Statistica Sinica 17(2007), 1165-1189

INFERENCE UNDER PEAKEDNESS RESTRICTIONS

Javier Rojo, José Batún-Cutz and Ramón Durazo-Arvizu

Rice University, CIMAT and Northwestern University

Abstract: A distribution function $F$ is more peaked about a known point $a$ than the distribution $G$ is about the known point $b$ if $F((x+a)^{-})-F(-x+a)\ge G((x+b)^{-})-G(-x+b)$ for every $x$. The statistical concept of dispersion plays an important role in the theory and practice of statistics. For example, in statistical genetics, the effect of a gene on a phenotype of interest can be ascertained by regressing the squared phenotypical differences on the proportion of identical by descent alleles shared by pairs of siblings (Haseman-Elston (1972)). This paper proposes estimators for the distribution functions $F$ and/or $G$, when $F$ is more ``peaked'' than $G$. The estimators are shown to be strongly uniformly consistent, their asymptotic distribution theory is discussed, and an asymptotic test for equality in peakedness is provided. The case of censored data is also considered. Data from various national and international studies are used to illustrate the new procedures.

Key words and phrases: Kaplan-Meier, peakedness ordering, Stochastic ordering, symmetry, weak convergence.