## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 1 (1983), 267-274.

### Testing Whether New is Better than Used with Randomly Censored Data

Yuan Yan Chen, Myles Hollander, and Naftali A. Langberg

#### Abstract

A life distribution $F$, with survival function $\bar{F} \equiv 1 - F$, is new better than used (NBU) if $\bar{F}(x + y) \leq \bar{F}(x)\bar{F}(y)$ for all $x, y \geq 0$. We propose a test of $H_0 : F$ is exponential, versus $H_1 : F$ is NBU, but not exponential, based on a randomly censored sample of size $n$ from $F$. Our test statistic is $J^c_n = \int \int \bar{F}_n(x + y) dF_n(x) dF_n(y)$, where $F_n$ is the Kaplan-Meier estimator. Under mild regularity on the amount of censoring, the asymptotic normality of $J^c_n$, suitably normalized, is established. Then using a consistent estimator of the null standard deviation of $n^{1/2}J^c_n$, an asymptotically exact test is obtained. We also study, using tests for the censored and uncensored models, the efficiency loss due to the presence of censoring.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 1 (1983), 267-274.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346077

**Digital Object Identifier**

doi:10.1214/aos/1176346077

**Mathematical Reviews number (MathSciNet)**

MR684884

**Zentralblatt MATH identifier**

0504.62086

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 62G10: Hypothesis testing

**Keywords**

Classes of life distributions efficiency loss exponentiality

#### Citation

Chen, Yuan Yan; Hollander, Myles; Langberg, Naftali A. Testing Whether New is Better than Used with Randomly Censored Data. Ann. Statist. 11 (1983), no. 1, 267--274. doi:10.1214/aos/1176346077. https://projecteuclid.org/euclid.aos/1176346077

#### Corrections

- See Correction: Yuan Yan Chen, Myles Hollander, Naftali A. Langberg. Corrections: Testing Whether New is Better than Used with Randomly Censored Data. Ann. Statist., Volume 11, Number 4 (1983), 1267--1267.Project Euclid: euclid.aos/1176346341