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.
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Mukhopadhyay, N., Chattopadhyay, B. On a new interpretation of the sample variance. Stat Papers 54, 827–837 (2013). https://doi.org/10.1007/s00362-012-0465-y
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DOI: https://doi.org/10.1007/s00362-012-0465-y
Keywords
- Actuarial mathematics
- Economic theory
- Gini’s mean difference
- Mathematical finance
- Sample variance
- U statistics