Abstract: Herein strong consistency of recursive extended least squares is established under considerably weaker assumptions than previously assumed in the literature. The argument used to establish consistency also leads to certain basic properties of adaptive predictors based on these recursive estimators. Making use of these properties of the adaptive predictors, simple modifications of the Astrom-Wittenmark self-tuning regulator are proposed and shown to be asymptotically optimal.

Abstract: This paper introduces some algorithms to solve crash-failure, failure-by-omission and Byzantine failure versions of the Byzantine Generals or consensus problem, where non-faulty processors need only arrive at values that are close together rather than identical. For each failure model and each value ofS, we give at-resilient algorithm usingS rounds of communication. IfS=t+1, exact agreement is obtained. In the algorithms for the failure-by-omission and Byzantine failure models, each processor attempts to identify the faulty processors and corrects values transmited by them to reduce the amount of disagreement. We also prove lower bounds for each model, to show that each of our algorithms has a convergence rate that is asymptotic to the best possible in that model as the number of processors increases.

Abstract: : Some selection rules based on monotone empirical Bayes estimators of the binomial parameters are proposed. First, it is shown that, under the squared error loss, the Bayes risks of the proposed monotone empirical Bayes estimators converge to the related minimum Bayes risks with rates of convergence at least of order 0(nsub -n), where n is the number of accumulated past experiences at hand. Further, for the selection problem, the rates of convergence of the proposed selection rules are shown to be at least of order 0(exp(-cn)) for some c 0. Keywords: Asymptotically optimal.

Abstract: This article shows that the likelihood ratio test for the equality of location parameters of k (k > 2) two-parameter exponential distributions with unequal scale parameters, in general, depends on the unknown scale parameters under the null hypothesis. A modification of the likelihood ratio test is suggested, which is applicable to Type II censored or complete data. The suggested test is consistent and asymptotically optimal in the sense of Bahadur efficiency. For k = 2, the power function of the test is obtained, and the test is shown to be unbiased. A set of data is analyzed.

Abstract: A method is proposed for testing normality, in the case of general Type II censored data—that is, data for which only a subset of the order statistics are available. Three test statistics are proposed, which are generalizations of the statistics proposed in LaRiccia (1986), and have many of the same properties. Specifically, they are designed to be asymptotically optimal with respect to specific alternatives and are easily adjusted to be asymptotically optimal with respect to many other types of alternatives. Under the null hypothesis, irrespective of the type or amount of censoring, the proposed test statistics are asymptotically distributed as chi-squared random variables. Further, results of a simulation study are presented, indicating that these statistics converge quite rapidly in distribution to the appropriate chi-squared random variables and that the asymptotic critical values provide a useful approximation to the small sample critical values even for n = 25. The results of a simulation s...

Abstract: Lower bounds for sorting on mesh-connected arrays of processors are presented For sorting N = n1n2nr elements on an n1 × n2 × × nr array 2(n1 + + nr−1) + nr data interchange steps are needed asymptotically For two dimensions these bounds are asymptotically best possible provided that n1 and n2 are powers of 2 In this case the generalized s2-way merge sort of Thompson and Kung turns out to be asymptotically optimal The minimal asymptotic bound of 2√2N interchange steps can be obtained only by sorting algorithms suitable for √N/2 × √2N meshes For r ≥ 3 dimensions an analysis of aspect-ratios also indicates that there might be mesh-connected architectures which are better suited for sorting than simple r-dimensional cubes

Abstract: For a class of stationary Gaussian sources and the squared-error distortion measure, an asymptotically optimal tree coding scheme is derived using tree codes with finite branch length The distribution of the reproduction process is derived in an explicit form, and using the random coding argument and results from the theory of branching processes with stationary ergodic environmental processes, the source coding theorem is proved The theorem is applicable to all Gaussian sources with a power spectrum that satisfies the Lip 1 condition with no restriction on the coding rate

Abstract: Snedecor and Cochran (1967, p. 324) observed that an application of different treatments to otherwise homogeneous experimental units often results in groups that are different not only in means but also in variances. The usual one-way classification procedure assumes a priori the homogeneity of variances among different groups and tests the equality of means only. Thus motivated by the problem of simultaneously testing the equality of means and the equality of variances of several normal populations, I suggest in this article a simple and an asymptotically optimal test. The suggested test is based on the combination of two independent tests, namely (a) the test for the equality of means given that all variances are equal but unspecified and (b) the test for the equality of variances when all means, not necessarily all equal, are unspecified. Tests (a) and (b) are combined by the Fisher method (1950, p. 99). Littell and Folks (1973) showed that the Fisher method of combining two or more independen...

