Estimation of a star-shaped distribution function

ABSTRACT

A life distribution function (DF) F is said to be star-shaped if is nondecreasing on its support. This generalises the model of a convex DF, even allowing for jumps. The nonparametric maximum likelihood estimation is known to be inconsistent. We provide a uniformly strongly consistent least-squares estimator. We also derive the convergence in distribution of the estimator at a point where is increasing using the arg max theorem.

KEYWORDS:

• Star-shaped distribution functions
• uniform consistency
• convergence in distribution
• argmax theorem

AMS SUBJECT CLASSIFICATION:

• 62P10
• 62G05
• 62E10

Acknowledgments

The authors are grateful to four anonymous reviewers for thorough reviews and comments, and especially to one reviewer who pointed out several mistakes in the proof of Theorem 4.1 and many other careful observations and suggestions. Hammou El Barmi acknowledges the support of the City University of New York through a PSC-CUNY grant and by Ministerio de Economia y Competitividad and FEDER under research grant MTM2013-40941-P.

Disclosure statement

No potential conflict of interest was reported by the author(s).