Kernel Estimators for Distribution Functions on Dependent Random Fields

Abstract

Consider observations (representing lifelengths) taken on a random field indexed by lattice points. Estimating the distribution function F(x) = P(X _i  ≤ x) is an important problem in survival analysis. We propose to estimate F(x) by kernel estimators, which take into account the smoothness of the distribution function. Under some general mixing conditions, our estimators are shown to be asymptotically unbiased and consistent. In addition, the proposed estimator is shown to be strongly consistent and sharp rates of convergence are obtained.

Keywords:

• Distribution function
• Mixing conditions
• Random field

Mathematics Subject Classification:

• Primary 62N02, 62G05
• Secondary 62G20

Acknowledgment

The author is grateful to the referees for their useful comments that have significantly improved the presentation of the article.