Estimation Error and Partial Moments

While partial moments like semivariance, lower and upper partial moments have seen acceptance by both academics and investment professionals, there are some who consider these measures to be ad hoc measures of investment performance. This paper seeks to provide the academic foundation for the use of partial moments as a field of nonparametric statistics. The basic foundation is Chebychev’s inequality. Furthermore, the semivariance exhibits a strong boundary equivalence to Chebychev’s inequality. Estimation error (or sample size sensitivity) has been a problem when estimating inputs to the optimizers used in modern portfolio theory. Using a resampling simulation methodology, the paper demonstrates that the partial moments provide asymptotic behavior similar to the mean and variance for a normal distribution and superior asymptotic behavior to the mean and variance for the chi-square distribution which exhibits a significant skewness and leptokurtotic distribution