A finite mixture model for multiple dependent competing risks with applications of automotive warranty claims data

Abstract

This paper introduces a parametric finite mixture model (FMM) approach to analyze the dependent competing risks data subjected to progressive first-failure censoring and multiple causes of failure. The cause-specific failure times are assumed to be flexibly modeled by the Lehmann family of distributions (also known as the exponentiated distributions) with variation in both distribution parameters. Application of the expectation maximization (EM) algorithm facilitates the maximum likelihood estimation of the model parameters and illuminates the contribution of the censored data. For interval estimation purposes, we resort to using the asymptotic confidence intervals based on the observed Fisher information matrix. Practitioners often prefer employing simpler lifetime distribution in order to facilitate the data modeling process while knowing the true distribution. In this context, the effects of model misspecification are studied based on the p-th quantile when the true distribution is misspecified. An extensive simulation study is performed to validate our proposed model. Finally, an automotive warranty claims data set is used as an illustration to study the effectiveness of our proposed model, assuming some important members of the Lehmann family, like generalized exponential and exponentiated Pareto distributions.

Appendix: Elements of Fisher Information Matrix

1.1 General Case

Suppose $\varvec{\theta }=(\theta _1,\theta _2,\ldots ,\theta _{3S-1})=(\phi _1,\phi _2,\ldots ,\phi _{S-1},\alpha _{1},\alpha _{2},\ldots ,\alpha _{S},\lambda _{1},\lambda _{2},\ldots ,\lambda _{S}),$ and for $i,j=1,2,\ldots ,3S-1,$ $I(\varvec{\theta }) = \Bigg (\Bigg (\dfrac{\partial ^2\,l(\varvec{\theta }|{\varvec{t,\,\delta }})}{\partial \theta _i \partial \theta _j }\Bigg )\Bigg )$ be the observed Fisher information matrix. Note that here $l=l(\varvec{\theta }|\varvec{t,\,\delta })$ is the log-likelihood function as defined in (2.5). For $j,k=1,2,\ldots ,S-1,\,j\ne k,$ and $x,\,y=1,2,\ldots ,S,\,x \ne y,$ the elements of the observed Fisher information matrix can then be expressed as follows:

$$\begin{aligned}{} & {} \frac{\partial ^2 l}{\partial \phi _x^2} = -\frac{m_x}{\phi _x^2}-\frac{m_{S}}{(1-\displaystyle \sum _{i=1}^{S-1}\phi _i)^2} \\{} & {} \qquad -\displaystyle \sum _{i=1}^{m}I_x(i)\dfrac{\{k(R_i+1)-1\}\{1-[G_0(t_i;\lambda _x)]^{\alpha _x}\} ^2}{[\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}]^2},\\{} & {} \frac{\partial ^2 l}{\partial \phi _x \partial \phi _y}=\frac{\partial ^2 l}{\partial \phi _y \partial \phi _x} =-\frac{m_{S}}{(1-\displaystyle \sum _{i=1}^{S-1}\phi _i)^2} -\displaystyle \sum _{i=1}^{m}I_x(i)\\{} & {} \quad \dfrac{\{k(R_i+1)-1\}\{1-[G_0(t_i;\lambda _x)]^{\alpha _x}\} \{1-[G_0(t_i;\lambda _y)]^{\alpha _y}\}}{[\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}]^2},\\{} & {} \frac{\partial ^2 l}{\partial \phi _x \partial \alpha _x} = \frac{\partial ^2 l}{\partial \alpha _x \partial \phi _x } = \sum _{i=1}^{m} I_x(i)(K(R_i+1)-1) \frac{\partial l}{\partial \alpha _x} \\{} & {} \bigg [ \frac{1-[G(t_i; \lambda _x)]^{\alpha _x}}{\sum _{j=1}^{S}\phi _i(1-[G(t_i; \lambda _j)]^{\alpha _j})}\bigg ] \\{} & {} \frac{\partial ^2 l}{\partial \phi _y \partial \alpha _x} = \frac{\partial ^2 l}{\partial \alpha _x \partial \phi _y } = \sum _{i=1}^{m} I_y(i)(K(R_i+1)-1)\frac{\partial l}{\partial \alpha _x}\\{} & {} \bigg [ \frac{1-[G(t_i; \lambda _y)]^{\alpha _y}}{\sum _{j=1}^{S}\phi _i(1-[G(t_i; \lambda _j)]^{\alpha _j})}\bigg ]\\{} & {} \frac{\partial ^2 l}{\partial \phi _x \partial \lambda _x} = \frac{\partial ^2 l}{\partial \lambda _x \partial \phi _x } = \sum _{i=1}^{m} I_x(i)(K(R_i+1)-1) \frac{\partial l}{\partial \lambda _x} \\{} & {} \bigg [ \frac{1-[G(t_i; \lambda _x)]^{\alpha _x}}{\sum _{j=1}^{S}\phi _i(1-[G(t_i; \lambda _j)]^{\alpha _j})}\bigg ]\\{} & {} \frac{\partial ^2 l}{\partial \phi _y \partial \lambda _x} = \frac{\partial ^2 l}{\partial \lambda _x \partial \phi _y } = \sum _{i=1}^{m} I_y(i)(K(R_i+1)-1) \frac{\partial l}{\partial \lambda _x} \\{} & {} \bigg [ \frac{1-[G(t_i; \lambda _y)]^{\alpha _y}}{\sum _{j=1}^{S} \phi _i(1-[G(t_i; \lambda _j)]^{\alpha _j})}\bigg ]\\{} & {} \frac{\partial ^2 l}{\partial \lambda _x^2} = \displaystyle \sum _{i=1}^{m}I_x(i) \Bigg \{\frac{\partial }{\partial \lambda _x} \Big [\dfrac{g_0^{'}(t_i;\lambda _x)}{g_0(t_i;\lambda _x)}\Big ]+(\alpha _x-1) \frac{\partial }{\partial \lambda _x}\Big [\dfrac{G_0^{'}(t_i;\lambda _x)}{G_0(t_i;\lambda _x)}\Big ]\\{} & {} -\{k(R_i+1)-1\}\phi _x\alpha _x \frac{\partial }{\partial \lambda _x}\\{} & {} \quad \Bigg [\dfrac{[G_0(t_i;\lambda _x)]^{\alpha _x-1} G_0^{'}(t_i;\lambda _x)}{\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}} \Bigg ]\Bigg \},\\{} & {} \frac{\partial ^2 l}{\partial \alpha _x \partial \lambda _x} =\frac{\partial ^2 l}{ \partial \lambda _x \partial \alpha _x}\,=\,\displaystyle \sum _{i=1}^{m}I_x(i) \Bigg \{\dfrac{G_0^{'}(t_i;\lambda _x)}{G_0(t_i;\lambda _x)}\\{} & {} -\{k(R_i+1)-1\}\phi _x G_0^{'}(t_i;\lambda _x)\frac{\partial }{\partial \alpha _x}\\{} & {} \quad \Bigg [\dfrac{\alpha _x [G_0(t_i;\lambda _x)]^{\alpha _x-1} }{\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}} \Bigg ] \Bigg \}, \end{aligned}$$

