Inference for the bivariate Birnbaum–Saunders lifetime regression model and associated inference

Abstract

In this paper, we discuss a regression model based on the bivariate Birnbaum–Saunders distribution. We derive the maximum likelihood estimates of the model parameters and then develop associated inference. Next, we briefly describe likelihood-ratio tests for some hypotheses of interest as well as some interval estimation methods. Monte Carlo simulations are then carried out to examine the performance of the estimators as well as the interval estimation methods. Finally, a numerical data analysis is performed for illustrating all the inferential methods developed here.

Appendix: Derivation of the Fisher information matrix

Let us define

$$\begin{aligned} z_{ji}&= \frac{1}{\alpha _j}\left( \sqrt{\frac{t_{ji}}{\theta _{ji}}}-\sqrt{\frac{\theta _{ji}}{t_{ji}}}\right) ,\\ z'_{jki}&= \frac{\partial z_{ji}}{\partial \beta _{jk}}= -\frac{1}{2\alpha _j}\left( \sqrt{\frac{t_{ji}}{\theta _{ji}}}+\sqrt{\frac{\theta _{ji}}{t_{ji}}}\right) x^{(j)}_{ki},\\ z''_{jkli}&= \frac{\partial ^2 z_{ji}}{\partial \beta _{jk}\partial \beta _{jl}}= \frac{1}{4\alpha _j}\left( \sqrt{\frac{t_{ji}}{\theta _{ji}}}-\sqrt{\frac{\theta _{ji}}{t_{ji}}}\right) x^{(j)}_{ki}x^{(j)}_{li},\\ \left( z^2_{jki}\right) '&= \frac{\partial (z^2_{ji})}{\partial \beta _{jk}}= -\frac{1}{\alpha ^2_j}\left( \frac{t_{ji}}{\theta _{ji}}-\frac{\theta _{ji}}{t_{ji}}\right) x^{(j)}_{ki},\\ \left( z^2_{jkli}\right) ''&= \frac{\partial ^2 (z^2_{ji})}{\partial \beta _{jk}\partial \beta _{jl}}= \frac{1}{\alpha ^2_j}\left( \frac{t_{ji}}{\theta _{ji}}+\frac{\theta _{ji}}{t_{ji}}\right) x^{(j)}_{ki}x^{(j)}_{li}, \end{aligned}$$

and $x^{(0)}_{0i}=x^{(1)}_{0i}=1$ for $i=1,\ldots ,n$, $j=1(2)$, $k=0,\ldots ,p(q)$, $l=0,\ldots ,p(q)$.

The first-order derivatives of the log-likelihood function with respect to the parameters, required for the Newton-Raphson method, are given by

$$\begin{aligned} \frac{\partial \ln L}{\partial \alpha _j}&= -\frac{n}{\alpha _j}+\frac{1}{(1-\rho ^2)\alpha _j} \sum _{i=1}^n z_{ji}^2 -\frac{\rho }{(1-\rho ^2)\alpha _j}\sum _{i=1}^n z_{ji}z_{j_1i}, \end{aligned}$$

(10.1)

$$\begin{aligned} \frac{\partial \ln L}{\partial \rho }&= \frac{n\rho }{1-\rho ^2} -\frac{\rho }{(1-\rho ^2)^2} \sum _{i=1}^n(z_{1i}^2+z_{2i}^2) +\frac{1+\rho ^2}{(1-\rho ^2)^2} \sum _{i=1}^n z_{1i}z_{2i}, \end{aligned}$$

(10.2)

$$\begin{aligned} \frac{\partial \ln L}{\partial \beta _{jk}}&= -\frac{1}{2}\sum _{i=1}^n\left( \frac{t_{ji}-\theta _{ji}}{t_{ji} +\theta _{ji}}\right) x_{ki}^{(j)} -\frac{1}{2(1-\rho ^2)} \sum _{i=1}^n(z^2_{jki})' +\frac{\rho }{1-\rho ^2} \sum _{i=1}^n z_{j_1i} z'_{jki},\nonumber \\ \end{aligned}$$

(10.3)

for $j\ne j_1\in \{1,2\}$. Next, the negative of all the second-order derivatives of the log-likelihood function with respect to the parameters are obtained from (10.1)–(10.3) as follows:

