New estimators of distribution functions

ABSTRACT

This article considers the estimation of a distribution function F_X(x) based on a random sample X₁, X₂, …, X_n when the sample is suspected to come from a close-by distribution F₀(x). The new estimators, namely the preliminary test (PTE) and Stein-type estimator (SE) are defined and compared with the “empirical distribution function” (edf) under local departure. In this case, we show that Stein-type estimators are superior to edf and PTE is superior to edf when it is close to F₀(x). As a by-product similar estimators are proposed for population quantiles.

KEYWORDS:

• Empirical distribution function
• Local alternatives
• Mean Square error
• Preliminary test estimator
• Shrinkage estimator.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05, 62G20

Acknowledgment

The authors are grateful to the referees whose comments were constructive to improve the presentation of this article.