Response-adaptive randomization has recently attracted a lot of attention in the literature. In this paper, we propose a new and simple family of response-adaptive randomization procedures that attain the Cramer–Rao lower bounds on the allocation variances for any allocation proportions, including optimal allocation proportions. The allocation probability functions of proposed procedures are discontinuous. The existing large sample theory for adaptive designs relies on Taylor expansions of the allocation probability functions, which do not apply to nondifferentiable cases. In the present paper, we study stopping times of stochastic processes to establish the asymptotic efficiency results. Furthermore, we demonstrate our proposal through examples, simulations and a discussion on the relationship with earlier works, including Efron's biased coin design.

We investigate the asymptotic behavior of a nonparametric M-estimator of a regression function for stationary dependent processes, where the explanatory variables take values in some abstract functional space. Under some regularity conditions, we give the weak and strong consistency of the estimator as well as its asymptotic normality. We also give two examples of functional processes that satisfy the mixing conditions assumed in this paper. Furthermore, a simulated example is presented to examine the finite sample performance of the proposed estimator.

In this paper we obtain some strong laws of large numbers (SLLNs) for arrays of rowwise independent (not necessary identically distributed) random compact sets and fuzzy random sets whose underlying spaces are separable Banach spaces.

Theories of random sets and fuzzy random sets are useful concepts which are frequently applied in scientific areas including information science, probability and statistics. In this paper strong limit theorems are derived for random sets and fuzzy random sets with slowly varying weights in separable Banach spaces. Both independent and dependent cases are covered to provide a wide range of applications.

There has been growing interest in exploiting potential forecast gains from the nonlinear structure of self-exciting threshold autoregressive (SETAR) models. Statistical tests have been proposed in the literature to help analysts check for the presence of SETAR-type nonlinearities in observed time series. However, previous studies show that classical nonlinearity tests are not robust to additive outliers. In practice, time series outliers are not uncommonly encountered. It is important to develop a more robust test for SETAR-type nonlinearity in time series analysis and forecasting. In this paper we propose a new robust nonlinearity test and the asymptotic null distribution of the test statistic is derived. A Monte Carlo experiment is carried out to compare the power of the proposed test with other existing tests under the infl uence of time series outliers. The effects of additive outliers on nonlinearity tests with misspecifi cation of the autoregressive order are also studied. The results indicate that the proposed method is preferable to the classical tests when the observations are contaminated with outliers. Finally, we provide illustrative examples by applying the statistical tests to three real datasets.