Abstract: SUMMARY Incomplete block designs for estimation of treatment-control differences are studied. A class of designs for this purpose, called balanced treatment incomplete block designs, has recently been proposed and studied by several authors. Here, a relatively small subclass of these designs is shown to contain the asymptotically optimal design under the criterion of minimizing the length of simultaneous confidence intervals for treatmentcontrol differences. These results can be used to easily construct good large designs. Some specific finite cases are considered.

Abstract: In this paper we consider the problem of spectral estimation through ARMA modeling of stationary time series witch missing observations. We consider estimators based on the sample covariances, and present an asymptotically optimal estimator among this group. The algorithm is based on a nonlinear least squares fit of the sample covariances computed from the data with missing observations to the true covariances of the assumed ARMA model. The statistical properties of the algorithm are shown to be asymptotically optimal. The performance of the algorithm is illustrated by a numerical example.

Abstract: Etude des schemas d'echantillonnage des fonctions periodiques en vue de la meilleure approximation

Abstract: Multistage point estimation, with a loss function that includes a cost for each stage of sampling, as well as a cost for each observation, is considered. It is shown that there exists an optimal (Bayes) policy when the loss function and prior satisfy certain mild conditions. For the case when the loss consists of squared error, fixed cost per observation and fixed cost per stage, the observations have density belonging to a one–parameter exponential family, the natural parameter has a conjugate prior distribution, and the estimand is the (conditional) mean of the observations, a one–stage look–ahead policy is proposed. Its asymptotic performance relative to the optimal policy is evaluated for the exponential and (two–para–meter) normal cases.

Abstract: This paper deals with the problems of constructing iterative methods with Chebyshev parameters for solving operator equations and eigenvalue problems. Asymptotically optimal methods of two types are analysed. Methods of the first type take into account the a priori information on the distribution of the initial error over the operator spectrum. Methods of the second type are employed when information on generalized eigenvectors is available. Iterative methods with Chebyshev parameters [14] for solving operator equations and eigenvalue problems with operator A have maximum effectiveness when: (a) operator A has a real spectrum and the bounds of the spectrum are known; (b) operator A has a complete system of eigenvectors {φη}; (c) optimization is carried out on a sufficiently wide class of initial error vectors ε° = £e°(/>„, |ε°| < c\\ (d) the norm of error vector π ε = Σ^Φη at the Nth iteration step is given by the formula ||ε|| = max|fi*|. A sufficiently wide class of problems can, however, be indicated for which these conditions are either violated (operator A has generalized eigenvectors in addition to eigenvectors) or too crude: the class of initial errors has a certain smoothness which corresponds to some law governing the decrease of the coefficients ε? as w-> oo. The main iterative process is often preceded by an algorithm for finding the required bounds of the spectrum. For example, one may begin by looking for the first eigenvalue and for the bounds of the remaining eigenvalues, and then use this information for finding the eigenvector [12]. If the result of iterations of the preceding algorithm is taken for the initial guess, the initial error has a specific spectral distribution. In this paper we construct two types of asymptotically optimal iterative methods. Methods of the first type take into account the spectral distribution of the coefficients ε°, assuming the system of eigenvectors {φη} of operator A to be complete. Methods of the second type may be used when information on generalized eigenvectors is available, assuming |ε°| B be an invertible linear operator, its spectrum Sp (A) being a bounded set such that Οφ5ρ(Α). Let A have a system {φη} of eigenvectors corresponding to eigenvalues /lneSp(>4), and let {φη} form a basis in B. For solving equation (1.1), consider two iterative methods (1.2) k(u-u~) fc = 0, 1,2,..., Ν (1.3) where αί? af and bf are parameters. Let ε = u — u. Then ε = ΡΝ(Λ)ε (1.4) where is a polynomial of degree Ν in a real variable f; ̂ are calculated from the parameters of method (1.2) or (1.3); e.g. in the case of (1.2) we find yt = a,·. Let the initial error have the form ε° = Σε?φπ; then η λη)ε°ηΦη· (1-6) Methods of improving the rate of convergence of iterations are based on algorithms which minimize max \\ΡΝ(λη)ε°\\ on an appropriate class of initial errors. Optimal η algorithms were obtained in the case of |ε°| < c, for Sp (A) belonging either to one closed interval of the real axis [4, 14] or to a system of closed intervals [10, 14]. In the present paper we construct optimal algorithms which enable us to take into account the a priori information on the spectral distribution of the initial error, information on the smoothness of the initial error was taken into account in the methods analysed in [1 1]. Let Ω be a closed finite domain in the complex plane, Sp (A) c Ω, 0£Ω; p(t) ^ 0 is a continuous function on Ω. Let us formulate the problem of optimization of the methods (1.2), (1.3). Problem. Given a subsequence of positive integers, find polynomials PN(t) of form (1.5) for which the quantity μΝ(ΡΝ9 Ω) = inf max \\PN(t)p(t)\\ (1.7) Vi is attained. When choosing p(t\\ we take into account various spectral characteristics of initial error vectors, their smoothness, the norm in the spaces in which it is desirable to estimate ε and ε°, and how ε° is affected by the computational algorithms which precede the application of methods (1.2) and (1.3) (e.g. iterations for determining the spectrum bounds. In the general case, solving problem (1.7) is a complicated task. In the case p(t) = 1, generalization of the results of [10] yields Asymptotically optimal iterative methods 279 Theorem 1.1. Let Ω consist of m simply connected disjoint components symmetrical with respect to the real axis. Let there exist a polynomial Qm(t) of degree m with real coefficients, mapping Ω onto a domain bounded by an ellipse Bd with foci at the points ± 1 and semi-axes ratio d < 1. If Ν = ml (I = 1, 2, . . .), the solution to problem (1.7) is a polynomial

