Generalized proportional reversed hazard rate distributions with application in medicine

Abstract

Proportional reversed hazard rate distribution family plays an important role in the reliability and lifetime data modelling. In this manuscript this family of distributions is generalized in order to enhance its modelling capability. Considering a special case, we introduce the extended generalized exponential distribution. Its mathematical properties are also studied. New model is applied in fitting levels of TSH hormone and remission times of bladder cancer patients. We believe these results may attract applied statisticians especially those who are in charge with life time data analysis.

Appendices

Appendix A. Proof of the Theorem 1

The conditional CDF of the random variable $X=Z|Y\le 1$ is

$$\begin{aligned} F_{X=Z|Y\le 1}(x)=\frac{ \int ^{x}_{0} \int ^{1}_{0} f_{Z,Y}(z,y) dydz }{\int ^{1}_{0}f_{Y}(y)dy}. \end{aligned}$$

Using the result obtained in Lemma 2 and by substitution $G(x;\varvec{\xi })=u,$ the marginal PDF of the random variable Y is obtained as

$$\begin{aligned} f_{Y}(y;\varvec{\alpha },\theta ,\varvec{\xi })=&\alpha _{1}\alpha _{2} \left\{ (1+\theta )\int _0^1 u^{\alpha _1-1}\,Q(u;\varvec{\xi })\,g\left( yQ(u;\varvec{\xi });\varvec{\xi }\right) \,{\left( G\left( yQ(u;\varvec{\xi })\right) \right) }^{\alpha _2-1}\,\rm{d}u\right. \\&-\left. 2\theta \int _0^1 u^{\alpha _1-1}\,Q(u;\varvec{\xi })\,g\left( yQ(u;\varvec{\xi });\varvec{\xi }\right) \,{\left( G\left( yQ(u;\varvec{\xi });\varvec{\xi }\right) \right) }^{2\alpha _2-1}\,\rm{d}u\right. \\&-\left. 2\theta \int _0^1 u^{2\alpha _1-1}\,Q(u;\varvec{\xi })\,g\left( yQ(u;\varvec{\xi });\varvec{\xi }\right) \,{\left( G\left( yQ(u;\varvec{\xi });\varvec{\xi }\right) \right) }^{\alpha _2-1}\,\rm{d}u \right. \\&+\left. 4\theta \int _0^1 u^{2\alpha _1-1}\,Q(u;\varvec{\xi })\,g\left( yQ(u;\varvec{\xi });\varvec{\xi }\right) \,{\left( G\left( yQ(u;\varvec{\xi });\varvec{\xi }\right) \right) }^{2\alpha _2-1}\,\rm{d}u \right\} \,, \end{aligned}$$

where $Q(u;\varvec{\xi })=G^{-1}(u;\varvec{\xi })\,.$

Now we need to calculate $F_Y(1)=P\left\{ Y<1\right\} .$ So,

$$\begin{aligned}&\int ^{1}_{0}f_{Y}(y;\varvec{\alpha },\theta ,\varvec{\xi })dy=\alpha _1\,\alpha _2\,\left\{ \frac{1+\theta }{\alpha _2} \int _0^1\,u^{\alpha _1-1}\,{\left( G(\alpha Q(u;\varvec{\xi }));\varvec{\xi }\right) }^{\alpha _2}\rm{d}u\right. \\&\qquad \left. -\frac{\theta }{\alpha _2}\int _0^1\,u^{\alpha _1-1}\,\left[ G(\alpha Q(u;\varvec{\xi });\varvec{\xi })\right] ^{2\alpha _2}\rm{d}u-\frac{2\theta }{\alpha _2}\int _0^1\,u^{2\alpha _1-1}\,{\left( G(\alpha Q(u;\varvec{\xi });\varvec{\xi })\right) }^{\alpha _2}\rm{d}u\right. \\&\qquad \left. +\frac{2\theta }{\alpha _2}\int _0^1\,u^{2\alpha _1-1}\,{\left( G(\alpha Q(u;\varvec{\xi });\varvec{\xi })\right) }^{2\alpha _2}\rm{d}u\right\} \,. \end{aligned}$$

