Estimating the distribution function of symmetric pairs

Abstract

We discuss three classes of bivariate symmetry models and study the estimation of their distribution functions (DFs). Under radial symmetry, an estimator based on the mean of the empirical and survival DFs is considered. For exchangeable symmetry, an estimator based on the mean of the empirical DF and its exchangeable image is presented. At their intersection, we define radial exchangeability and study estimation of its DF. The symmetrized estimators coincide with the non parametric maximum likelihood estimators of the DF under each model. We obtain their mean and variance and state their asymptotic normality. The relative efficiency of the estimators for the bivariate normal distribution is obtained.

KEYWORDS:

• Bootstrap
• Distribution function
• Exchangeability
• Radial symmetry

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H12
• 62G05
• G2g10

Acknowledgment

The author would like to thank two anonymous referees and the editor for comments that led to an improved manuscript.