Benzion Boukai

Abstract: The objective of this paper is to derive explicit expressions to distributions of the supremum of Brownian motion processes having a change-point. Such processes are characterized by a drift parameter which is subjected to a change over time. For that purpose, we use the distribution of the supremum of Brownian bridge process, via conditioning arguments. Several different cases are considered and the resulting distributions are illustrated through their p.d.f.'s.

Abstract: Retrospective tests are constructed to detect a local shift of parameter of a distribution function occuring at unknown point of time between consecutive independent observations. Three parametric models of distributions are considered; the one parameter exponential family, the location parameter family and the scale parameter family. The class of tests considered is based on the Generalized Likelihood Ratio (GLR) tests, appropriately adapted for such a change-point problem. Asymptotic techniques are used to obtain the limiting distribution of the test statistics, under both the null hypothesis of no change and the change-point alternative. The test statistics, being maximum likelihood type statistics, are shown to converge (in distribution) to the supremum of Brownian motion process, with or without a change-point according to the alternative and the null hypothesis respectively. Analytical expressions for the asymptotic power functions of the proposed tests are provided. These results are then used to provide power comparisons of the tests with those of the Chernoff-Zacks' quasi-Bayesian test.

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Abstract: The first-exit time process of an inverse Gaussian Levy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail...

Abstract: Five types of change-point problems concerning change in mean, variance, slope, hazard rate, and space-time distribution are briefly reviewed and a list of comprehensive bibliography is provided. Directions for future studies are discussed.

Abstract: The objective of this paper is to derive explicit expressions to distributions of the supremum of Brownian motion processes having a change-point. Such processes are characterized by a drift parameter which is subjected to a change over time. For that purpose, we use the distribution of the supremum of Brownian bridge process, via conditioning arguments. Several different cases are considered and the resulting distributions are illustrated through their p.d.f.'s.

Abstract: We consider the FCFS $GI/GI/n$ queue in the Halfin-Whitt heavy traffic regime, and prove bounds for the steady-state probability of delay (s.s.p.d.) for generally distributed processing times. We prove that there exist $\epsilon_1, \epsilon_2 > 0$, depending on the first three moments of the inter-arrival and processing time distributions, such that the s.s.p.d. is bounded from above by $\exp\big(-\epsilon_1 B^2\big)$ as the associated excess parameter $B \rightarrow \infty$; and by $1 - \epsilon_2 B$ as $B \rightarrow 0$. We also prove that the tail of the steady-state number of idle servers has a Gaussian decay. We provide explicit bounds in all cases, in terms of the first three moments of the inter-arrival and service distributions, and use known results to show that our bounds correctly capture various qualitative scalings. \\\indent Our main proof technique is the derivation of new stochastic comparison bounds for the FCFS $GI/GI/n$ queue, which are of a structural nature, hold for all $n$ and times $t$, and significantly generalize the work of \citet{GG.10c} (e.g. by providing bounds for the queue length to exceed any given level, as opposed to any given level strictly greater than the number of servers as acheived in \citet{GG.10c}). Our results do not follow from simple comparison arguments to e.g. infinite-server systems or loss models, which would in all cases provide bounds in the opposite direction.