Using $G(Q(u;\varvec{\xi });\varvec{\xi })=u$, we easily find

$$\begin{aligned} F_Y(1)&=\frac{\alpha _1}{\alpha _1+\alpha _2}+\theta \,\frac{\alpha _1\,\alpha _2\,(\alpha _1-\alpha _2)}{(\alpha _1+\alpha _2)(2\alpha _1+\alpha _2)(\alpha _1+2\alpha _2)}\,. \end{aligned}$$

(14)

Using substitution $G(yz;\varvec{\xi })=u$, one can obtain

$$\begin{aligned} \int ^{1}_{0}f_{Z,Y}(z,y)dy=&\alpha _1\left\{ (1+\theta )\,g(z;\varvec{\xi })\, {\left( G(z;\varvec{\xi })\right) }^{\alpha _1+\alpha _2-1}\,-\theta \, g(z;\varvec{\xi })\,{\left( G(z;\varvec{\xi })\right) }^{\alpha _1+2\alpha _2-1}\right. \nonumber \\&\left. -2\theta g(z;\varvec{\xi }){\left( G(z;\varvec{\xi })\right) }^{2\alpha _1+\alpha _2-1}+ 2\theta {\left( G(z;\varvec{\xi })\right) }^{2\alpha _1+2\alpha _2-1}\right\} \,. \end{aligned}$$

Finally we have

$$\begin{aligned}&\int ^{x}_{0} \left[ \int ^{1}_{0}f_{Z,Y}(z,y)dy\right] dz\nonumber \\&\quad =\alpha _1\left\{ \frac{1+\theta }{\alpha _1+\alpha _2}\,{\left( G(x;\varvec{\xi })\right) }^{\alpha _1+\alpha _2} -\frac{\theta }{\alpha _1+2\alpha _2}{\left( G(x;\varvec{\xi })\right) }^{\alpha _1+2\alpha _2} \right. \nonumber \\&\qquad \left. -\frac{2\theta }{2\alpha _1+\alpha _2}{\left( G(x;\varvec{\xi })\right) }^{2\alpha _1+\alpha _2} +\frac{\theta }{\alpha _1+\alpha _2}{\left( G(x;\varvec{\xi })\right) }^{2\alpha _1+2\alpha _2}\right\} \,. \end{aligned}$$

(15)

The interchanging the order of integration is allowed by Tonelli’s theorem.

By combining (14)–(15), we obtain the CDF of $X=Z|Y\le 1.$ If we differentiate it with respect to x, one can verify that the conditional PDF $f_{X=Z|Y\le 1}(x; \varvec{\alpha }, \theta ,\varvec{\xi } )$ coincides with (5).

Appendix B. The likelihood equations

$$\begin{aligned} \dfrac{\partial l_k}{\partial \alpha _1} =&\dfrac{1}{\alpha _1}+\dfrac{1}{a_1} \cdot \dfrac{\partial a_1 }{\partial \alpha _1} -\dfrac{1}{a_2 + \theta a_3} \left[ \dfrac{\partial a_2 }{\partial \alpha _1} + \theta \,\, \dfrac{\partial a_3 }{\partial \alpha _1} \right] \\&+\log \left( G(x_{k};\varvec{\xi }) \right) + \dfrac{\theta }{1+\theta c} \cdot \dfrac{\partial c}{\partial \alpha _1}\\ \dfrac{\partial l_k}{\partial \alpha _2} =&\dfrac{\partial a_1 }{\partial \alpha _2} -\dfrac{1}{a_2 + \theta a_3}\left[ \dfrac{\partial a_2 }{\partial \alpha _2} + \theta \dfrac{\partial a_3 }{\partial \alpha _2} \right] \\&+\log \left( G(x_{k};\varvec{\xi }) \right) + \dfrac{\theta }{1+\theta c}\cdot \dfrac{\partial c}{\partial \alpha _2}\\ \dfrac{\partial l_k}{\partial \theta } =&-\dfrac{a_3}{a_2 + \theta a_3} + \dfrac{c}{1+\theta c}\\ \dfrac{\partial l_k}{\partial \xi _j} =&\dfrac{1}{ g(x_{k};\varvec{\xi }) } \cdot \dfrac{\partial g(x_{k};\varvec{\xi })}{\partial \xi _j} + \dfrac{\alpha _1 + \alpha _2 -1}{ G(x_{k};\varvec{\xi }) } \cdot \dfrac{\partial G(x_{k};\varvec{\xi })}{\partial \xi _j} + \dfrac{\theta }{1+\theta c} \cdot \dfrac{\partial c}{\partial \xi _j} \end{aligned}$$

