Some Inferential Studies on Inliers in Gompertz Distribution

Abstract

This paper deals with the inferential studies of inliers in Gompertz distribution. The inliers are inconsistent observations, which are generally the resultant of instantaneous and early failures. These situations are generally modeled using a non-standard mixture of distributions with a failure time distribution (FTD) for positive observations. Considering FTD as Gompertz distribution, we have studied various methods of estimating parameters including the uniformly minimum variance unbiased estimate of some parametric functions. An application of inliers prone models is illustrated with a real data set.

Appendix: Asymptotic Distribution of MLE

For inlier prone Gompertz distribution $g\left( {x;p,\alpha ,\theta } \right)$ given by (4) with $\alpha$ known,

$$\begin{aligned} \frac{\partial ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial p}=\left\{ {{\begin{array}{ll} 0, &{}\quad x<d \\ -\frac{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}{1-p\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }},&{} \quad x=d \\ \frac{1}{p},&{} \quad x>d \\ \end{array}},} \right. \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial \theta }=\left\{ {{\begin{array}{ll} 0,&{} \quad x<d \\ \frac{p\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }\frac{\left( {e^{\alpha d}-1} \right) }{\alpha }}{1-p\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}, &{}\quad x=d \\ \frac{1}{\theta }-\frac{\left( {e^{\alpha x}-1} \right) }{\alpha }, &{} \quad x>d \\ \end{array}}.} \right. \end{aligned}$$

One can verify that $E\left( {\frac{\partial ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial p}} \right) =0$   and    $E\left( {\frac{\partial ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial \theta }} \right) =0$.

Also,

$$\begin{aligned}&\frac{\partial ^{2}ln\,g\left( x;p,\alpha ,\theta \right) }{\partial p^{2}}=\left\{ {{\begin{array}{ll} 0, &{}\quad x<d \\ -\frac{e^{-\frac{2\,\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}{\left[ {1-p\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }} \right] ^{2}},&{} \quad x=d \\ -\frac{1}{p^{2}}, &{} \quad x>d \\ \end{array}}} \right. , \\&\quad \frac{\partial ^{2}ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial \theta ^{2}}=\left\{ {{\begin{array}{ll} 0, &{} \quad x<d \\ -\frac{p\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }\left[ {\frac{\left( {e^{\alpha d}-1} \right) }{\alpha }} \right] ^{2}}{\left[ {1-p\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }} \right] ^{2}},&{} \quad x=d \\ -\frac{1}{\theta ^{2}}, &{} \quad x>d \\ \end{array} }} \right. , \\&\quad \frac{\partial ^{2}ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial p\,\partial \theta }=\left\{ {{\begin{array}{ll} 0, &{} \quad x<d \\ \frac{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}\,-\,1} \right) }\frac{\left( {e^{\alpha d}\,-\,1} \right) }{\alpha }}{\left[ {1-p\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }} \right] ^{2}}, &{} \quad x=d \\ 0,&{} \quad x>d \\ \end{array} }}\right. . \end{aligned}$$

Hence, the Fisher information is:

$$\begin{aligned} I_{pp}= & {} E\left( {-\frac{\partial ^{2}ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial p^{2}}} \right) =\frac{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}{p\,\left( {1-pe^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }} \right) }=\frac{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}{p\,p^{*}}, \\ I_{\theta \theta }= & {} E\left( {-\frac{\partial ^{2}\,ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial \theta ^{2}}} \right) =\frac{\left( {1-p^{*}} \right) \left\{ {\theta ^{2}\left[ {\frac{\left( {e^{\alpha d}-1} \right) }{\alpha }} \right] ^{2}+p^{*}} \right\} }{\theta ^{2}\,p^{*}}, \\ I_{p\theta }= & {} E\left( {-\frac{\partial ^{2}ln\,g\left( {x;p,\alpha ,\theta } \right) }{\partial p\,\partial \theta }} \right) =-\frac{\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}\,-\,1} \right) }\,\, \frac{\left( {e^{\alpha d}-1} \right) }{\alpha }}{p^{*}} \end{aligned}$$

where, $p^{*}=1-pe^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }$.

Therefore, the Fisher information matrix $I_g \left( {p,\theta } \right)$ is given by:

$$\begin{aligned} I_g \left( {p,\theta } \right) =\left[ {{\begin{array}{ll} {I_{pp} } &{} {I_{p\theta } } \\ {I_{\theta p} }&{} {I_{\theta \theta } } \\ \end{array} }} \right] =\left[ {{\begin{array}{ll} {\frac{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}{p\,p^{*}}} &{} {-\frac{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }\,\,\frac{\left( {e^{\alpha d}-1} \right) }{\alpha }}{p^{*}}} \\ {-\frac{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }\,\,\frac{\left( {e^{\alpha d}-1} \right) }{\alpha }}{p^{*}}}&{} {\frac{\left( {1-p^{*}} \right) \left\{ {\theta ^{2}\frac{\left( {e^{\alpha d}-1} \right) ^{2}}{\alpha ^{2}}+p^{*}} \right\} }{\theta ^{2}p^{*}}} \\ \end{array} }} \right] . \end{aligned}$$

The inverse matrix $I_g^{-1} \left( {p,\theta } \right)$ is given by:

$$\begin{aligned} I_g^{-1} \left( {p,\theta } \right) =\left[ {{\begin{array}{ll} {\frac{p\left( {\theta ^{2}\frac{\left( {e^{\alpha d}-1} \right) ^{2}}{\alpha ^{2}}+p^{*}} \right) }{e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}}&{} {-\frac{\theta ^{2}\left( {e^{\alpha d}-1} \right) }{\alpha \,\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}} \\ {-\frac{\theta ^{2}\left( {e^{\alpha d}-1} \right) }{\alpha \,\,e^{-\frac{\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}}&{} {\frac{\theta ^{2}}{1-p^{*}}} \\ \end{array} }} \right] . \end{aligned}$$

and the determinant of $I_g \left( {p,\theta } \right)$ is given by ${\Delta }$ is ${\Delta }=\frac{e^{-\frac{2\theta }{\alpha }\left( {e^{\alpha d}-1} \right) }}{\theta ^{2}p^{*}}$.

Using the standard result of MLE, we have

$$\begin{aligned} \left( {\hat{p},\hat{\theta }} \right) ^{{\prime }}\sim AN^{\left( 2 \right) }\left[ {\left( {p,\theta } \right) ^{{\prime }},\,\frac{1}{n}\,I_g^{-1} \left( {p,\theta } \right) } \right] . \end{aligned}$$

Using the estimated variances, one can also propose large sample tests for p and $\theta$.