Asymptotically minimax tests of composite hypotheses

Summary X ₁,,X> _n are independent, identically distributed random variables with common density function f(x¦ ₁ ,, _k , _k+1 ), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that _k+1 =b against the alternative that _k+1 =b+cn ^–1/2 . ₁ ,, _k are nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦ ₁ -a _i ¦n ^–1/2 M(n) for i=1,,k, where a ₁ , ,a _k are known, and M(n) approaches infinity as n increases but n ^–1/2 M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to _k+1 =b, and weight 1-A to _k+1 =b+cn ^–1/2 , in each case the weight being spread uniformly over the possible values of ₁ ,, _k in the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of ₁ ,..., _k are substituted for a ₁ ..., a _k respectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a ₁ ..., a _k are.Research supported by the U.S. Air Force under Grant AF-AFOSR-68-1472.     

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 testing ofr>2 simple hypotheses

In this paper, we consider the problem of asymptotically minimax testing ofr≥2 simple hypotheses when a general stochastic process is observed. We establish general conditions for the exponential decrease of maximal probability errors of minimax tests as the number of observations increases. At the present time, similar results for testing several multinomial schemes were obtained by Salihov [8]. Similar results for testing two simple hypotheses were obtained in [5]. In the proofs of the main results, we use the theory of large deviations ([3], [2]). In Sec. 1, the main result is proved. In Secs. 2–4, we analyze the i.i.d. case, nonhomogeneous Poisson processes, and renewal processes as examples. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 313–320, July–September, 2000.     

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 distribution-free aligned rank order tests for composite hypotheses for general multivariate linear models

For general multivariate linear models, a composite hypothesis does not usually induce invariance of the joint distribution under appropriate groups of transformations, so that genuinely distribution-free tests do not usually exist. For this purpose, some aligned rank order statistics are incorporated in the proposal and study of a class of asymptotically distribution-free tests. Tests for the parallelism of several multiple regression surfaces are also considered. Finally the optimal properties of these tests are discussed.Work supported by the Air Force Office of Scientific Research, AFSC, USAF, Contract Nos: AFOSR-74-2736 and AFOSR-76-2927. Reproduction in whole or part is permitted for any purpose of the U.S. Government     

On optimal asymptotic tests of composite hypotheses with several constraints

Summary A locally asymptotically most powerful test for composite hypotheses with t independent linear constraints on the parameters has been developed for the case where the observed random variables {X _nk , k=1,2,...,n} are independently but not necessarily identically distributed. However, their distributions depend on two vector parameters, one = ( ₁, ₂, ..., _t) being under test, and the other = ( ₁, ₂, ..., _s) being the nuisance parameter.This investigation was supported (in part) by a research grant (No. GM-10525(2)) from the National Institutes of Health, Public Health Service.On leave from UniversitÄt Heidelberg, Germany, and supported by a NATO research scholarship.     

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.     

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.     

On minimax robust testing of composite hypotheses on Poisson process intensity

Statistical Inference for Stochastic Processes - The problem on the minimax testing of a Poisson process intensity is considered. For a given disjoint sets $${{\mathcal {S}}}_T$$ and $${{\mathcal...     

Asymptotically optimal sequential estimation of regular functionals of several distributions based on generalized U-statistics

For the several sample problem, a vector of estimable parameters is considered. For a fixed total sample size, a multistage (sequential) procedure based on generalized U-statistics is developed for choosing a partition of this sample size into individual sample size for which the generalized variance of the estimator of the parameter vector is asymptotically minimized.     

Asymptotically optimal cubature weight formulas

Krasnoyarsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 2, pp. 151–160, March–April, 1989.     

Asymptotically optimal open-loop load balancing

In many distributed computing systems, stochastically arriving jobs need to be assigned to servers with the objective of minimizing waiting times. Many existing dispatching algorithms are basically included in the SQ(d) framework: Upon arrival of a job, \(d\ge 2\) servers are contacted uniformly at random to retrieve their state and then the job is routed to a server in the best observed state. One practical issue in this type of algorithm is that server states may not be observable, depending on the underlying architecture. In this paper, we investigate the assignment problem in the open-loop setting where no feedback information can flow dynamically from the queues back to the controller, i.e., the queues are unobservable. This is an intractable problem, and unless particular cases are considered, the structure of an optimal policy is not known. Under mild assumptions and in a heavy-traffic many-server limiting regime, our main result proves the optimality of a subset of deterministic and periodic policies within a wide set of (open-loop) policies that can be randomized or deterministic and can be dependent on the arrival process at the controller. The limiting value of the scaled stationary mean waiting time achieved by any policy in our subset provides a simple approximation for the optimal system performance.     

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     

Stability of testing hypotheses

The stability of testing hypotheses is discussed. Differing from the usual tests measured by Neyman-Pearson lemma, the regret and correction of the tests are considered. After the decision is made based on the observationsX ₁,X ₂, ⋅⋅⋅,X _n, one more piece of datumX _n+1 is picked and the test is done again in the same way but based onX ₁,X ₂, ⋅⋅⋅,X _n,X _n+l There are three situations: (i) The previous decision is right but the new decision is wrong; (ii) the previous decision is wrong but the new decision is right; (iii) both of them are right or both of them are wrong. Of course, it is desired that the probability of the occurrence of (i) is as small as possible and the probability of the occurrence of (ii) is as large as possible. Since the sample size is sometimes not chosen very precisely after the type I error and the type II error are determined in practice, it seems more urgent to consider the above problem. Some optimal plans are also given. Project supported by the National Natural Science Foundation of China and the Doctoral Programme Foundation.     

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.     

Criteria for testing nonparametric hypotheses for random processes

A criterion is constructed for testing the hypothesis on the homogeneity of the distributions of two random processes and some other hypotheses, consistent in a large class of alternatives.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 8–13, 1990.     

A characterization of a minimax test in the problem of testing two composite hypotheses

We obtain a characterization of a minimax test in the problem of testing two composite hypotheses via a solution of a dual problem. In comparison to previous results, our characterization is valid under much weaker assumptions, which became possible, in particular, due to a modification of a dual problem.