Regular Article

Statistical Papers

, Volume 54, Issue 3, pp 827-837

On a new interpretation of the sample variance

  • Nitis MukhopadhyayAffiliated withDepartment of Statistics, University of Connecticut Email author 
  • , Bhargab ChattopadhyayAffiliated withDepartment of Statistics, University of Connecticut

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Abstract

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.

Keywords

Actuarial mathematics Economic theory Gini’s mean difference Mathematical finance Sample variance U statistics

Mathematics Subject Classification (2000)

62G05 62G99 62F10