Comparing Bayesian and Frequentist Estimators under Asymmetric Loss

Abstract

While many estimation problems involving one or more parameters are treated using a symmetric loss function which gives equal weight to estimation errors that are the same “distance” from the true parameter value, there are clearly problems in which estimation errors in a particular direction are considered more serious than errors in another direction. In univariate problems, it may well be the case that overestimation has potential repercussions that underestimation does not (or, of course, vice versa). For example, Varian (1975) motivated the use of asymmetric loss functions in estimation problems arising in real estate assessment, where the overestimation of a property’s value might cause it to remain on the market unsold for an extended period, ultimately costing the seller inordinate and unnecessary expenses. The estimation of peak water flow in the construction of dams or levies clearly has asymmetric consequences; overestimation might lead to increased construction costs while underestimation might lead to the much more serious consequence of subsequent overflows which can seriously threaten lives and property in adjacent communities. Further examples of contexts requiring the asymmetric treatment of estimation errors are given in papers by Shao and Chow (1991), who treat estimation problems regarding release dates of certain pharmaceutical products, by Thompson and Basu (1996), who treat asymmetric problems arising in reliability and by Zellner and Palm (1974), who considers a variety of problems in the area of econometrics.