Outlier discordancy tests based on saddlepoint approximations

Abstract

When testing for the discordancy of a single observed value, a test based on large values of the maximum absolute studentized residual (MASR) or maximum squared studentized residual (MSSR) is known to be optimal, by maximizing the probability of correctly identifying an outlying value, while controlling the risk of a false identification to α. The exact distribution of MASR and MSSR is not known. In its place, the first Bonferroni bound on the distribution of these statistics is commonly used as an outlier test; see Grubbs (1950). We present new approximations to the distribution of MASR or MSSR, based on saddlepoint approximations of the density of statistics calculated from truncated normal random variables. These approximations are developed in three settings: a one-sample case, univariate regression, and multivariate regression. In comparisons with three versions of Bonferroni bounds and a Monte Carlo simulation, the saddlepoint approximations are shown to perform well in a wide range of situations, especially at larger sample size. The saddlepoint approximations also calculate faster than the improved versions of Bonferroni bounds.