Asymptotic null distribution of person fit statistics with estimated person parameter

Person fit statistics are considered for dichotomous item response models. The asymptotic null distribution is derived for statistics which are linear in the item responses, and in which the ability parameter is replaced by an estimate. This allows the asymptotically correct standardization of linear person fit statistics with estimated ability parameter. The fact that the ability parameter is estimated usually decreases the asymptotic variance.I am indebted to Herbert Hoijtink and three anonymous referees for their comments on a previous version.