## The Annals of Statistics

- Ann. Statist.
- Volume 14, Number 4 (1986), 1388-1398.

### Testing Exponentiality Versus a Trend Change in Mean Residual Life

Frank Guess, Myles Hollander, and Frank Proschan

#### Abstract

Given that an item is of age $t$, the expected value of the random remaining life is called the mean residual life (MRL) at age $t$. We propose two new nonparametric classes of life distributions for modeling aging based on MRL. The first class of life distributions consists of those with "increasing initially, then decreasing mean residual life" (IDMRL). The IDMRL class models aging that is initially beneficial, then adverse. The second class, "decreasing, then increasing mean residual life" (DIMRL), models aging that is initially adverse, then beneficial. We propose two testing procedures for $H_0$: constant MRL (i.e., exponentiality) versus $H_1$: IDMRL, but not constant MRL (or $H'_1$: DIMRL, but not constant MRL). The first testing procedure assumes the turning point, $\tau$, from IMRL to DMRL is specified by the user or is known. The second procedure assumes knowledge of the proportion, $\rho$, of the population that "dies" at or before the turning point (knowledge of $\tau$ itself is not assumed).

#### Article information

**Source**

Ann. Statist., Volume 14, Number 4 (1986), 1388-1398.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350165

**Digital Object Identifier**

doi:10.1214/aos/1176350165

**Mathematical Reviews number (MathSciNet)**

MR868307

**Zentralblatt MATH identifier**

0613.62122

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62N05: Reliability and life testing [See also 90B25]

Secondary: 62G10: Hypothesis testing

**Keywords**

Nonparametric classes of distributions mean residual life exponentiality increasing then decreasing mean residual life distributions hypothesis testing differentiable statistical functions $L$-statistics

#### Citation

Guess, Frank; Hollander, Myles; Proschan, Frank. Testing Exponentiality Versus a Trend Change in Mean Residual Life. Ann. Statist. 14 (1986), no. 4, 1388--1398. doi:10.1214/aos/1176350165. https://projecteuclid.org/euclid.aos/1176350165