Generalized confidence limits for the performance index of the exponentially distributed lifetime

ABSTRACT

Under a two-parameter exponential distribution, this study constructs the generalized lower confidence limit of the lifetime performance index C_L based on type-II right-censored data. The confidence limit has to be numerically obtained; however, the required computations are simple and straightforward. Confidence limits of C_L computed under the generalized paradigm are compared with those of C_L computed under the classical paradigm, citing an illustrative example with real data and two examples with simulated data, to demonstrate the merits and advantages of the proposed generalized variable method over the classical method.

KEYWORDS:

• Classical lower confidence bound
• Coverage probability
• Generalized lower confidence bound
• Lifetime performance index
• Shifted-exponential distribution.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F03;, 62F10

Acknowledgments

We are greatly indebted to the Editor-in-Chief and Reviewers for the very constructive suggestions and the useful comments which improved the earlier version of the manuscript.

Notes

1 Two confidence intervals based on the parametric bootstrap methods for estimating C_L are proposed:

•    (i) percentile bootstrap method (Efron 1979) (we call it from now on as boot-p), and

•   (ii) studentized bootstrap method (Hall 1988) (we call it from now on as boot-t).

2 The concept of generalized confidence intervals has been widely applied to a variety of practical settings such as regression, analysis of variance, analysis of reciprocals, analysis of covariance, analysis of frequency, multivariate analysis of variance, mixed models, and growth curves where standard methods failed to produce satisfactory results obliging practitioners to settle for asymptotic results and approximate solutions. For example, out of a large number of seminal papers on the generalized variable method, see Weerahandi (1993, 1995, 2004), Gunasekera (2015, 2016, 2017, 2018), and Gunasekera and Ananda (2015). For a recipe of constructing generalized pivotal quantities, see Hanning, Iyer and Patterson (2006).

3 Gail and Gastwirth (1978) described four methods of goodness-fit-test for exponential distributions: Lorenz Statistic: $L_n={\sum }_{i=1}^{⌞np⌟}{X}_{\left(i\right)}/{\sum }_{i=1}^{⌞np⌟}{Y}_i;$ Gini Statistic: G_n $\left[{\sum }_{i=1}^{n-1}\left\{i(n-i)\right\}(Y_{(i+1)}-Y_{\left(i\right)})\right]/{\sum }_{i=1}^n(n-1)Y_i;$ Pietra Statistic: $P_n=\left[{\sum }_{i=1}^{n-1}\left\{i(n-1)\right\}(Y_{(i+1)}-\bar{Y})\right]/{\sum }_{i=1}^n(n-1)Y_i;$ Scale-Free Statistic$:R_n=\bar{Y}{\{(n-1)/{\sum }_{i=1}^n{(Y_i-\bar{Y})}^2\}}^{1/2}$

4 The actual (empirical) coverage 1 − α_a for the nominal (intended) 1 − α confidence interval is:

1 − α_a =$\frac{\text{number}\text{of}1-\alpha \text{confidence}\text{intervals}\text{that}\text{contains}\text{the}\text{true}\text{lifetime}\text{performance}\text{index}{C}_L}{\text{The}\text{total}\text{number}\text{of}\text{simulations}}$

Also, see the Algorithm 4.

5 Both the expected length and the average expected length of the intervals are very important in evaluation of a confidence interval. By definition, Expected  Length $=E_{\theta ,\beta }\left(Length\left(CI\right)\right)={\int }_0^{\infty }(C_L^{UB}\left(y\right)-C_L^{LB}\left(y\right))\frac{1}{\beta }exp\{-(y-\theta )/\beta )dy,$ where C^UB_L(y) and C^LB_L(y) are the upper and lower limits of the CI of C_L, respectively. The average expected length is then given by Average Expected  Length $=E_{\theta ,\beta }\left(E_{\theta ,\beta }Length\left(CI\right)\right))={\int }_0^{\infty }{E}_{\theta ,\beta }\left(Length\left(CI\right)\right)dy$.