A robust Birnbaum–Saunders regression model based on asymmetric heavy-tailed distributions

Abstract

Skew-normal/independent distributions provide an attractive class of asymmetric heavy-tailed distributions to the usual symmetric normal distribution. We use this class of distributions here to derive a robust generalization of sinh-normal distributions (Rieck in Statistical analysis for the Birnbaum–Saunders fatigue life distribution, 1989), we then propose robust nonlinear regression models, generalizing the Birnbaum–Saunders regression models proposed by Rieck and Nedelman (Technometrics 33:51–60, 1991) that have been studied extensively. The proposed regression models have a nice hierarchical representation that facilitates easy implementation of an EM algorithm for the maximum likelihood estimation of model parameters and provide a robust alternative to estimation of parameters. Simulation studies as well as applications to a real dataset are presented to illustrate the usefulness of the proposed model as well as all the inferential methods developed here.

Appendix A: Special cases of sinh-SNI models

• i) The sinh-SCN distribution

  This distribution is based on the SCN model, which is called sinh-skew-contaminated normal (sinh-SCN) distribution. Here, U is a discrete random variable with pdf $h(u;\nu ,\gamma )=\nu {\mathbb {I}}_{\{\gamma \}}(u)+(1-\nu ) {\mathbb {I}}_{\{1\}}(u)$, with $0< \nu< 1, 0< \gamma < 1$, where ${\mathbb {I}}_A(\cdot )$ denotes the indicator function of the set A. Then, the pdf of Y is

  $$\begin{aligned} f_Y(y)= & {} \frac{2}{\sigma } \left[ \nu \phi \big (\xi _{2y};0,\frac{1}{\gamma }\big ) \Phi \big (\gamma ^{1/2}\,\lambda \xi _{2y}\big ) +(1-\nu )\phi \big (\xi _{2y}\big )\Phi \big (\lambda \xi _{2y}\big )\right] \xi _{1y}. \end{aligned}$$

  This distribution is denoted by $Y\sim \text{ sinh-SCN }(\alpha ,\mu ,\sigma ,\lambda ;\nu ,\gamma )$. In this case,

  $$\begin{aligned} \kappa _r= & {} \frac{2\,\xi _{1y}}{ \sigma \,f_Y(y)}\left[ \nu \gamma ^r\phi \left( \xi _{2y};0,\gamma ^{-1}\right) \Phi \left( \gamma ^{1/2}\,\lambda \xi _{2y}\right) +(1-\nu )\phi \left( \xi _{2y}\right) \Phi \left( \lambda \xi _{2y}\right) \right] ,\\ \tau _r= & {} \frac{2\,\xi _{1y}}{\sigma \,f_Y(y)}\left[ \nu \gamma ^{r/2}\phi \left( \xi _{2y};0,\gamma ^{-1}\right) \Phi \left( \gamma ^{1/2}\,\lambda \xi _{2y}\right) +(1-\nu )\phi \left( \xi _{2y}\right) \Phi \left( \lambda \xi _{2y}\right) \right] . \end{aligned}$$

  Finally, d(Y) has its cdf as $Pr(d(Y)\le \upsilon ) = \nu Pr(\chi ^2_1 \le \gamma \upsilon )+(1-\nu )Pr(\chi _1^2 \le \upsilon )$;

• ii) The sinh-SSL distribution

  In this case, $U \sim \mathrm{Beta}(\nu ,1)$ distribution, and the resulting distribution is called the sinh-skew-slash (sinh-SSL) distribution, denoted by $Y\sim \text{ sinh-SSL }(\alpha ,\mu ,\sigma ,\lambda ;\nu )$, and its pdf is given by

  $$\begin{aligned} f_Y(y)= \frac{2\,\nu }{\sigma }\int ^1_0u^{\nu -1}\phi (\xi _{2y};0,{1}/{u})\Phi (u^{1/2}\lambda \xi _{2y})du\, \xi _{1y}. \end{aligned}$$

  (19)