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \alpha _j^2}&= -\frac{n}{\alpha _j^2}+\frac{3}{(1-\rho ^2)\alpha _j^2} \sum _{i=1}^n z^2_{ji} -\frac{2\rho }{(1-\rho ^2)\alpha ^2_j}\sum _{i=1}^n z_{ji}z_{j_1i}, \end{aligned}$$

(10.4)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \rho ^2}&= -\frac{n(1+\rho ^2)}{(1-\rho ^2)^2} +\frac{1+3\rho ^2}{(1-\rho ^2)^3} \sum _{i=1}^n (z_{1i}^2+z_{2i}^2) -\frac{2\rho (3+\rho ^2)}{(1-\rho ^2)^3} \sum _{i=1}^n z_{1i}z_{2i},\nonumber \\\end{aligned}$$

(10.5)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \beta ^2_{jk}}&= -\sum _{i=1}^n\frac{\theta _{ji}t_{ji}}{(t_{ji}+\theta _{ji})^2}[x^{(j)}_{ki}]^2 +\frac{1}{2(1-\rho ^2)} \sum _{i=1}^n (z_{jkki}^2)''\nonumber \\&-\frac{\rho }{1-\rho ^2} \sum _{i=1}^n (z_{jkki})''z_{j_1i},\end{aligned}$$

(10.6)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \alpha _1\partial \alpha _2}&= -\frac{\rho }{(1-\rho ^2)\alpha _1\alpha _2}\sum _{i=1}^n z_{ji}z_{j_1i},\end{aligned}$$

(10.7)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \alpha _j\partial \rho }&= -\frac{2\rho }{(1-\rho ^2)^2\alpha _j} \sum _{i=1}^n z_{ji}^2 +\frac{1+\rho ^2}{(1-\rho ^2)^2\alpha _j}\sum _{i=1}^n z_{1i}z_{2i},\end{aligned}$$

(10.8)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \alpha _j\partial \beta _{jk}}&= \frac{\rho }{(1-\rho ^2)\alpha _j}\sum _{i=1}^n z'_{jki}z_{j_1i} -\frac{1}{(1-\rho ^2)\alpha _j} \sum _{i=1}^n[z_{jki}^2]', \end{aligned}$$

(10.9)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \alpha _j\partial \beta _{j_1k}}&= \frac{\rho }{(1-\rho ^2)\alpha _j}\sum _{i=1}^n z_{ji}z_{j_1ki}', \end{aligned}$$

(10.10)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \rho \partial \beta _{jk}}&= \frac{\rho }{(1-\rho ^2)^2} \sum _{i=1}^n [z^2_{jki}]' - \frac{1+\rho ^2}{(1-\rho ^2)^2} \sum _{i=1}^n z_{jki}'z_{j_1i}, \end{aligned}$$

(10.11)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \beta _{jk}\partial \beta _{jl}}&= -\sum _{i=1}^n\frac{\theta _{ji}t_{ji}}{(t_{ji}+\theta _{ji})^2}x^{(j)}_{ki}x^{(j)}_{li} +\frac{1}{2(1-\rho ^2)} \sum _{i=1}^n[z_{jkli}^2]'' \nonumber \\&-\frac{\rho }{1-\rho ^2}\sum _{i=1}^nz_{jkli}''z_{j_1i},\quad k\ne l=0,\ldots ,p(q), \end{aligned}$$

(10.12)

$$\begin{aligned} -\frac{\partial ^2\ln L}{\partial \beta _{1k_1}\partial \beta _{2k_2}}&= -\frac{\rho }{1-\rho ^2} \sum _{i=1}^n z_{1k_1i}'z_{2k_2i}', \qquad k_1=0,\ldots ,p,~ k_2=0,\ldots ,q.\nonumber \\ \end{aligned}$$

(10.13)

By using Property 3.2, we then obtain the expected values of the second-order derivatives of the log-likelihood with respect to the parameters, in (10.4)–(10.13), as follows:

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \alpha _j^2}\right]&= \frac{n(2-\rho ^2)}{\alpha _j^2(1-\rho ^2)},\end{aligned}$$

