This is a very interesting paper reviewing the technique for testing semiparametric hypotheses using GLR tests. I’d like to supplement Fan and Jiang’s review with some cautions and a somewhat different point of view. 1 The Wilks phenomenon Rigorous results for smooth parametric models, see, for example, Bickel and Doksum (2005, Chap. 6) do say that, if θ̂ , η̂ or, equivalently, (θ̂η̂, η̂) are MLE’s,...

The generalized likelihood ratio test (GLRT), which is commonly used in composite hypothesis testing problems, is investigated. Conditions for asymptotic optimality of the GLRT in the NeymanPearson sense are studied and discussed. First, a general necessary and sufficient condition is established, and then based on this, a sufficient condition, which is easier to verify, is derived. A counterex...

We are very grateful to the Editors, Maria Angeles Gil and Leandro Pardo, for organizing this stimulating discussion. We would like to take this opportunity to thank all discussants for their insightful and constructive comments regarding our paper, opening new avenues for the GLR tests. They have made valuable contributions to the understanding of various testing problems. As stressed in our p...

We propose the degenerate-generalized likelihood ratio test DGLRT for one-sided composite hypotheses in cases of independent and dependent observations. The theoretical results show that the DGLRT has controlled error probabilities and stops sampling with probability 1 under some regularity conditions. Moreover, its stopping boundaries are constants and can be easily determined using the provid...

We propose an algorithm for simultaneously detecting and locating changepoints in a time series, and a framework for predicting the distribution of the next point in the series. The kernel of the algorithm is a system of equations that computes, for each index i, the probability that the last (most recent) change point occurred at i. We evaluate this algorithm by applying it to the change point...

Data collected from response adaptive designs are dependent. Traditional statistical methods need to be justified for the use in response adaptive designs. This paper generalizes the Rao’s score test to response adaptive designs and introduces a generalized score statistic. Simulation is conducted to compare the statistical powers of the Wald, the score, the generalized score and the likelihood...

In this paper, we consider the problem of testing two separate families of hypotheses via a generalization of the sequential probability ratio test. In particular, the generalized likelihood ratio statistic is considered and the stopping rule is the first boundary crossing of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense tha...