Nonparametric change-point estimation for dependent sequences

We present a class of nonparametric change-point estimators for a possibly nonstationary sequence. The estimators are defined using the empirical measures and a semi-norm on the space of measures defined via a family of functions. Using a general setting we prove the rate of $1/n$ convergence in probability. Surprisingly, this optimal rate holds for independent, short-range dependent and long-range dependent sequences. To cite this article: S. Ben Hariz et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Asymptotically optimal Bayesian sequential change detection and identification rules

We study the joint problem of sequential change detection and multiple hypothesis testing. Suppose that the common distribution of a sequence of i.i.d. random variables changes suddenly at some unobservable time to one of finitely many distinct alternatives, and one needs to both detect and identify the change at the earliest possible time. We propose computationally efficient sequential decision rules that are asymptotically either Bayes-optimal or optimal in a Bayesian fixed-error-probability formulation, as the unit detection delay cost or the misdiagnosis and false alarm probabilities go to zero, respectively. Numerical examples are provided to verify the asymptotic optimality and the speed of convergence.     

Asymptotically optimal estimating equation with strongly consistent solutions for longitudinal data

In this article, we introduce a conditional marginal model for longitudinal data, in which the residuals form a martingale difference sequence. This model allows us to consider a rich class of estimating equations which contains several estimating equations proposed in the literature. A particular sequence of estimating equations in this class contains a random matrix R _i−1^*(β) as a replacement for the “true” conditional correlation matrix of the ith individual. Using the approach of [12], we identify some sufficient conditions under which this particular sequence of equations is asymptotically optimal (in our class). In the second part of the article, we identify a second set of conditions under which we prove the existence and strong consistency of a sequence of estimators of β defined as roots of estimation equations which are martingale transforms (in particular, roots of the sequence of asymptotically optimal equations).     

Asymptotically minimax tests of composite hypotheses for nonergodic type processes

Asymptotically efficient tests satisfying a minimax type criterion are derived for testing composite hypotheses involving several parameters in nonergodic type stochastic processes. It is shown, in particular, that the analogue of the usual Neyman's C (α) type test (i.e., the score test) is not efficient for the nonergodic case. Moreover, the likelihood-ratio statistic is not fully efficient for the model discussed in the paper. The efficient statistic derived here is a modified version of the score-statistic discussed previously by Basawa and Koul (1979).     

Asymptotically minimax tests for nonparametric hypotheses concerning the distribution density

Asymptotically minimax goodness-of-fit, as well as symmetry, homogeneity, and independence tests are constructed, when it is known that the norm of the difference of the densities of the hypothesis and the alternative are bounded away from zero by a constant _n and the densities satisfy some smoothness conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 44–53, 1988.     

Asymptotically optimal estimators for moments of change

We consider the problem of finding asymptotically optimal estimators for many moments of change in the case of incomplete information on distributions. We prove that if the maximum-likelihood estimator is asymptotically optimal, then, under certain conditions, it preserves this property after the replacement of actual values by density estimators. We solve the problem for the case of one moment of change and generalize the results obtained to the case of several moments of change. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 406–416, March, 2006.     

A pointwise quasi-Newton method for unconstrained optimal control problems

Summary For a class of unconstrained optimal control problems we propose a quasi-Newton method that exploits the structure of the problem. We define a new type of superlinear convergence for sequences in function spaces and prove superlinear convergence of the iterates generated by the quasi-Newton method in this sense.This author supported by NSF grants # DMS-8300841 and # DMS-8500844     

Constrained minimax approximation and optimal preconditioned for Toeplitz matrices

A good preconditioner is extremely important in order for the conjugate gradients method to converge quickly. In the case of Toeplitz matrices, a number of recent studies were made to relate approximation of functions to good preconditioners. In this paper, we present a new result relating the quality of the Toeplitz preconditionerC for the Toeplitz matrixT to the Chebyshev norm (f– g)/f, wheref and g are the generating functions forT andC, respectively. In particular, the construction of band-Toeplitz preconditioners becomes a linear minimax approximation problem. The case whenf has zeros (but is nonnegative) is especially interesting and the corresponding approximation problem becomes constrained. We show how the Remez algorithm can be modified to handle the constraints. Numerical experiments confirming the theoretical results are presented.     

