## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 40, Number 4 (1969), 1216-1235.

### Tests for Monotone Failure Rate Based on Normalized Spacings

Peter J. Bickel and Kjell A. Doksum

#### Abstract

Let $F$ be a distribution with density $f$, and let $q(t) = f(t)/\lbrack 1 - F(t) \rbrack$ be the failure rate of $F$. Tests for constant versus monotone increasing failure rate based on the ranks of the normalized spacings between the ordered observations have been considered by Proschan and Pyke (1967). They show that these statistics are asymptotically normally distributed for fixed alternatives $F$ and compute the ratios of the efficacies of one of their rank tests to the best statistics for Weibull and Gamma alternatives. In this paper, it is shown that asymptotic normality holds also for sequences of alternatives $\{F_{\theta_n}\}$ that approach the $H_0$ distribution $1 - \exp (-\lambda t), t \geqq 0,$ as $n \rightarrow \infty$; and that the above mentioned ratios of efficacies are in fact Pitman efficiencies. Let $R_1, \cdots, R_n$ be the ranks of the normalized spacings, $T_1 = \sum iR_i$ and $T_2 = - \sum i \log \lbrack 1 - R_i/(n + 1) \rbrack$. Then $T_1$ is asymptotically equivalent to the Proschan Pyke statistic. It is shown that the Pitman efficiency satisfies \begin{equation*} \tag{1.1} e(T_1, T_2) \equiv \frac{3}{4}\end{equation*} for all sequences of alternatives $\{F_{\theta_n}\}$ and thus $T_1$ is asymptotically inadmissible. Statistics that are linear in the normalized spacings and asymptotically most powerful for parametric alternatives $\{F_{\theta_n}\}$ if the scale parameter $\lambda$ is known, are derived, and it is shown that the rank statistics that are asymptotically most powerful in the class of linear rank tests, are nowhere most powerful in the class of all tests, when $\lambda$ is known. If $\lambda$ is unknown, studentizing of the linear normalized spacing tests which are asymptotically most powerful for $\lambda$ known leads to procedures which have only the same asymptotic power as the most powerful linear rank tests. Unbiasedness is shown for tests that are monotone in the normalized spacings, and Monte Carlo power estimates are used to compare the various statistics with the likelihood ratio tests considered by Barlow (1967).

#### Article information

**Source**

Ann. Math. Statist., Volume 40, Number 4 (1969), 1216-1235.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177697498

**Digital Object Identifier**

doi:10.1214/aoms/1177697498

**Mathematical Reviews number (MathSciNet)**

MR242301

**Zentralblatt MATH identifier**

0191.50504

**JSTOR**

links.jstor.org

#### Citation

Bickel, Peter J.; Doksum, Kjell A. Tests for Monotone Failure Rate Based on Normalized Spacings. Ann. Math. Statist. 40 (1969), no. 4, 1216--1235. doi:10.1214/aoms/1177697498. https://projecteuclid.org/euclid.aoms/1177697498