Consider testingH0 : F ∈ ω0against H1 : F ∈ ω1for a random sampleX1, ..., Xnfrom F, where ω0 andω1 are two disjoint sets of cdfs onℝ = (−∞, ∞). Two non-local types of efficiencies, referred to as thefixed-α and fixed-β efficiencies, are introduced forthis two-hypothesis testing situation. Theoretical tools are developed to evaluate theseefficiencies for some of the most usual goodness of fit tests (including theKolmogorov–Smirnov tests). Numerical comparisons are provided using several examples.

Statistical inference for Markov processes is commonly based on the maximum likelihood method of estimation and the likelihood ratio criterion for testing hypotheses. Construction of estimators and test statistics by these methods require that a model be chosen in the form of a family of transition density functions. In this paper, asymptotic properties of the maximum likelihood estimator and of the likelihood ratio statistic λn are examined when the model chosen for their construction is incorrect—that is, when no density in the model is a density for the transition probability distribution of the Markov process. It is shown that if and λn are constructed from a ‘regular’ incorrect model, then is consistent and asymptotically normally distributed and the asymptotic null distribution of −2 log λn is that of a linear combination of independent chi-squared random variables. These results are applied to propose measures of the performance of the test based on λn when the statistic is constructed from an incorrect model.