On the Extended Birnbaum–Saunders Distribution Based on the Skew-t-Normal Distribution

Abstract

In this article, a generalized version of the univariate Birnbaum–Saunders distribution based on the skew-t-normal distribution is introduced and its characterizations, properties are studied. Maximum likelihood estimation of the parameters via the ECM algorithm evaluated by Monte Carlo simulations is also discussed. Finally, two real datasets are analyzed for illustrative purposes.

Appendix

This appendix presents the proof of Propositions 2.2 and 2.3.

Proof of Proposition 2.2

By using Eq.(5) the cdf of $T \mid \left( \gamma ,\tau \right)$ is

$$\begin{aligned} F(T\le t\mid \gamma ,\tau )=P(Y\le a(t; \alpha ,\beta )\mid \gamma ,\tau )=P\left( Z_{2}\le \sqrt{\tau +\lambda ^{2}}a(t; \alpha ,\beta )-\frac{\lambda \gamma }{\sqrt{\tau +\lambda ^{2}}}\mid \gamma, \tau \right), \end{aligned}$$

then the pdf of T$\mid \gamma ,\tau$ is

$$\begin{aligned} f(t\mid \gamma ,\tau )=\phi \left( \sqrt{\tau +\lambda ^{2}} a(t;\alpha ,\beta )-\frac{\lambda \gamma }{\sqrt{\tau +\lambda ^{2}}}\right) \sqrt{\tau +\lambda ^{2}}A(t;\alpha ,\beta ). \end{aligned}$$

The conditional distribution of $\gamma \mid \tau$ can be easily obtained from the definitions. $\square$

Proof of Proposition 2.2

From Proposition 2.2, the joint pdf of $T,\gamma$ and $\tau$ is given by

$$\begin{aligned} f(t,\gamma ,\tau )=\frac{1}{\pi }\exp \left\{ -\frac{1}{2}\left( \left( \gamma -\frac{\lambda }{\alpha }\varepsilon (t;\beta )\right) ^{2}+\frac{\tau }{\alpha ^{2} }\varepsilon ^{2}(t;\beta )+\upsilon \tau \right) \right\} A(t;\alpha ,\beta )\frac{\left( \frac{\upsilon }{2} \right) ^{\frac{\upsilon }{2}}}{\Gamma \left( \frac{ \upsilon }{2}\right) }\tau ^{\frac{\upsilon -1}{2}}. \end{aligned}$$

(6)

where $\varepsilon (t;\beta )=\sqrt{\frac{t}{\beta }}-\sqrt{\frac{\beta }{t}} .$

By integrating on $\gamma$ in (6), we get

$$\begin{aligned} f(t,\tau )=\sqrt{\frac{2}{\pi }}\frac{\left( \frac{\upsilon }{2}\right) ^{\frac{\upsilon }{2}}}{\Gamma \left( \frac{\upsilon }{2}\right) }\tau ^{\frac{\upsilon -1}{2}}A(t;\alpha ,\beta )\exp \left\{ \left( -\frac{\upsilon +\frac{\varepsilon ^{2}(t;\beta )}{\alpha ^{2}}}{2}\right) \tau \right\} \Phi \left( \lambda \frac{\varepsilon (t,\beta )}{\alpha }\right) . \end{aligned}$$

(7)

Dividing (7) by (3) gives

$$\begin{aligned} f(\tau \mid t)=\frac{\left( \frac{\upsilon +a^{2}(t;\alpha ,\beta )}{2}\right) ^{\frac{ \upsilon +1}{2}}}{\Gamma \left( \frac{\upsilon +1}{2}\right) }\tau ^{\frac{\upsilon +1}{2} -1}\exp \left( -\frac{\upsilon +a^{2}(t;\alpha ,\beta )}{2}\right) \tau , \end{aligned}$$

so

$$\begin{aligned} \tau \mid \left( T=t\right) \sim \Gamma \left( \frac{\upsilon +1}{ 2},\frac{\upsilon +a^{2}(t;\alpha ,\beta )}{2}\right) , \end{aligned}$$

which concludes parts (a) and (b),  and dividing (6) by (7), gives

$$\begin{aligned} f(\gamma \mid t,\tau )=\frac{1}{\Phi \left( \lambda \frac{\varepsilon (t; \beta )}{ \alpha }\right) }\frac{1}{\sqrt{2\pi }}\exp \left\{ -\frac{1}{2}\left( \gamma - \frac{\lambda }{\alpha }\varepsilon (t; \beta )\right) ^{2}\right\} I(0,+\infty )=f(\gamma \mid t), \end{aligned}$$

so $\gamma$ and $\tau$ are conditionally independent given $T=t$ and the conditional distribution of $\gamma$ given t is

$$\begin{aligned} \gamma \mid \left( T=t\right) \sim TN(\lambda a(t; \alpha , \beta ), 1, (0, \infty )), \end{aligned}$$

which concludes the part (c). $\square$