(10.14)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \rho ^2}\right]&= \frac{n(1+\rho ^2)}{(1-\rho ^2)^2}, \end{aligned}$$

(10.15)

$$\begin{aligned} \!-\!E\left[ \frac{\partial ^2\ln L}{\partial \beta ^2_{jk}}\right]&\!=\!&\frac{1}{1\!-\!\rho ^2}\sum _{i=1}^n\left( \frac{1}{\alpha _j^2}\!+\!\frac{1}{2}-\frac{\rho ^2}{4}\right) [x^{(j)}_{ki}]^2 -\sum _{i=1}^nx^2_{ki}E\left[ \frac{V_j}{(1+V_j)^2}\right] ,\nonumber \\ \end{aligned}$$

(10.16)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \alpha _1\partial \alpha _2}\right]&= -\frac{n\rho ^2}{(1-\rho ^2)\alpha _1\alpha _2}, \end{aligned}$$

(10.17)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \alpha _j\partial \rho }\right]&= -\frac{n\rho }{(1-\rho ^2)\alpha _j}, \end{aligned}$$

(10.18)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \alpha _j\partial \beta _{jk}}\right]&= 0,\end{aligned}$$

(10.19)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \alpha _j\partial \beta _{j_1k}}\right]&= 0,\end{aligned}$$

(10.20)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \rho \partial \beta _{jk}}\right]&= 0,\end{aligned}$$

(10.21)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \beta _{jk}\partial \beta _{jl}}\right]&= \frac{2+\alpha _1^2}{2(1-\rho ^2)\alpha _1^2}\sum _{i=1}^nx^{(j)}_{ki}x^{(j)}_{li} -\sum _{i=1}^nx^{(j)}_{ki}x^{(j)}_{li}E\left[ \frac{V_j}{(1+V_j)^2}\right] \nonumber \\&-\frac{\rho ^2}{4(1-\rho ^2)}\sum \limits _{i=1}^nx_{ki}^{(j)}x_{li}^{(j)}, \end{aligned}$$

(10.22)

$$\begin{aligned} -E\left[ \frac{\partial ^2\ln L}{\partial \beta _{1k_1}\partial \beta _{2k_2}}\right]&= -\frac{\rho I_{2}}{(1-\rho ^2)\alpha _1\alpha _2} \sum _{i=1}^nx^{(1)}_{k_1i}x^{(2)}_{k_2i}, \end{aligned}$$

(10.23)

where $V_j\sim BS(\alpha _j,1)$.

For computing some of the above expressions, we need the value of $E \big [\frac{V}{(1+V)^2}\big ]$, which cannot be obtained analytically. So, we may either approximate it by the Monte Carlo method or by directly using integration function in R, or utilize the binomial approximation to obtain

$$\begin{aligned} E\left[ \frac{V}{(1+V)^2}\right]&\approx E\left[ V(1-2V+3V^2)\right] \nonumber \\&= E\left[ V\right] -2E\left[ V^2\right] +3E\left[ V^3\right] \nonumber \\&= \frac{45}{2}\alpha ^6+24\alpha ^4+10\alpha ^2. \end{aligned}$$

(10.24)

Moreover, if we use $\mathbf{J_1}$ to denote the information matrix for $\alpha _1$, $\alpha _2$ and $\rho$, and use $\mathbf{J_2}$ to denote the information matrix for $\varvec{\beta }_1$ and $\varvec{\beta }_2$, then we observe from the expressions in (10.14)–(10.23) that

$$\begin{aligned} Var(\hat{\varvec{\eta }})= \left[ \begin{array}{cc} \mathbf{J}_1 &{}\quad \mathbf{0} \\ \mathbf{0} &{}\quad \mathbf{J}_2 \\ \end{array} \right] ^{-1} = \left[ \begin{array}{cc} \mathbf{J}_1^{-1} &{}\quad \mathbf{0} \\ \mathbf{0} &{}\quad \mathbf{J}_2^{-1} \\ \end{array} \right] , \end{aligned}$$

(10.25)

where the last equality holds if the matrix $Var(\hat{\varvec{\eta }})$ is positive definite.