Further results involving percentile inactivity time order and its inference
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Further results involving percentile inactivity time order and its inference Mervat Mahdy
Received: 8 August 2012 / Accepted: 5 September 2013 © Sapienza Universitá di Roma 2013
Abstract This paper studies a family of stochastic orders of random variables defined via the comparison of their percentile inactivity time functions. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. The estimator of the percentile inactivity time is introduced. Finally, some applications in reliability theory and finance are described. Keywords The percentile reversed residual function · Mean inactivity time class · The empirical distribution
1 Introduction The median, or other percentiles, of the inactivity lifetime of random variables are useful alternatives to the mean inactivity lifetime of that random variable (it has been studied by many researchers recently (c.f. [2,3,13,15–20,22–24,26,27]). But only a few papers studied comparisons of such percentiles of different random variables. This paper presents in depth stochastic orders that compare such percentiles. Some surprising properties of these orders will be discovered. Assume T as a random variable with distribution function FT , and denote by F T its reliability function. Let T(t) = [t − T | T ≤ t],
(1.1)
be the inactivity time (IT) at time t > 0; then the distribution function of T(t) is given by FT(t) (s) =
FT (t) − FT (t − s) , s ≥ 0, FT (t)
M. Mahdy (B) Department of Statistics, Mathematics, & Insurance, College of Commerce, Benha University, Benha, Egypt e-mail: [email protected]; [email protected]
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M. Mahdy
and its reliability functions can be represented as F T(t) (s) =
FT (t − s) , s ≥ 0. FT (t)
The inactivity time is of interest in many areas of applied probability and statistics such as actuarial studies, biometry, survival analysis, economics, risk management, and reliability. The inactivity time function m T (t) that is associated with T is given by ⎧ ⎨ t FT (u) du ; t ≥ 0; 0 . m T (t) = FT (t) ⎩0 otherwise. provided the expectation existence. It is a useful tool for analyzing important properties of T when it exists. However, the mean inactivity time function may not exist. Even when it exists it may have some practical shortcomings, especially in situations where the data are censored, or when the underlying distribution is skewed or heavy- tailed. In such cases, either the empirical mean inactivity time function cannot be calculated, or a single short-term survivor can have a marked effect upon it, which will tend to be unstable due to its strong dependence on very short durations. An alternative to the mean inactivity time function is the percentile inactivity time function δT,θ (t), where θ is some number between 0 and 1. This function is defined for any t ≥ 0 by letting δT,θ (t) be the percentile inactivity time of T(t) . The δT,θ (t) can be represented as δT,θ (t) =
FT−1 (θ ) , t ≥ 0; (t)
0, t < 0.
A straight
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