A Note on the Construction of a Sample Variance

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