## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 4 (1972), 1136-1146.

### Testing Whether New is Better than Used

Myles Hollander and Frank Proschan

#### Abstract

A $U$-statistic $J_n$ is proposed for testing the hypothesis $H_0$ that a new item has stochastically the same life length as a used item of any age (i.e., the life distribution $F$ is exponential), against the alternative hypothesis $H_1$ that a new item has stochastically greater life length $(\bar{F}(x)\bar{F}(y) \geqq \bar{F}(x + y)$, for all $x \geqq 0, y \geqq 0$, where $\bar{F} = 1 - F). J_n$ is unbiased; in fact, under a partial ordering of $H_1$ distributions, $J_n$ is ordered stochastically in the same way. Consistency against $H_1$ alternatives is shown, and asymptotic relative efficiencies are computed. Small sample null tail probabilities are derived, and critical values are tabulated to permit application of the test.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 4 (1972), 1136-1146.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692466

**Digital Object Identifier**

doi:10.1214/aoms/1177692466

**Mathematical Reviews number (MathSciNet)**

MR348909

**Zentralblatt MATH identifier**

0241.62055

**JSTOR**

links.jstor.org

#### Citation

Hollander, Myles; Proschan, Frank. Testing Whether New is Better than Used. Ann. Math. Statist. 43 (1972), no. 4, 1136--1146. doi:10.1214/aoms/1177692466. https://projecteuclid.org/euclid.aoms/1177692466