Asymptotically optimal empirical bayes procedures for selecting good

Let ∏₁,…,∏_k denote k independent populations, where a random observation from population ∏ _i has a uniform distribution over the interval (0,θ _i ) and θ _i is a realization of a random variable having an unknown prior distribution G _i . Population ∏ _i is said to be a good population if θ _i ≥θ₀, where θ₀ is a given, positive number. This paper provides a sequence of empirical Bayes procedures for selecting the good populationsamong ∏₁,…,∏ _k . It is shown that these procedures are asymptotically optimal and that the order of associated convergence rates is O(n^-r/4) for some r, 0<r<2, where n is the number of accumulated past observations

at hand     

Algorithms for the quickest path problem and the reliable quickest path problem

The quickest path problem consists of finding a path in a directed network to transmit a given amount of items from an origin node to a destination node with minimal transmission time, when the transmission time depends on both the traversal times of the arcs, or lead time, and the rates of flow along arcs, or capacity. In telecommunications networks, arcs often also have an associated operational probability of the transmission being fault free. The reliability of a path is defined as the product of the operational probabilities of its arcs. The reliability as well as the transmission time are of interest. In this paper, algorithms are proposed to solve the quickest path problem as well as the problem of identifying the quickest path whose reliability is not lower than a given threshold. The algorithms rely on both the properties of a network which turns the computation of a quickest path into the computation of a shortest path and the fact that the reliability of a path can be evaluated through the reliability of the ordered sequence of its arcs. Other constraints on resources consumed, on the number of arcs of the path, etc. can also be managed with the same algorithms.     

极大极小随机规划逼近问题最优解集和最优值的稳定性

研究了特殊的二层极大极小随机规划逼近收敛问题. 首先将下层初始随机规划最优解集拓展到非单点集情形, 且可行集正则的条件下, 讨论了下层随机规划逼近问题最优解集关于上层决策变量参数的上半收敛性和最优值函数的连续性. 然后把下层随机规划的epsilon-最优解向量函数反馈到上层随机规划的目标函数中, 得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性和最优值的连续性.     

Asymptotically periodic solution and optimal harvesting policy for Gompertz system

In this paper, the single species modelled by (asymptotically) periodic Gompertz equation is investigated. It is shown that the (asymptotically) periodic system has a unique (asymptotically) periodic solution which is globally asymptotically stable for the positive solution. When the nonautonomous Gompertz equation is subject to harvesting, we study the optimal harvesting policy for the periodic system and obtain the corresponding optimal population level and the maximum sustainable yield. Further, when the functions in the exploited Gompertz system are stably bounded functions, we study the ultimately optimal harvesting policy. By choosing the average limiting maximum sustainable yield as management objective, the corresponding optimal population level is determined.     

An algorithm for optimal minimax routing in ATM networks

Asynchronous Transfer Mode (ATM) has been adopted by the CCITT as the transport mode in which Broadband ISDN will be based. In this paper, we formulate the problem of routing cells in an ATM network as an optimization problem. The objective is to minimize the largest cell loss probability among all links. The constraints correspond to a multicommodity network flow problem with gains. An algorithm to determine a global optimal flow assignment is presented. The minimax routing algorithm was implemented and tested on several sample networks. The computational experiments show that the algorithm is computationally efficient.Supported by NSF grant NCR 92-23148.     

Asymptotically optimal dividend policy for regime-switching compound Poisson models

This work develops asymptotically optimal dividend policies to maximize the expected present value of dividends until ruin.Compound Poisson processes with regime switching are used to model the surplus and the switching（a continuous-time controlled Markov chain） represents random environment and other economic conditions.Assuming the switching to be fast varying together with suitable conditions,it is shown that the system has a limit that is an average with respect to the invariant measure of a related Markov chain.Under simple conditions,the optimal policy of the limit dividend strategy is a threshold policy.Using the optimal policy of the limit system as a guide,feedback control for the original surplus is then developed.It is demonstrated that the constructed dividend policy is asymptotically optimal.     

Maximum principle and second-order conditions for minimax problems of optimal control

In this paper, we study the optimal control problem of minimizing the functionalJ(x, u)=max_t1tt2(x(t),t). We formulate and prove necessary optimality conditions for this problem. We establish the equivalence between the initial minimax problem and a problem involving a terminal functional and phase constraints.     

Some results on pointwise second-order necessary conditions for stochastic optimal controls

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.     

Asymptotically optimal tests for parametric functions against ordered functional alternatives

There are hypothesis testing problems for (nonlinear) functions of parameters against functional ordered alternatives for which a reduction to a conventional order-restricted hypothesis testing problem may not be feasible. While such problems can be handled in an asymptotic setup, among the available choices, it is shown that the union-intersection principle may have certain advantages over the likelihood principle or its ramifications. An application to a genomic model is also considered.     

Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regularization parameter are derived.