Abstract: The problem of comparing several experimental treatments to a control arises frequently in clinical trials. Various multi-stage randomized phase II/III designs have been proposed for the purpose of selecting one or more promising experimental treatments and comparing them with a control, while controlling overall Type I and Type II error rates. In this paper, a hybrid selection and testing design for comparing the means of several experimental normal populations among themselves and with the mean of a control normal population is proposed. It is assumed that the variances of the experimental and the control normal populations are unknown and unequal. A Stein-type two-sample selection approach is used at both the selection and testing stages to solve the heteroscedastic problems caused by the unknown variances. The hybrid two-stage design allows for dropping the poorly performing treatments early on the basis of interim analysis results and for early termination if none of the experimental treatments seems promising. Numerical computations are given to show the advantage of the proposed procedure over a pure selection procedure. An example is provided to illustrate the use of the new procedure.

Abstract: The present chapter proposes a portmanteau-type test, based on a sort of likelihood ratio statistic, useful to test general parametric hypotheses inherent to statistical models, which includes the classical portmanteau tests and Whittle-type portmanteau test provided in Chap. 2 as special cases. Sufficient conditions for the statistic to be asymptotically chi-square distributed are elucidated in terms of the Fisher information matrix, and the results have very clear implications for the relationships between the parameter of interest and nuisance parameter. In addition, the power of the test is investigated when local alternative hypotheses are considered. The bias adjustment procedure of the portmanteau-type test is also proposed when the sufficient conditions for asymptotic chi-squared distribution fail to hold, and we relax the conditions in Akashi et al. (2018). Some interesting applications of the proposed test to various problems are illustrated. Since portmanteau tests are widely used in many fields, it appears essential to elucidate the fundamental mechanism in a unified view.

Abstract: In the general risk model (or the Sparre-Andersen model), it is well-known that the following assertion holds: if the claim size is exponentially distributed then the non-ruin probability distribution is a mixture of exponential distributions. In this paper, under some general conditions, we prove that the converse statement of the previous assertion is also true. Besides, we define a new non-ruin measure associated with the aggregate logarithms of the claim-over-profit ratios and obtain a result on Pareto-type distributions.