}\\ \end{array} \right) \left( \begin{array}{cc} \sigma _{k}^{11} &{} \sigma _{k}^{12}\\ \sigma _{k}^{21} &{} \sigma _{k}^{22}\\ \end{array} \right) \left( \begin{array}{cc} a_{k} &{} c_{k}\\ b_{k} &{} d_{k}\\ \end{array} \right) = \left( \begin{array}{cc} \left( \frac{u_{k2}-u_{k1}^2}{4u_{k1}^2}\right) \alpha _{k}^2 &{} 0\\ 0 &{} \frac{u_{k1}+\frac{u_{k2}}{4}\alpha _{k}^2}{\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^2}\alpha _{k}^2\beta _{k}^2\\ \end{array} \right) . \end{aligned}$$

Appendix 2: Asymptotic distribution of $\widetilde{\alpha }_{k}^{*}$

Note that

$$\begin{aligned} {\mathrm{E}}\left[ \overline{Y}_{k}\right] =\frac{1}{2{\big (\begin{array}{l}n\\ 2 \end{array}\big )}}\sum _{1\le {i}\ne {j}\le {n}}{\mathrm{E}}\left[ Y_{kij}\right] =\left( 1+ \frac{u_{k1}}{2}\alpha _{k}^2\right) ^2, \quad k=1,2, \end{aligned}$$

and

$$\begin{aligned} {\mathrm{E}}\left[ \,\overline{Y}_{k}^{2}\,\right]= & {} \frac{1}{n^{2}(n-1)^{2}}{\mathrm{E}}\left[ \sum _{1\le {i}\,\ne \,{j}\,\ne \,{h}\,\ne \,{l}\,\le \,{n}}\frac{{T}_{ki}T_{kj}}{T_{kh}T_{kl}} +\sum _{1\,\le \,{i}\,\ne \,{j}\,\ne \,{h}\,\le \,{n}}\frac{T_{ki}^{2}}{T_{kj}T_{kh}}\right. \\&+\sum _{1\,\le \,{i}\,\ne \,{j}\,\ne \,{h}\,\le \,{n}}\frac{T_{kj}T_{kh}}{T_{ki}^{2}}\\&\left. +2\sum _{1\,\le \,{i}\,\ne \,{j}\,\ne \,{h}\,\le \,{n}}\frac{T_{ki}T_{kj}}{T_{ki}T_{kh}} + \sum _{1\,\le \,{i}\,\ne \,{j}\,\le \,{n}}\frac{T_{ki}^{2}}{T_{kj}^{2}}+ \sum _{1\,\le \,{i}\,\ne \,{j}\,\le \,{n}}\frac{T_{ki}T_{kj}}{t_{kj}T_{ki}} \right] \\= & {} \frac{(n-2)(n-3)}{n(n-1)} \left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^4\\&+ \frac{2(n-2)}{n(n-1)}\left\{ \frac{\alpha _{k}^2}{4}\left( 1+\frac{u_{k1}}{2} \alpha _{k}^2\right) ^2\varTheta _{k} +\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^4 \right\} \\&+\frac{2(n-2)}{n(n-1)}\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^2\\&+\frac{1}{n(n-1)}\left\{ \frac{\alpha ^{4}}{16}\varTheta _{k}^{2}+\frac{\alpha ^{2}}{2} \varTheta _{k}\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^2 +\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^4 \right\} \\&+\frac{1}{n(n-1)}, \end{aligned}$$

where $\varTheta _{k}=4u_{k1}+(2u_{k2}-u_{k1}^{2})\alpha _{k}^{2}$. From these results, we have

$$\begin{aligned} \text {Var}\left[ \overline{Y}_{k}\right]= & {} {\mathrm{E}}\left[ \overline{Y}_{k}^{2}\right] -\left( 1+\frac{u_{k1}}{2}\alpha _{k}^2\right) ^4. \end{aligned}$$

To obtain the distribution of $\widetilde{\alpha }^{*}_{k}$, we use a Taylor series expansion such that

$$\begin{aligned} \widetilde{\alpha }_{k}^{*}= & {} \left\{ \frac{2}{u_{k1}}\left[ \sqrt{\overline{y}_{k}} -1\right] \right\} ^{\frac{1}{2}} = g\left( \overline{y}_{k}\right) =g(\xi _{k}) +\left( \overline{y}_{k}-\xi _{k}\right) g'(\xi _{k}) \\&+\frac{\left( \overline{y}_{k}-\xi _{k}\right) ^2}{2}g''(\xi )+\cdots , \end{aligned}$$

where $g'(\cdot )$ and $g''(\cdot )$ denote the first and second derivatives of the function of $g(\cdot )$ and $\xi _{k}=\left( 1+ \frac{u_{k1}}{2}\alpha _{k}^2\right) ^2$. We thus obtain the asymptotic distribution of $\widetilde{\alpha }_{k}^{*}$ as

$$\begin{aligned} \sqrt{n}\left( \widetilde{\alpha }^{*}_{k}-{\alpha }_{k}\right) \xrightarrow [n\rightarrow \infty ]{}\text {N}\left( 0, {\alpha _{k}^2}\left[ \frac{u_{k2}-u_{k1}^2}{4u_{k1}^2}\right] \right) . \end{aligned}$$