Local asymptotic minimax properties of Pitman estimates

Summary It is shown that for Euclidean parameter spaces every sequence of Pitman estimates has local asymptotic minimax properties. The result generalizes a previous result of Hájek, 1972, which has been proved under the condition of local asymptotic normality. In the present paper it is only assumed that a sequence of experiments E _n, n, converges weakly to a translation invariant limit experiment. According to LeCam, 1973 b, this is nearly the most general case which may occur. There are two main results. The first result states that Pitman estimates are minimax for translation invariant experiments. This improves a theorem of Girshick and Savage, 1951, which is restricted to location parameter experiments. In the second result we prove that the distributions of Pitman estimates for E _n, n, converge weakly to the distribution of the Pitman estimate for the limit experiment. This improves previous assertions of this kind due to Ibragimov and Has'minskii, 1973, Inagaki and Ogata, 1975, or Großmann, 1979, since the condition of weak convergence of experiments used here is considerably weaker than the invariance principles for likelihood processes used by these authors.     

Minimax invariant estimator of a continuous distribution function

Consider the problems of the continuous invariant estimation of a distribution function with a wide class of loss functions. It has been conjectured for long that the best invariant estimator is minimax for all sample sizes n1. This conjecture is proved in this short note.Partially supported by National Science Foundation Grant DMS 9001194.     

Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates

k-out-of-n systems frequently appear in applications. They consist of n components of the same kind with independent and identically distributed life-lengths. The life-length of such a system is described by the (n–k+1)-th order statistic in a sample of size n when assuming that remaining components are not affected by failures. Sequential order statistics are introduced as a more flexible model to describe sequential k-out-of-n systems in which the failure of any component possibly influences the other components such that their underlying failure rate is parametrically adjusted with respect to the number of preceding failures. Useful properties of the maximum likelihood estimators of the model parameters are shown, and several tests are proposed to decide whether the new model is the more appropriate one in a given situation. Moreover, for specific distributions, e.g. Weibull distributions, simultaneous maximum likelihood estimation of the model parameters and distribution parameters is considered.     

Values of the Pukánszky invariant in free group factors and the hyperfinite factor

Let AMB(L²(M)) be a maximal abelian self-adjoint subalgebra (masa) in a type II₁ factor M in its standard representation. The abelian von Neumann algebra generated by A and JAJ has a type I commutant which contains the projection onto L²(A). Then decomposes into a direct sum of type I_n algebras for n{1,2,…,∞}, and those n's which occur in the direct sum form a set called the Pukánszky invariant, Puk(A), also denoted Puk_M(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S{∞} when M is both a free group factor and the hyperfinite factor, and where S is an arbitrary subset of . The only previously known values for masas in free group factors were {∞} and {1,∞}, and some values of the form S{∞} are new also for the hyperfinite factor.We also consider a more refined invariant (that we will call the measure-multiplicity invariant), which was considered recently by Neshveyev and Størmer and has been known to experts for a long time. We use the measure-multiplicity invariant to distinguish two masas in a free group factor, both having Pukánszky invariant {n,∞}, for arbitrary .     

Pure Gaussian limit distributions of trigonometric series with bounded gaps

This paper is mainly concerned with the limit distribution of \((\cos 2\pi n_{1}x+\cdots +\cos 2\pi n_{N}x)/\sqrt{N}\) on the unit interval when the increasing sequence {n _k } has bounded gaps, i.e., 1≤n _k+1?n _k =O(1). By Bobkov–Götze [4], it was proved that the limiting variance must be less than 1/2 in this case. They proved that the centered Gaussian distribution with variance 1/4 together with mixtures of Gaussian distributions belonging to a huge class can be limit distributions. In this paper it is proved that any Gaussian distribution with variance less than 1/2 can be a limit distribution.     

Estimation with Sequential Order Statistics from Exponential Distributions

The lifetime of an ordinary k-out-of-n system is described by the (n–k+1)-st order statistic from an iid sample. This set-up is based on the assumption that the failure of any component does not affect the remaining ones. Since this is possibly not fulfilled in technical systems, sequential order statistics have been proposed to model a change of the residual lifetime distribution after the breakdown of some component. We investigate such sequential k-out-of-n systems where the corresponding sequential order statistics, which describe the lifetimes of these systems, are based on one- and two-parameter exponential distributions. Given differently structured systems, we focus on three estimation concepts for the distribution parameters. MLEs, UMVUEs and BLUEs of the location and scale parameters are presented. Several properties of these estimators, such as distributions and consistency, are established. Moreover, we illustrate how two sequential k-out-of-n systems based on exponential distributions can be compared by means of the probability P(X < Y). Since other models of ordered random variables, such as ordinary order statistics, record values and progressive type II censored order statistics can be viewed as sequential order statistics, all the results can be applied to these situations as well.     

Joint distributions of runs in a sequence of higher-order two-state Markov trials

Consider a time homogeneous {0, 1}-valued m-dependent Markov chain . In this paper, we study the joint probability distribution of number of 0-runs of length and number of 1-runs of length in n trials. We study the joint distributions based on five popular counting schemes of runs. The main tool used to obtain the probability generating function of the joint distribution is the conditional probability generating function method. Further a compact method for the evaluation of exact joint distribution is developed. For higher-order two-state Markov chain, these joint distributions are new in the literature of distributions of run statistics. We use these distributions to derive some waiting time distributions.     

Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R ₁, R ₂, ..., R _m ), mn. The results presented in the paper generalize the results given by Joshi (1978, Sankhy Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.     

On the Stability of Eaton’s Characterization by the Properties of Linear Forms

We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics $(X_{1}+\cdots +X_{k_{1}})/k_{1}^{1/\alpha}We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics and have the same distribution as the monomial X ₁, then this monomial has a symmetric stable distribution of order α. The stability estimation in this theorem is investigated in the λ ₀-metric.      

On stochastic orders of absolute value of order statistics in symmetric distributions

Let Y₁,…,Y_n be the order statistics of a simple random sample from a finite or infinite population, having median =M. We compare the variables |Y_j−M| and |Y_m−M|, where Y_m is the sample median, that is, for odd n. The comparison is in terms of the likelihood ratio order, which implies stochastic order as well as other orders. The results were motivated by the study of best invariant and minimax estimators for the k/N quantile of a finite population of size N, with a natural loss function of the type , where F_N is the population distribution function, t is an estimate, and g is an increasing function.     

What portion of the sample makes a partial sum asymptotically stable or normal?

Summary Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<2 be given. We show that if at each stage n a number k _n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k _n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant     

4n − 10

We show that the maximal number K₂(n) of splits in a 2-compatible split system on an n-set is exactly 4n – 10, for every n > 3.Since K₂(n) = CF₃(n)/2 where CF₃(n) is the maximal number of members in any 3-cross-free collection of (proper) subsets of an n-set, this gives a definitive answer to a question raised in 1979 by A. Karzanov who asked whether CF₃(n) is, as a function of n, of type O(n).Karzanovs question was answered first by P. Pevzner in 1987 who showed K₂(n) 6n, a result that was improved by T. Fleiner in 1998 who showed K₂(n) 5n.The argument given in the paper below establishes that the even slightly stronger inequality K₂(n) 4n – 10 holds for every n > 3; the identity K₂(n) = 4n – 10 (n > 3) then follows in conjunction with a result from a previous paper that implies K₂(n) 4n – 10. In that paper, it was also mentioned that—in analogy to well known results regarding maximal weakly compatible split systems—2-compatible split systems of maximal cardinality 4n – 10 should be expected to be cyclic. Luckily, our approach here permits us also to corroborate this expectation. As a consequence, it is now possible to generate all 2-compatible split systems on an n-set (n > 3) that have maximal cardinality 4n – 10 (or, equivalently, all 3-cross-free set systems that have maximal cardinality 8n – 20) in a straight forward, systematic manner.Received March 19, 2003     

Exact functions on manifolds

It is proved that the property of a manifold Mⁿ possessing a smooth function with given numbers of critical points of each index is homotopic invariant if Wh( ₁ (Mⁿ)) = 0 and every Z( ₁ (Mⁿ))-stable free module is free.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 77–83, July, 1970.     

On the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics

Summary We shall disclose a relationship between the almost sure stability of weighted empirical distribution functions and sums of order statistics. First we obtain an extension of a theorem due to Csáki on the almost sure stability of the standardized uniform empirical distribution function. This result is then shown to be an essential tool to derive a characterization of the almost sure stability of the sum of k _nupper order statistics from a sample of n independent observations from a distribution with positive support in the domain of attraction of a non-normal stable law, where 1k _nn and k _n as n.Research performed while the author was at the Catholic University NijmegenResearch supported by the Alexander von Humboldt Foundation while the author was visiting the University of Munich on leave from the University of Delaware     

Remarks on properties of probability distributions determined by conditional moments

Summary In this paper we consider some properties of rotation — invariant distributions onR ⁿ , which are determined by a form of conditional moment of order >0. In particular we prove that the Gaussian distribution is determined uniquely by its conditional moments and we investigate the related question of finiteness of exponential moments. The case of general >0 appears to be more difficult to analyze than the case =2, studied previously by other authors.     

Interrelations Between Various Definitions of the Entropy of Decreasing Sequences of Partitions; Scaling

The present paper is devoted to the study of scaling sequences that occur in the definition of entropy type invariants. The necessity to distinguish nonstandard sequences with zero entropy leads to a generalization of the entropy of decreasing sequences of measurable partitions. A more refined entropy type invariant, the scaling entropy considered by A. M. Vershik is based on the notion of -entropy of a metric space with measure. In the present work it is shown that the scaling entropy is a generalization of the entropy of decreasing sequences if 2 ⁿ is taken as the scaling sequence. The scaling entropy of the partition into the pasts of the (T,T ^-1)-endomorphisms is calculated. Bibliography: 11 titles.     

The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

A ternary quasigroup (or 3‐quasigroup) is a pair (N, q) where N is an n‐set and q(x, y, z) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x)=y. A conjugation of a 3‐quasigroup, considered as an OA(3, 4, n), $({{N}},{\mathcal{B}})$, is a permutation of the coordinate positions applied to the 4‐tuples of ${\mathcal{B}}$. The subgroup of conjugations under which $({{N}},{\mathcal{B}})$ is invariant is called the conjugate invariant subgroup of $({{N}},{\mathcal{B}})$. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010     

On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data

We study the rate of weak convergence of the distributions of the statistics {t _λ (Y), λ ∈ ℝ} from the power divergence family of statistics to the χ ² distribution. The statistics are constructed from n observations of a random variable with three possible values. We show that