Limit Theorems for φ-Divergences Based on k-Spacings

Abstract

We study the asymptotic behavior of the divergence-based statistic

where X ₁ < X ₂ < ··· < X _n are the order statistics of a random sample of size n, G is a known continuous distribution, φ is a convex function on (0, + ∞), k ≥ 1 is a fixed integer, and N the smallest integer greater than or equal to (n + 1)/k. Laws of large numbers and central limit theorems for W _φ,n (G,k) are established under sharp conditions on φ and an information-type inequality is obtained to characterize the unknown fixed distribution which generated the data. Application to goodness-of-fit tests are discussed with respect to general consistency and asymptotic power for several widely used φ and various k.

Keywords:

• Goodness-of-fit
• Information-type inequalities
• k-spacings
• φ-divergences

Mathematics Subject Classification:

• 62B10
• Secondary 62E20

Acknowledgments

The authors would like to thank the referees for several valuable suggestions and references. Part of this research was carried out when the authors were visiting University of California at Berkeley; they would like to thank Professor Michael Klass for great hospitality. R. J. is supported in part by Spanish Grant ECO2008-05080.