Mis-Specification Analysis Between Normal and Extreme Value Distributions for a Linear Regression Model

Abstract

Normal and extreme value distributions are much alike. They may fit the data at hand well in practical applications, however, their predictions may lead to a significant difference. Recently, Kundu and Manglick (2004) considered the discrimination problem for normal and extreme value distributions for complete data. It is a pity that the impacts of mis-specification between normal and extreme value distributions on the estimation or inference in other statistical models (e.g., in regression analysis) are not studied. The main purpose of the present article is to address this issue. More specifically, for a linear regression model, the essence of this study is to investigate the impacts of mis-specification between normal and extreme value distributions on the estimation of the 100p th percentile of the response associated with a level of the independent variable. Four mean squared errors and two relative impact indexes corresponding to correct specification and mis-specification are computed. The results indicate that for both distributions, the estimation precision is significantly influenced by mis-specification. Surprisingly, this seems to shake violently the well-known assertion that with sufficiently large samples, the estimation for a linear regression model is rather robust even if the normality assumption on the error term is violated. Finally, an example is used to illustrate the proposed method.