The inverted exponentiated Chen distribution with application to cancer data

Abstract

This article defines a new inverted exponentiated Chen (IEC) distribution, which has upside-down bathtub shape and increasing hazard function. A detailed analysis (along with a graphical analysis) of the density and hazard rate function is carried out. The maximum likelihood and Bayesian methods are discussed to estimate the parameters of the IEC distribution based on a right censored scheme. The Lindley approximation is used to obtain the Bayes estimator under squared error and LINEX loss functions. Observed Fisher information matrix and asymptotic confidence intervals of parameters are derived. Also, we conduct the numerical simulation to compare the accuracy of the various estimators through the mean square error (MSE) values. Finally, four cancer datasets are analyzed to illustrate the applicability and flexibility of the IEC distribution.

Appendices

Appendix A: Elements of the observed information matrix

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{\partial ^2 \ell (\alpha , \beta , \lambda \mid {\varvec{D}})}{\partial \alpha ^2}=-\frac{r}{\alpha ^2}, \\ \frac{\partial ^2 \ell (\alpha , \beta , \lambda \mid {\varvec{D}}) }{\partial \beta ^2}=-\frac{r}{\beta ^2}-\sum _{i=1}^n \delta _i \left[ t_i^{-2 \beta } \left( \log t_i\right) ^2 \left( t_i^{\beta } \left[ \lambda e^{t_i^{-\beta }}-1\right] +\lambda e^{t_i^{-\beta }}\right) \right] \\ \quad \quad \quad \quad \quad \quad \quad \quad -\sum _{i=1}^{n} \frac{\lambda \left( \alpha -\delta _i\right) t_i^{-2 \beta } \left( \log t_i\right) ^2 e^{t_i^{-\beta }+\lambda } \left[ t_i^{\beta } \left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) +e^{\lambda e^{t_i^{-\beta }}} \left( \lambda e^{t_i^{-\beta }}-1\right) +e^{\lambda }\right] }{\left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) ^2}, \\ \frac{\partial ^2 \ell (\alpha , \beta , \lambda \mid {\varvec{D}}) }{\partial \lambda ^2}=-\frac{r}{\lambda ^2}-\sum _{i=1}^n \frac{\left( \alpha -\delta _i\right) \left( e^{t_i^{-\beta }}-1\right) ^2 e^{\lambda e^{t_i^{-\beta }}+\lambda }}{\left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) ^2}, \\ \frac{\partial ^2 \ell (\alpha , \beta , \lambda \mid {\varvec{D}}) }{\partial \alpha \partial \beta }=-\sum _{i=1}^n \frac{\lambda t_i^{-\beta } \log t_i e^{\lambda \left( 1-e^{t_i^{-\beta }}\right) +t_i^{-\beta }}}{1-e^{\lambda \left( 1-e^{t_i^{-\beta }}\right) }}, \\ \frac{\partial ^2 \ell (\alpha , \beta , \lambda \mid {\varvec{D}}) }{\partial \alpha \partial \lambda }=\sum _{i=1}^n \frac{e^{\lambda } \left( e^{t_i^{-\beta }}-1\right) }{e^{\lambda e^{t_i^{-\beta }}}-e^{\lambda }}, \\ \frac{\partial ^2 \ell (\alpha , \beta , \lambda \mid {\varvec{D}}) }{\partial \beta \partial \lambda }=\sum _{i=1}^n \delta _i e^{t_i^{-\beta }} t_i^{-\beta } \log t_i, \\ \quad \quad \quad \quad \quad \quad \quad \quad +\sum _{i=1}^n \frac{\left( \alpha -\delta _i\right) t_i^{-\beta } \log t_i e^{t_i^{-\beta }+\lambda } \left[ e^{\lambda e^{t_i^{-\beta }}} \left( \lambda \left[ e^{t_i^{-\beta }}-1\right] -1\right) +e^{\lambda }\right] }{\left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) ^2}. \end{array} \right. \end{aligned}$$

Appendix B: Elements of Eq. (20)

