A Simple Bivariate Example of Fast Convergence Rates for Maximum Likelihood Estimates

TL;DR

This paper introduces a bivariate distribution family where maximum likelihood estimates converge faster than the traditional rate, demonstrating flexible convergence speeds based on regular variation.

Contribution

It provides a simple example showing that MLE convergence rates can surpass the classic square root rate, with rates determined by regularly varying functions.

Findings

MLE convergence rate can be faster than sqrt(n)

Any rate given by a regularly varying function with index > 0.5 is achievable

Some rates with index 0.5 are also attainable

Abstract

We present a one-parameter family of bivariate absolutely continuous distributions based on location-scale family of variance Gaussian mixtures, with continuous densities with the same support (effective domain). The maximum likelihood estimation of the location parameter converges to the true value faster than the classic square root rate. In fact, we can obtain any convergence rate given by a regularly varying function with index greater than 0.5, and some convergence rates given by regularly varying functions with index 0.5 but faster than the classic square root rate.