Asymptotic bias of estimation methods caused by the assumption of false probability distribution

Abstract

Asymptotic bias in large quantiles and moments for four parameter estimation methods, including the maximum likelihood method (MLM), method of moments (MOM), method of L-moments (LMM), and least squares method (LSM), is derived when a probability distribution function (PDF) is falsely assumed. The first three estimation methods are illustrated using the lognormal and gamma distributions forming an alternative set of PDFs. It is shown that for every method when either the gamma or lognormal distribution serves as the true distribution, the relative asymptotic bias (RB) of moments and quantiles corresponding to the upper tail is an increasing function of the true value of the coefficient of variation (cv), except that RB of moments for MOM is zero. The value of RB is the smallest for MOM and largest for MLM. The bias of LMM occupies an intermediate position. The value of RB from MLM is larger for the lognormal distribution as a hypothetical distribution with the gamma distribution being assumed to be the true distribution than it would be in the opposite case. For cv=1 and MLM, it equals 30, 600, 320% for mean, variance and 0.1% quantile, respectively, while for MOM, the moments are asymptotically unbiased and the bias for 0.1% quantile amounts to 35%. An analysis of 39 70-year long annual peak flow series of Polish rivers provides an empirical evidence for the necessity to include bias in evaluation of the efficiency of PDF estimation methods.