A Consistent Estimator of Structural Distribution

Marijus Radavičius

doi:10.17713/ajs.v49i4.1135

Abstract

We consider sparse count data models with the sparsity rate ? = N/n = O(1) where N = N (n) is the number of observations and n ? ? is the number of cells. In this case the plug-in estimator of the structural distribution of expected frequencies is inconsistent. If ? = O(n ?? ) for some ? > 0, the nonparametric maximum likelihood estimator, in general, is also inconsistent. Assuming that some auxiliary information on the expected frequencies is available, we construct a consistent estimator of the structural distribution.

Highlights

Let us consider the multinomial sampling scheme y = (y1, . . . , yn), y ∼ M ultinomialn(N, p), p = (p1, . . . , pn) ∈ Pn, (1)in case of sparse asymptotics:p = p(n), N = N (n) → ∞ as n → ∞.Here Pn is the standard (n − 1)-simplex of probabilities p. Khmaladze (1988) proposed specifications of sparse asymptotics by introducing sampling schemes with large number of rare events
A consistent estimator of probabilities p does not exist for any reasonable metric (Kolchin, Sevastyanov, and Chistyakov 1978; Khmaladze 1988; Klaassen and Mnatsakanov 2000; Radavicius 2019)
Assuming that some auxiliary information on expected frequencies is available, we construct a consistent estimator of the structural distribution

Summary

Introduction

Khmaladze (1988) proposed specifications of sparse asymptotics by introducing sampling schemes with large number of rare events. A Consistent Estimator of Structural Distribution for Sparse Data for statistical inference. This means that components of the vector y are exchangeable, i.e. their distribution is invariant with respect to the coordinate permutations. We consider a hierarchical Poisson sampling scheme with a certain sparsity rate This enables us to cover the case of very sparse data where N = o(n).

Structural distribution estimator

Proofs