Estimation error and partial moments

Abstract

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 Chebyshev's inequality which underlies Markowitz's mean-variance modern portfolio theory. The semivariance exhibits a strong boundary equivalence to Chebyshev's inequality based on a proof provided by Berck and Hihn (1982). From this, we infer that partial moments provides a strong statistical analysis of risk and return for modern portfolio theory without assuming the underlying probability distribution of a security. The same probability statements can be made with partial moments as can be made with the normal distribution.Estimation error (or sample size sensitivity) has been a problem when estimating inputs to the optimizers used in modern portfolio theory. The issue is whether a sample statistic can be calculated that stabilizes asymptotically on the population value. Using a resampling simulation methodology to reduce estimation error, 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. In our study, the normal distribution acts as our control variable.