  In this case,

  $$\begin{aligned} \kappa _r= & {} \frac{2^{\nu +r+1}\nu \,\Gamma \left( \frac{2\nu +2r+1}{2}\right) \,\xi _{1y}}{f_Y(y)\sqrt{\pi }\,\sigma } P_1\Big (\frac{2\nu +2r+1}{2},\frac{\xi ^2_{2y}}{2}\Big ) \xi _{2y}^{-(2\nu +2r+1)}\\&\mathrm{E} \left[ \Phi \left( S^{1/2} \lambda \xi _{2y}\right) \right] ,\\ \tau _r= & {} \frac{2^{\nu +r/2+1/2}\,\nu \,\Gamma \left( \frac{2\nu +r+1}{2}\right) \,\xi _{1y} }{f_Y(y)\sqrt{\pi }^2\,\sigma } \left( \xi ^2_{2y}+\lambda ^2 \xi ^2_{2y}\right) ^{-\frac{2\nu +r+1}{2}} \\&P_1 \left( \frac{2\nu +r+1}{2},\frac{\xi ^2_{2y}+\lambda ^2 \xi ^2_{2y}}{2}\right) , \end{aligned}$$

  where $P_x \left( a,b\right)$ denotes the cdf of the $\mathrm{Gamma}(a,b)$ distribution evaluated at x and $S \sim \mathrm{Gamma} \left( \frac{2\nu +2r+1}{2},\frac{\xi ^2_{2y}}{2}\right) {\mathbb {I}}_{(0,1)}$, a truncated gamma distribution on (0, 1). Finally, the cdf of d(Y) is $Pr(d(Y)\le \upsilon ) = Pr(\chi ^2_1 \le \upsilon )-\displaystyle \frac{2^{\nu }\Gamma (\nu +1/2)}{\upsilon ^{\nu }\Gamma (1/2)} Pr(\chi _{2\nu +1}^2\le \upsilon )$;

• iii) The sinh-ST distribution

  This distribution is obtained when $U \sim \mathrm{Gamma}(\nu /2,\nu /2)$, and the resulting distribution is called the sinh-skew-Student-t (sinh-ST) distribution. This distribution is denoted by $Y\sim \text{ sinh-ST }(\alpha ,\mu ,\sigma ,\lambda ;\nu )$, and its pdf is given by

  $$\begin{aligned} f_Y(y)= \frac{2}{\sigma } \, t\big (\xi _{2y};\nu \big )\mathrm{T}\left( \sqrt{\frac{\nu +1}{\xi ^2_{2y}+\nu }}\,\, \lambda \xi _{2y};\nu +1\right) \xi _{1y}, \end{aligned}$$

  (20)

  where $\mathrm{t}(.;\nu )$ and $\mathrm{T}(.; \nu )$ denote, respectively, the pdf and cdf of the standard Student-t distribution. In this case, the conditional expectations are given by

  $$\begin{aligned} \kappa _r=\frac{2^{r+1}\,\nu ^{\nu /2} \Gamma \big (\frac{\nu +2r+1}{2}\big ) \big ( \xi ^2_{2y} + \nu \big )^{-\frac{\nu +2r+1}{2}}\,\xi _{1y}}{f_Y(y) \Gamma (\nu /2) \sqrt{\pi }\,\sigma } \mathrm{T} \left( \sqrt{\frac{\nu +2r+1}{\xi ^2_{2y}+\nu }}\, \lambda \xi _{2y};\nu +2r+1\right) \end{aligned}$$

  and

  $$\begin{aligned} \tau _r=\frac{2^{(r+1)/2}\,\nu ^{\nu /2}\,\Gamma \big (\frac{\nu +r+1}{2}\big ) \,\big ( \xi ^2_{2y} + \nu + \lambda ^2 \xi ^2_{2y} \big )^{-\frac{\nu +r+1}{2}}\,\xi _{1y}}{f_Y(y) \Gamma \left( \nu /2\right) \sqrt{(\pi )}^2\,\sigma }. \end{aligned}$$

  Finally, we have $d(Y)\sim \mathrm{F}(1,\nu )$.