It may not be an overstatement that one of the most widely reported measures of variation involves S2, 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 S2 (Theorem 3.1). While Theorem 3.1 is interesting and novel in its own right, it leads to a newer interpretation of S2 that is much broader than what is known in the statistical literature including economics, actuarial mathematics, and mathematical finance.
Actuarial mathematicsEconomic theoryGini’s mean differenceMathematical financeSample varianceU statistics