Second order asymptotics for the number of times an estimator is more than epsilon from its target value

TL;DR

This paper explores second order asymptotics of the frequency with which estimators deviate from their target, providing refined comparisons of estimator efficiency beyond traditional asymptotic analysis.

Contribution

It introduces second order asymptotic analysis for differences in estimator deviation counts, enabling more precise comparisons of estimator performance.

Findings

Derived limits for expected differences in deviation counts.

Identified optimal estimator adjustments for minimal error.

Connected second order results to exponential distributions related to Brownian motion.

Abstract

Suppose $\{\widehat\theta_n\colon n\ge1\}$ is a strongly consistent sequence of estimators for a parameter $\theta$, where $\widehat\theta_n$ is based on the first $n$ observations. Consider $Q_\varepsilon$, the number of times $|\widehat\theta_n-\theta|\ge\varepsilon$. In another paper (Hjort and Fenstad, 1992) we have shown that $\varepsilon^2 Q_\varepsilon$ has a limit distribution as $\varepsilon\rightarrow0$, depending only on $\sigma$, the standard deviation of the limit distribution for $\sqrt{n}(\widehat\theta_n-\theta)$, under natural regularity conditions. The present paper investigates some second order asymptotics for differences between $Q_\varepsilon$ variables. The limit of ${\rm E}(Q_{1,\varepsilon}-Q_{2,\varepsilon})$ is calculated in cases where ${\rm E} Q_{1,\varepsilon}/{\rm E} Q_{2,\varepsilon}$ goes to 1, leading to a notion of `asymptotic relative deficiency' in cases where the asymptotic relative efficiency is 1. This is used to distinguish between competing estimators with identical limit distributions. Thus using denominator $n-{1\over3}$ in the familiar formula for estimating a normal variance is better than both $n$ and $n-1$ and indeed all other choices, for example, in the sense of leading to the smallest possible expected number of $\varepsilon$ errors. Results of this type are found in a selection of familiar estimation problems, using limit results for expected differences, and are compared to corresponding asymptotic relative deficiency analysis in the sense of Hodges and Lehmann. Some second order distributional results are reached as well. It is shown how $\varepsilon$ times a $Q_\varepsilon$-difference tends to a variable which is related to some exponential distributions associated with Brownian motion, and that have recently been investigated by Hjort and Khasminskii (1993).