for $j=1,\dots ,q$, where

$$\begin{aligned} \dfrac{\partial a_1}{\partial \alpha _1} =\,&a_1 \left\{ \dfrac{2}{\alpha _1 + \alpha _2} + \dfrac{2}{2\alpha _1 + \alpha _2} + \dfrac{1}{\alpha _1 + 2\alpha _2} \right\} \,,\\ \dfrac{\partial a_1}{\partial \alpha _2} =\,&a_1 \left\{ \dfrac{2}{\alpha _1 + \alpha _2} + \dfrac{1}{2\alpha _1 + \alpha _2} + \dfrac{ 2 }{\alpha _1 + 2\alpha _2} \right\} \,,\\ \dfrac{\partial a_2}{\partial \alpha _1} =\,&a_2 \left\{ \dfrac{1}{\alpha _1} + \dfrac{ 1}{\alpha _1 + \alpha _2} + \dfrac{2}{2\alpha _1 + \alpha _2} + \dfrac{ 1}{\alpha _1 + 2\alpha _2} \right\} \,,\\ \dfrac{\partial a_2}{\partial \alpha _2} =\,&a_2 \left\{ \dfrac{1}{\alpha _1 + \alpha _2} + \dfrac{ 1}{2\alpha _1 + \alpha _2} + \dfrac{2 }{\alpha _1 + 2\alpha _2} \right\} \,,\\ \dfrac{\partial a_3}{\partial \alpha _1} =\,&\dfrac{a_3}{\alpha _1 } + 2 \alpha _{1}^{2} \alpha _{2}\,,\\ \dfrac{\partial a_3}{\partial \alpha _2} =\,&\dfrac{a_3}{\alpha _2 } - 2 \alpha _{1} \alpha _{2}^{2} \,,\\ \dfrac{\partial c}{\partial \alpha _1} =&- 2 \left( G(x_{k};\varvec{\xi }) \right) ^{\alpha _1} \log \left( G(x_{k};\varvec{\xi }) \right) \left[ 1- \left( G(x_{k};\varvec{\xi }) \right) ^{\alpha _2} \right] \,,\\ \dfrac{\partial c}{\partial \alpha _2} =\,&- \left( G(x_{k};\varvec{\xi }) \right) ^{\alpha _2} \log \left( G(x_{k};\varvec{\xi }) \right) \left[ 1- 2\left( G(x_{k};\varvec{\xi }) \right) ^{\alpha _1} \right] \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial c}{\partial \xi _j}&= - \dfrac{\partial G(x_{k};\varvec{\xi })}{\partial \xi _j} \left\{ 2\alpha _1 \left( G(x_{k};\varvec{\xi }) \right) ^{\alpha _1 -1} + \alpha _{2} \left( G(x_{k};\varvec{\xi }) \right) ^{\alpha _2 -1} \right. \\&\left. - 2(\alpha _1 + \alpha _2) \left( G(x_{k};\varvec{\xi }) \right) ^{\alpha _1 + \alpha _2 -1} \right\} \,. \end{aligned}$$