## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 32, Number 2 (2018), 422-436.

### Some unified results on stochastic properties of residual lifetimes at random times

#### Abstract

The residual life of a random variable $X$ at random time $\Theta$ is defined to be a random variable $X_{\Theta}$ having the same distribution as the conditional distribution of $X-\Theta$ given $X>\Theta$ (denoted by $X_{\Theta}=(X-\Theta|X>\Theta)$). Let $(X,\Theta_{1})$ and $(Y,\Theta_{2})$ be two pairs of jointly distributed random variables, where $X$ and $\Theta_{1}$ (and, $Y$ and $\Theta_{2}$) are not necessarily independent. In this paper, we compare random variables $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ by providing sufficient conditions under which $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ are stochastically ordered with respect to various stochastic orderings. These comparisons have been made with respect to hazard rate, likelihood ratio and mean residual life orders. We also study various ageing properties of random variable $X_{\Theta_{1}}$. By considering this generalized model, we generalize and unify several results in the literature on stochastic properties of residual lifetimes at random times. Some examples to illustrate the application of the results derived in the paper are also presented.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 32, Number 2 (2018), 422-436.

**Dates**

Received: July 2016

Accepted: December 2016

First available in Project Euclid: 17 April 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1523952021

**Digital Object Identifier**

doi:10.1214/16-BJPS348

**Mathematical Reviews number (MathSciNet)**

MR3787760

**Zentralblatt MATH identifier**

06914681

**Keywords**

Hazard rate order likelihood ratio order mean residual life order reversed hazard rate order

#### Citation

Misra, Neeraj; Naqvi, Sameen. Some unified results on stochastic properties of residual lifetimes at random times. Braz. J. Probab. Stat. 32 (2018), no. 2, 422--436. doi:10.1214/16-BJPS348. https://projecteuclid.org/euclid.bjps/1523952021