Evaluating Statistical Hypotheses Using Weakly‐Identifiable Estimating Functions

Abstract. Many statistical models arising in applications contain non‐ and weakly‐identified parameters. Due to identifiability concerns, tests concerning the parameters of interest may not be able to use conventional theories and it may not be clear how to assess statistical significance. This paper extends the literature by developing a testing procedure that can be used to evaluate hypotheses under non‐ and weakly‐identifiable semiparametric models. The test statistic is constructed from a general estimating function of a finite dimensional parameter model representing the population characteristics of interest, but other characteristics which may be described by infinite dimensional parameters, and viewed as nuisance, are left completely unspecified. We derive the limiting distribution of this statistic and propose theoretically justified resampling approaches to approximate its asymptotic distribution. The methodology's practical utility is illustrated in simulations and an analysis of quality‐of‐life outcomes from a longitudinal study on breast cancer.