A class of asymptotically efficient estimators based on sample spacings

Abstract

In this paper, we consider general classes of estimators based on higher-order sample spacings, called the Generalized Spacings Estimators. Such classes of estimators are obtained by minimizing the Csiszár divergence between the empirical and true distributions for various convex functions, include the "maximum spacing estimators" as well as the maximum likelihood estimators (MLEs) as special cases, and are especially useful when the latter do not exist. These results generalize several earlier studies on spacings-based estimation, by utilizing non-overlapping spacings that are of an order which increases with the sample size. These estimators are shown to be consistent as well as asymptotically normal under a fairly general set of regularity conditions. When the step size and the number of spacings grow with the sample size, an asymptotically efficient class of estimators, called the "Minimum Power Divergence Estimators," are shown to exist. Simulation studies give further support to the performance of these asymptotically efficient estimators in finite samples and compare well relative to the MLEs. Unlike the MLEs, some of these estimators are also shown to be quite robust under heavy contamination.

Some parametric tests based on sample spacings

“…The functions φ defined in (10) satisfy the assumption lim m→∞ σ 2 φ,m = 1 (cf. Ekström et al, 2020). Theorem 3 suggests that, if m → ∞ such that m = o(n), then all the tests based on φ with lim m→∞ σ 2 φ,m = 1, have same asymptotic power.…”