Estimates Based on Sequential Order Statistics with the Two-Parameter Pareto Distribution

Abstract

This paper deals with the concept of sequential order statistics (SOS) for modeling the sequential r-out-of-n: F systems under a conditional proportional hazard rates drawn from a two-parameter Pareto distribution. Maximum likelihood estimates of parameters of interest are investigated based on multiple SOSs. Proposing various priors for the parameters of the assumed model, the Bayes estimates of the parameters of interest are derived and admissibility of obtained estimators is investigated. Due to the lack of explicit expressions for some of the obtained Bayes estimates, Lindley’s approximations are derived. To assess the performance of the obtained estimators, we conducted extensive simulation studies.

Appendix

To prove admissibility of ${\hat{\beta }}_{1,0,0}$, we show that the conditions of Theorem 13 in [11] (p. 547) hold. Notice that, following Theorem 3.1 of [13], we conclude that $-\ln T_2 \sim \Gamma (rs, \beta )$. Let consider $\{\pi _m; m\ge 1\} = \{\Gamma (1/m,1/m); m\ge 1\}$ be a sequence of proper priors for $\beta$. The Bayes estimate of $\beta$ under the SEL function is ${\hat{\beta }}_{1,1/m,1/m}=(rs+1/m)/(1/m-\ln T_2)$ which tends to ${\hat{\beta }}_{1,0,0}$ as $m \rightarrow \infty$. The risk difference of ${\hat{\beta }}_{1,1/m,1/m}$ and ${\hat{\beta }}_{1,0,0}$ is

$$\begin{aligned} \nonumber D_{m}= & {} (rs+1/m)^2 E\left[ \left( \frac{1}{1/m-\ln T_2}\right) ^2\right] -(rs)^2 E\left[ \left( \frac{1}{-\ln T_2}\right) ^2\right] \\&\; -\,2\beta (rs+1/m)E\left[ \frac{1}{1/m-\ln T_2}\right] +2\beta rs E\left[ \frac{1}{-\ln T_2}\right] . \end{aligned}$$

(36)

Notice that $E\left( -\ln T_2\right) ^{-1}=\beta /(rs-1)$, $E\left( -\ln T_2\right) ^{-2}=\beta ^2/(rs-1)(rs-2)$, $E\left( m^{-1}-\ln T_2\right) ^{-1}=\beta ^{rs}e^{\beta m^{-1}} \Phi _1(rs,1,\beta ,m^{-1})/\Gamma (rs)$ and $E\left( m^{-1}-\ln T_2\right) ^{-2}=\beta ^{rs}e^{\beta /m} \Phi _1(rs,2,\beta ,1/m)/\Gamma (rs)$ where $\Phi _1(n,,k,h,\eta )=\int _{\eta }^{\infty } \frac{(x-\eta )^{n-1}}{x^k}e^{-h x}\, dx$.

Substituting these values into (36) and using the monotone convergence theorem [28] at p 87, $D_{m}$ tends to zero as $m\rightarrow \infty$. From (36), Condition (c) of Theorem 13 in [11] (p. 547) satisfies by taking expectation w.r.t. the proper prior $\pi _m$. The estimator ${\hat{\beta }}_{1,1/m,1/m}$ is the unique Bayes estimate under the SEL function w.r.t. the proper prior $\pi _m$, and hence, the Bayes risk $r({\hat{\beta }}_{1,1/m,1/m},\beta )=E^{\pi _m}[R({\hat{\beta }}_{1,1/m,1/m},\beta )]$ is finite. This implies that the Bayes risk of the estimator ${\hat{\beta }}_{1,0,0}$ is finite, and this fulfill of Condition (a) of Theorem 13 in [11] (p. 547). On the other hand, any nondegenerate convex subset $C\subseteq \Theta =[0,\infty )$ is an interval, say [a, b], then it is sufficient to show that, there exist a $K>0$ and integer M such that, for $m\ge M$, $\int _a^b \pi _m(\beta ) d\beta \ge K$. Since the function $(1/m)^{1/m}$ is increasing in m for $m\ge 4$, we have

$$\begin{aligned} \int _a^b \pi _m(\beta ) d\beta= & {} \int _a^b \frac{(1/m)^{1/m}}{\Gamma (1/m)} \beta ^{1/m-1}e^{-\beta /m} d\beta \nonumber \\\ge & {} \int _a^{b^*} \frac{(1/4)^{1/4}}{\Gamma (1/4)} \beta ^{1/m-1}e^{-\beta /4} d\beta \nonumber \\= & {} \frac{(1/4)^{1/4}}{\Gamma (1/4)}e^{-b^*/4}\int _a^{b^*} \beta ^{1/m-1}d\beta \nonumber \\= & {} A\,m\left[ (b^*)^{1/m}-a^{1/m}\right] \ge A\,m\,(b^*)^{1/m} \end{aligned}$$

(37)

where $A=\frac{(1/4)^{1/4}}{\Gamma (1/4)}e^{-b^*/4}$, $b^*=\min (a+1,b)$ and $a>0$. For $b^*<1$, it is easy to see that $m\,(b^*)^{1/m}$ is an increasing function w.r.t. m and hence $A\,m\,(b^*)^{1/m}\ge 4A\,(b^*)^{1/4}=K_1$. For $b^*\ge 1$, $A\,m\,(b^*)^{1/m}\ge 4A=K_2 \ge K_1$. Assuming $K= K_1$, Condition (b) of Theorem 13 in [11] (p. 547).