We propose a geometric approach for bounding average stopping times for stopped random walks in discrete and continuous time. We consider stopping times in the hyperspace of time indexes and stochastic processes. Our techniques relies on exploring ge
ometric properties of continuity or stopping regions. Especially, we make use of the concepts of convex sets and supporting hyperplane. Explicit formulae and efficiently computable bounds are obtained for average stopping times. Our techniques can be applied to bound average stopping times involving random vectors, nonlinear stopping boundary, and constraints of time indexes. Moreover, we establish a stochastic characteristic of convex sets and generalize Jensens inequality, Walds equations and Lordens inequality, which are useful for investigating average stopping times.
We explore the applications of our previously established likelihood-ratio method for deriving concentration inequalities for a wide variety of univariate and multivariate distributions. New concentration inequalities for various distributions are de
veloped without the idea of minimizing moment generating functions.
We propose new generalized multivariate hypergeometric distributions, which extremely resemble the classical multivariate hypergeometric distributions. The proposed distributions are derived based on an urn model approach. In contrast to existing met
hods, this approach does not involve hypergeometric series.
We derive simple concentration inequalities for bounded random vectors, which generalize Hoeffdings inequalities for bounded scalar random variables. As applications, we apply the general results to multinomial and Dirichlet distributions to obtain multivariate concentration inequalities.
We propose a new approach for deriving probabilistic inequalities based on bounding likelihood ratios. We demonstrate that this approach is more general and powerful than the classical method frequently used for deriving concentration inequalities su
ch as Chernoff bounds. We discover that the proposed approach is inherently related to statistical concepts such as monotone likelihood ratio, maximum likelihood, and the method of moments for parameter estimation. A connection between the proposed approach and the large deviation theory is also established. We show that, without using moment generating functions, tightest possible concentration inequalities may be readily derived by the proposed approach. We have derived new concentration inequalities using the proposed approach, which cannot be obtained by the classical approach based on moment generating functions.
A large class of problems in sciences and engineering can be formulated as the general problem of constructing random intervals with pre-specified coverage probabilities for the mean. Wee propose a general approach for statistical inference of mean v
alues based on accumulated observational data. We show that the construction of such random intervals can be accomplished by comparing the endpoints of random intervals with confidence sequences for the mean. Asymptotic results are obtained for such sequential methods.
We first review existing sequential methods for estimating a binomial proportion. Afterward, we propose a new family of group sequential sampling schemes for estimating a binomial proportion with prescribed margin of error and confidence level. In pa
rticular, we establish the uniform controllability of coverage probability and the asymptotic optimality for such a family of sampling schemes. Our theoretical results establish the possibility that the parameters of this family of sampling schemes can be determined so that the prescribed level of confidence is guaranteed with little waste of samples. Analytic bounds for the cumulative distribution functions and expectations of sample numbers are derived. Moreover, we discuss the inherent connection of various sampling schemes. Numerical issues are addressed for improving the accuracy and efficiency of computation. Computational experiments are conducted for comparing sampling schemes. Illustrative examples are given for applications in clinical trials.
In this paper, we develop a general approach for probabilistic estimation and optimization. An explicit formula and a computational approach are established for controlling the reliability of probabilistic estimation based on a mixed criterion of abs
olute and relative errors. By employing the Chernoff-Hoeffding bound and the concept of sampling, the minimization of a probabilistic function is transformed into an optimization problem amenable for gradient descendent algorithms.
In this article, we derive a new generalization of Chebyshev inequality for random vectors. We demonstrate that the new generalization is much less conservative than the classical generalization.
In this paper, we have developed new multistage tests which guarantee prescribed level of power and are more efficient than previous tests in terms of average sampling number and the number of sampling operations. Without truncation, the maximum samp
ling numbers of our testing plans are absolutely bounded. Based on geometrical arguments, we have derived extremely tight bounds for the operating characteristic function. To reduce the computational complexity for the relevant integrals, we propose adaptive scanning algorithms which are not only useful for present hypothesis testing problem but also for other problem areas.
