It is often important to incorporate covariate information in the design of clinical trials. In literature there are many designs of using stratification and covariate-adaptive random- ization to balance certain known covariate. Recently, some covariate-adjusted response-adaptive (CARA) designs have been proposed and their asymptotic properties have been studied (Ann. Statist. 2007). However, these CARA designs usually have high variabilities. In this paper, a new family of covariate-adjusted response-adaptive (CARA) designs is presented. It is shown that the new designs have less variables and therefore are more efficient.
In this paper, we study the compound binomial model in Markovian environment, which is proposed by Cossette, et al. (2003). We obtain the recursive formula of the joint distributions of T,X(T-1) and |X(T)|(i.e., the time of ruin, the surplus before ruin and the deficit at ruin) by the method of mass function of up-crossing zero points, as given by Liu and Zhao (2007). By using the same method, the recursive formula of supremum distribution is obtained. An example is included to illustrate the results of the model.
In this paper,we prove a general law of the iterated logarithm (LIL) for independent non-identically distributed B-valued random variables.As an interesting application,we obtain the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes.
Empirical Euclidean likelihood for general estimating equations for association dependent processes is investigated. The strong consistency and asymptotic normality of the blockwise maximum empirical Euclidean likelihood estimator are presented. We show that it is more efficient than estimator without blocking. The blockwise empirical Euclidean log-likelihood ratio asymptotically follows a chi-square distribution.
CHEN You-you ZHANG Li-xin Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 > 0 and E|X| 2+ε < ∞ for some ε> 0, then for any r > 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).
