Estimation of Location Parameter from Two Biased Samples

Leonid I Piterbarg

doi:10.4236/am.2013.49171

Abstract

We consider a problem of estimating an unknown location parameter from two biased samples. The biases and scale parameters of the samples are not known as well. A class of non-linear estimators is suggested and studied based on the fuzzy set ideas. The new estimators are compared to the traditional statistical estimators by analyzing the asymptotical bias and carrying out Monte Carlo simulations.

Highlights

IntroductionIn the general formulation (1) with biased observations a choice of an appropriate measure of the estimation skill is a challenge because (nuisance) parameters the b1, bbi2aswhBic ˆh depends on unknown never can be identified from the available observations
The problem is to estimate an unknown scalar parameter from two different independent samples of size n1 and n2x1i b1 1 i, i 1, 2, n1, (1)x2i b2 2 i, i 1, 2, n2 where bj and j are the bias and scale parameter of the j-th sample respectively, j = 1, 2, and i, i are zero mean independent random noises.A novelty in our set up is that the biases bj, j = 1, 2 are assumed to be unknown which makes unidentifiable from the classical statistics viewpoint, e.g. [1]
We suggest to use fuzzy set ideas [6,7] to construct non-linear estimators for diminishing the bias comparing to the aforementioned approach

Summary

Introduction

In the general formulation (1) with biased observations a choice of an appropriate measure of the estimation skill is a challenge because (nuisance) parameters the b1, bbi2aswhBic ˆh depends on unknown never can be identified from the available observations. We construct such a measure as follows. 2 2 one cannot estimate at all under the given observations In such a situation one of the ways to order estimators according to their biases is to accept that ˆ 1 is better than ˆ 2 (under the identifiable parameters being fixed) if. Where the angle brackets mean averaging over parameters defined in (3) and the ratio (7)

Estimators and Their Asymptotical Bias

F du u du

C2 1 2 1 2 2

Simulations

Conclusions and Discussion