$$\begin{aligned}{} & {} \frac{\partial ^2 l}{\partial \alpha _y \partial \lambda _x} =\frac{\partial ^2 l}{ \partial \lambda _x \partial \alpha _y}\\{} & {} \,=\,-\displaystyle \sum _{i=1}^{m}I_x(i)\Bigg \{\{k(R_i+1)-1\}\phi _x G_0^{'}(t_i;\lambda _x)\alpha _x [G_0(t_i;\lambda _x)]^{\alpha _x-1}\\{} & {} \times \frac{\partial }{\partial \alpha _y}\Bigg [\dfrac{ 1}{\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}} \Bigg ] \Bigg \},\\{} & {} \frac{\partial ^2 l}{\partial \lambda _y \partial \lambda _x} =\frac{\partial ^2 l}{ \partial \lambda _x \partial \lambda _y}\\{} & {} \,=\,-\displaystyle \sum _{i=1}^{m}I_x(i)\Bigg \{\{k(R_i+1)-1\}\phi _x G_0^{'}(t_i;\lambda _x)\alpha _x [G_0(t_i;\lambda _x)]^{\alpha _x-1}\\{} & {} \times \frac{\partial }{\partial \lambda _y}\Bigg [\dfrac{ 1}{\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}} \Bigg ] \Bigg \},\\{} & {} \frac{\partial ^2{l}}{\partial \alpha _x^2} = -\dfrac{m_x}{\alpha _x^2} -\displaystyle \sum _{i=1}^{m}I_x(i)\\{} & {} \Bigg \{\{k(R_i+1)-1\}\phi _x \ln G_0^{}(t_i;\lambda _x)\frac{\partial }{\partial \alpha _x}\\{} & {} \quad \Bigg [\dfrac{[G_0(t_i;\lambda _x)]^{\alpha _x} }{\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}} \Bigg ] \Bigg \}, \\{} & {} \frac{\partial ^2{l}}{\partial \alpha _x \alpha _y} = \frac{\partial ^2{l}}{\partial \alpha _y \alpha _x} \\{} & {} = -\displaystyle \sum _{i=1}^{m}I_x(i)\Bigg \{\{k(R_i+1)-1\}\phi _x [G_0(t_i;\lambda _x)]^{\alpha _x}\\{} & {} \quad \ln G_0^{}(t_i;\lambda _x) \\{} & {} \times \frac{\partial }{\partial \alpha _y}\Bigg [\dfrac{1 }{\displaystyle \sum _{j=1}^{S}\phi _j\{1-[G_0(t_i;\lambda _j)]^{\alpha _j}\}} \Bigg ]\Bigg \}, \end{aligned}$$

where

$$\begin{aligned} g_0^{'}(t_i;\lambda _x)= & {} \frac{\partial }{\partial \lambda _x}g_0^{}(t_i;\lambda _x),\,\,G_0^{'}(t_i;\lambda _x)= \frac{\partial }{\partial \lambda _x}G_0^{}(t_i;\lambda _x). \end{aligned}$$