$$\begin{aligned}{} & {} \left\{ \begin{array}{lll} \theta _{1}=\frac{\partial \theta }{\partial \alpha },\quad \theta _{2}=\frac{\partial \theta }{\partial \beta },\quad \theta _{3}=\frac{\partial \theta }{\partial \lambda },\\ \theta _{11}=\frac{\partial ^2 \theta }{\partial \alpha ^2},\quad \theta _{12}=\frac{\partial ^2 \theta }{\partial \alpha \partial \beta },\quad \theta _{13}=\frac{\partial ^2 \theta }{\partial \alpha \partial \lambda },\\ \theta _{33}=\frac{\partial ^2 \theta }{\partial \lambda ^2},\quad \theta _{22}=\frac{\partial ^2 \theta }{\partial \beta ^2}, \quad \theta _{23}=\frac{\partial ^2 \theta }{\partial \beta \partial \lambda }, \end{array} \right. \\{} & {} \left\{ \begin{array}{lll} \rho =\log \pi (\alpha , \beta , \lambda ),\\ \rho _{1}=\frac{\partial \rho }{\partial \alpha }=\frac{a_1-1}{\alpha }-b_1,\\ \rho _{2}=\frac{\partial \rho }{\partial \beta }=\frac{a_2-1}{\beta }-b_2,\\ \rho _{3}=\frac{\partial \rho }{\partial \lambda }=\frac{a_3-1}{\lambda }-b_3, \end{array} \right. \\{} & {} \left\{ \begin{array}{lll} \ell _{300}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D})}{\partial \alpha ^3}=\frac{2 r}{\alpha ^3},\\ \ell _{120}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D})}{\partial \alpha \partial \beta ^2}=-\sum _{i=1}^{n}\frac{\lambda t_i^{-2 \beta } \log ^2\left( t_i\right) e^{t_i^{-\beta }+\lambda } \left( t_i^{\beta } \left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) +e^{\lambda e^{t_i^{-\beta }}} \left( \lambda e^{t_i^{-\beta }}-1\right) +e^{\lambda }\right) }{\left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) {}^2},\\ \ell _{102}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D}) }{\partial \alpha \partial \lambda ^2}=-\sum _{i=1}^{n}\frac{\left( e^{t_i^{-\beta }}-1\right) ^2 e^{\lambda e^{t_i^{-\beta }}+\lambda }}{\left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) ^2},\\ \ell _{210}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D})}{\partial \alpha ^2 \partial \beta }=0,\\ \ell _{201}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D})}{\partial \alpha ^2 \partial \lambda }=0,\\ \ell _{012}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D}) }{\partial \beta \partial \lambda ^2}=-\sum _{i=1}^{n} \frac{\left( \alpha -\delta _i\right) \left( e^{t_i^{-\beta }}-1\right) t_i^{-\beta } \log \left( t_i\right) e^{\lambda e^{t_i^{-\beta }}+t_i^{-\beta }+\lambda }}{\left( e^{\lambda e^{t_i^{-\beta }}}-e^{\lambda }\right) {}^3}\\ \quad \quad \quad \ \ \times e^{\lambda e^{t_i^{-\beta }}} \left( \lambda \left( e^{t_i^{-\beta }}-1\right) -2\right) +e^{\lambda } \left( \lambda \left( e^{t_i^{-\beta }}-1\right) +2\right) , \end{array} \right. \\{} & {} \left\{ \begin{array}{lll} \ell _{021}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D}) }{\partial \beta ^2 \partial \lambda }=\sum _{i=1}^n \delta _i \left( -e^{t_i^{-\beta }} t_i^{-2 \beta } \left( t_i^{\beta }+1\right) \log ^2\left( t_i\right) \right) \\ \quad \quad \quad \quad -\sum _{i=1}^{n}\frac{(\alpha -\delta _i)t_i^{-2 \beta }\left( \log t_i\right) ^2 e^{t_i^{-\beta }+\lambda }}{\left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) ^3}\\ \quad \quad \quad \quad \times \Bigg [e^{2 \lambda e^{t_i^{-\beta }}} \left( \lambda e^{t_i^{-\beta }} \left( \lambda \left( e^{t_i^{-\beta }}-1\right) -3\right) +\lambda +1\right) +e^{2 \lambda }\\ \quad \quad \quad \quad +e^{\lambda e^{t_i^{-\beta }}+\lambda } \left( \lambda \left( e^{t_i^{-\beta }} \left( \lambda \left( e^{t_i^{-\beta }}-1\right) +3\right) -1\right) -2\right) \\ \quad \quad \quad \quad +t_i^{\beta } \left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) \left( e^{\lambda e^{t_i^{-\beta }}} \left( \lambda \left( e^{t_i^{-\beta }}-1\right) -1\right) +e^{\lambda }\right) \Bigg ],\\ \ell _{111}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D}) }{\partial \alpha \partial \beta \partial \lambda }=\sum _{i=1}^{n}\frac{t_i^{-\beta } \log \left( t_i\right) e^{t_i^{-\beta }+\lambda } \left( e^{\lambda e^{t_i^{-\beta }}} \left( \lambda \left( e^{t_i^{-\beta }}-1\right) -1\right) +e^{\lambda }\right) }{\left( e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right) ^2}\\ \ell _{030}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D})}{ \partial \beta ^{3}}=\sum _{i=1}^n \delta _i \left[ t_i^{-3 \beta } \left( \log t_i\right) ^3 \left( 3 \lambda e^{t_i^{-\beta }} t_i^{\beta }+t_i^{2 \beta } \left[ \lambda e^{t_i^{-\beta }}-1\right] +\lambda e^{t_i^{-\beta }}\right) \right] \\ \quad \quad \quad \quad +\sum _{i=1}^{n} \frac{\lambda \left( \alpha -\delta _i\right) (\log t_i)^3 e^{t_i^{-\beta }+\lambda }}{t_i^{3 \beta } \big (e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\big )^3} \times \Bigg \{e^{2 \lambda }+e^{2 \lambda e^{t_i^{-\beta }}} \left[ \lambda e^{t_i^{-\beta }} \left( \lambda e^{t_i^{-\beta }}-3\right) +1\right] \\ \quad \quad \quad \quad +e^{\lambda e^{t_i^{-\beta }}+\lambda } \left[ \lambda e^{t_i^{-\beta }} \left( \lambda e^{t_i^{-\beta }}+3\right) -2\right] \\ \quad \quad \quad \quad +t_i^{\beta } \left[ e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right] \left( t_i^{\beta } \left[ e^{\lambda }-e^{\lambda e^{t_i^{-\beta }}}\right] +3 \left[ e^{\lambda e^{t_i^{-\beta }}} \left( \lambda e^{t_i^{-\beta }}-1\right) +e^{\lambda }\right] \right) \Bigg \}+\frac{2 r}{\beta ^3},\\ \ell _{003}=\frac{\partial ^3 \ell (\alpha , \beta , \lambda \mid \varvec{D})}{ \partial \lambda ^{3}}=\frac{2 r}{\lambda ^3}+\sum _{i=1}^n\frac{\left( \alpha -\delta _i\right) \big (e^{t_i^{-\beta }}-1\big )^3 \big (e^{\lambda e^{t_i^{-\beta }}+\lambda }\big ) \big (e^{\lambda e^{t_i^{-\beta }}}+e^{\lambda }\big )}{\big (e^{\lambda e^{t_i^{-\beta }}}-e^{\lambda }\big )^3}. \end{array} \right. \end{aligned}$$