A Note on the Construction of a Sample Variance

Nitis Mukhopadhyay,Bhargab Chattopadhyay

doi:10.4038/sljastats.v15i1.6795

Nitis Mukhopadhyay, Bhargab Chattopadhyay

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https://doi.org/10.4038/sljastats.v15i1.6795

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Abstract

Mukhopadhyay and Chattopadhyay (2013) proposed a new approach to construct unbiased estimators of by U-statistics starting with a class of non-symmetric initial kernels of degree m(> 2): But, surprisingly all symmetrized final estimators in the form of U-statistics reduced to when the parent distribution function (d.f.) F remained unknown. Now, we exhibit another class of non-symmetric initial kernels to come up with U-statistics of degree m(> 2) and unbiased for : Interestingly, the associated symmetrized final U-statistics again coincide with (Theorem 2.1), settling the open question from Remark 3.1 of Mukhopadhyay and Chattopadhyay (2013). DOI: http://dx.doi.org/10.4038/sljastats.v15i1.6795

Highlights

A popular distribution-free measure of variation is the sample variance
The associated symmetrized final U-statistics again coincide with S 2 (Theorem 2.1), settling the open question from Remark 3.1 of Mukhopadhyay and Chattopadhyay (2013)
Suppose that X1, ..., Xn are independent and identically distributed random variables observed from a population distribution function (d.f.) F with μ ≡ μ(F) = xdF(x) and σ2 ≡ σ2(F) = (x−μ)2dF(x)

Summary

Introduction

A popular distribution-free measure of variation is the sample variance. Suppose that X1, ..., Xn are independent and identically distributed random variables observed from a population distribution function (d.f.) F with μ ≡ μ(F) = xdF(x) and σ2 ≡ σ2(F) = (x−μ)2dF(x). The latter will capture everyone’s income difference from everyone else, and not merely from the mean, which might not be anybody’s income whatsoever.” Mukhopadhyay and Chattopadhyay (2013) elegantly utilized non-symmetric initial kernels of order m > 2 to unbiasedly estimate σ2 when F remained completely unknown. Such a general method, led them back to S 2. The sample variance compares the average income of any subgroup of individuals (of size k) with any other subgroup of individual’s (of size l) average income, k and l are fixed but arbitrary, k + l = m < n.

The Motivation for the Present Work An important question came up

A New Class of Initial Kernels

Further Thoughts