## The Annals of Statistics

### On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint

#### Abstract

We identify the asymptotic behavior of the estimators proposed by Rojo and Samaniego and Mukerjee of a distribution $F$ assumed to be uniformly stochastically smaller than a known baseline distribution $G$.We show that these estimators are $\sqrt{n}$-convergent to a limit distribution with mean squared error smaller than or equal to the mean squared error of the empirical survival function. By examining the mean squared error of the limit distribution, we are able to identify the optimal estimator within Mukerjee’s class under a variety of different assumptions on $F$ and $G$. Similar theoretical results are developed for the two-sample problem where$F$ and $G$ are both unknown. The asymptotic distribution theory is applied to obtain confidence bands for the survival function $\bar{F}$ based on published data from an accelerated life testing experiment.

#### Article information

Source
Ann. Statist., Volume 28, Number 1 (2000), 116-150.

Dates
First available in Project Euclid: 14 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1016120367

Digital Object Identifier
doi:10.1214/aos/1016120367

Mathematical Reviews number (MathSciNet)
MR1762906

Zentralblatt MATH identifier
1106.62332

Subjects
Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing 62E20

#### Citation

Arcones, Miguel A.; Samaniego, Francisco J. On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint. Ann. Statist. 28 (2000), no. 1, 116--150. doi:10.1214/aos/1016120367. https://projecteuclid.org/euclid.aos/1016120367

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