Abstract: In this paper the author gives nice assumptions (i.e. easy to verify) to obtain asymptotically optimal test for the problem of testing hypothesis of a real parameter

Abstract: In a paper of Knuth and Yo [Kibern. Sb.,19 (1983)] the question is posed which “intermediate” model of DG-algorithms (transforming uniformly distributed random bits into random variables with arbitrary distributions) best corresponds to the practically important distributions. The goal of the present paper is the study of one such model for absolutely continuous distributions, which apparently includes both finite and general DG-algorithms.

Abstract: We consider discrete-time stochastic control systems in which the disturbance process is a sequence of iid random elements with unknown distribution We introduce three asymptotically optimal adaptive policies and for each of them we obtain almost surely uniform approximations

Abstract: The problem of detecting a change from one given stationary and ergodic stochastic process to another such process is considered. It is assumed that both stochastic processes are processes with memory and that they are mutually independent. A sequential test is proposed and analyzed. It is proved that the proposed test is asymptotically optimal in a mathematically precise sense.

Abstract: Time discretisations of teh vector stochastic differential equation are considered, where (y t) is a continuous scalar process whose distribution is absolutely continuous with respect to Wiener measure. Among approximations to x T that depend on (y t) only at the discretisation points, the conditional mean is asymptotically optimal in the sense that it minimises all symmetrical conditional moments of the error. A conditional central limit theorem is derived for this minimal error, and a finite difference formula is developed which yields appromimations with the same asymptotically optimal properties. This formula is necessarily more complex than the familiar Milshtein scheme (Milshtein [7]). The latter has the maximum order of convergence but its error, considered as a power serioes in the discretisation parameter h, does not have the minimal leading coefficient. The results generalise for a special class of equations with multi-dimensional forcing terms

Abstract: The article examines the properties of generalized method of moments GMM estimators of utility function parameters. The research strategy is to apply the GMM procedure to generated data on asset returns from stochastic exchange economies; discrete methods and Markov chain models are used to approximate the solutions to the integral equations for the asset prices. The findings are as follows: (a) There is variance/bias trade-off regarding the number of lags used to form instruments; with short lags, the estimates of utility function parameters are nearly asymptotically optimal, but with longer lags the estimates concentrate around biased values and confidence intervals become misleading, (b) The test of the overidentifying restrictions performs well in small samples; if anything, the test is biased toward acceptance of the null hypothesis.

Abstract: : This paper makes two important contributions to the theory of bandwidth selection for kernel density estimators under right censorship. First, an asymptotic representation of the integrated squared error into easily understood variance and squared bias components is given. Second, it is shown that if the bandwidth is chosen by the data-based method of least squares cross-validation, then it is asymptotically optimal in a compelling sense. A by-product of the first part is an interesting comparison of the two most popular kernel estimators. Keywords: Nonparametric density estimation; Smoothing parameter.