On a new interpretation of the sample variance

摘要

It may not be an overstatement that one of the most widely reported measures of variation involves S~ 2, the sample variance, which is also well-known to be alternatively expressed in the form of an U statistic with a symmetric kernel of degree 2 whatever be the population distribution function. We propose a very general new approach to construct unbiased estimators of a population variance by U statistics with symmetric kernels of degree higher than two. Surprisingly, all such estimators ultimately reduce to S~ 2 (Theorem 3.1). While Theorem 3.1 is interesting and novel in its own right, it leads to a newer interpretation of S~ 2 that is much broader than what is known in the statistical literature including economics, actuarial mathematics, and mathematical finance.