Unbiased estimators in the sense of Lehmann and their discrimination rates (II): Multi-parameter cases

Summary The notion ofdiscrimination rate of any unbiased estimator in the sense of Lehmann is, as defined by the author (1982,Ann. Inst. Statist. Math.,34, A, 19–37), extended to multi-parameter cases. This research was supported in part by a Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Science and Culture. The Institute of Statistical Mathematics     

Unbiased estimators of p (X<Y) and their variances in the case of uniform and two-parameter exponential distributions

In this paper, we consider the two-parameter exponential distribution with unknown shift and known scale parameters as well as the uniform distribution on a segment with one end known. For these cases unbiased estimators for (1) are found. We also obtain unbiased estimators for the variances of these unbiased estimators. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 58–63, Perm, 1993.     

Closer estimators of a common mean in the sense of Pitman

Consider the problem of estimating the common mean of two normal populations with different unknown variances. Suppose a random sample of sizem is drawn from the first population and a random sample of sizen is drawn from the second population. The paper gives a family of estimators closer than the sample mean of the first population in the sense of Pitman (1937,Proc. Cambridge Phil. Soc.,33, 212–222). In particular, the Graybill-Deal estimator (1959,Biometrics,15, 543–550) is shown to be closer than each of the sample means ifm5 andn5.     

Consistency of infinitesimal perturbation analysis estimators with rates

We study a class of infinitesimal perturbation analysis (IPA) algorithms for queueing systems with load-dependent service and/or arrival rates. Such IPA algorithms were originally motivated by applications to large queueing systems in conjunction with aggregation algorithms. We prove strong consistency of these estimators through a type of birth and death queue. This work was supported in part by the NSF under Grants Nos. ECS85-15449 and CDR-8803012, by ONR under Contracts Nos. N00014-89-J-0075 and N00014-90-K-1093, and by the US Army under Contract No. DAAL-03-83-K-0171. This paper was written while the author was with the Division of Applied Sciences at Harvard University.     

Optimal rates of convergence in the Weibull model based on kernel-type estimators

Let F be a distribution function in the maximal domain of attraction of the Gumbel distribution such that −log(1−F(x))=x^1/θL(x) for a positive real number θ, called the Weibull tail index, and a slowly varying function L. It is well known that the estimators of θ have a very slow rate of convergence. We establish here a sharp optimality result in the minimax sense, that is when L is treated as an infinite dimensional nuisance parameter belonging to some functional class. We also establish the rate optimal asymptotic property of a data-driven choice of the sample fraction that is used for estimation.     

Sharp rates of convergence of maximum likelihood estimators in nonparametric models

Summary Sharp rates of convergence of maximum likelihood estimators are established in models which are defined by probability densities having bounded derivatives. This result is achieved by making use of local properties of the empirical distribution function.     

On the L 1 convergence of kernel estimators of regression functions with applications in discrimination

Summary An estimate m _n of a regression function m(x)=E{Y|X=x} is weakly (strongly) consistent in L ₁ if ¦m _n (x)-m(x)¦(dx) converges to 0 in probability (w.p. 1) as the sample size grows large ( is the probability measure of X).We show that the well-known kernel estimate (Nadaraya, Watson) and several recursive modifications of it are weakly (strongly) consistent in L ₁ under no conditions on (X, Y) other than the boundedness of Y and the absolute continuity of . No continuity restrictions are put on the density corresponding to . We further notice that several kernel-type discrimination rules are weakly (strongly) Bayes risk consistent whenever X has a density.Research of both authors was sponsored by AFOSR Grant 77-3385     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

A family of minimax rates for density estimators in continuous time

In continuous time, rates of convergence of density estimators fluctuate with the nature of observed sample paths. In this paper, we give a family of rates reached by the kernel estimator and we show that these rates are minimax. Finally, we study applications of these results for specific classes of processes including the Gaussian ones     

New kernel estimators of the hazard ratio and their asymptotic properties

We propose a kernel estimator of a hazard ratio that is based on a modification of Ćwik and Mielniczuk (Commun Stat-Theory Methods 18(8):3057–3069, 1989)’s method. A naive nonparametric estimator is Watson and Leadbetter (Sankhyā: Indian J Stat Ser A 26(1):101–116, 1964)’s one, which is naturally given by the kernel density estimator and the empirical distribution estimator. We compare the asymptotic mean squared error (AMSE) of the hazard estimators, and then, it is shown that the asymptotic variance of the new estimator is usually smaller than that of the naive one. We also discuss bias reduction of the proposed estimator and derived some modified estimators. While the modified estimators do not lose nonnegativity, their AMSE is small both theoretically and numerically.

     

On exponential rates of likelihood ratio estimators for location parameters

In this paper the exponential rates, bounds, and local exponential rates for likelihood ratio estimators are studied. Under certain regularity conditions, a family of likelihood ratio estimators is shown to be admissible in exponential rate. It is also shown that the maximum likelihood estimator is the limit of this family of estimators.     

Strongly consistent estimators of k-th order regression curves and rates of convergence

Summary Let X and Y be two jointly distributed real valued random variables, and let the conditional distribution of X given Y be either in a Lebesgue exponential family or in a discrete exponential family. Let r_k be the k-th order regression curve of Y on X. Let X _n=(X ₁,..., X_n) be a random sample of size n on X. For a subset S of the real line R, statistics based on X_n are exhibited and sufficient conditions are given under which is close to O(n ^–1/2) with probability one. To obtain this result, with uf (u known and f unknown) denoting the unconditional (on y) density of X, the problem of estimating r _k (·) is reduced to the one of estimating f ^(k) (·)/f(·) if the density is wrt the Lebesgue measure on R and f ^(k) is the k-th order derivative of f; and to the one of estimating f(·+k)/f(·) if the density is wrt the counting measure on a countable subset of R.     

Weighted allocations,their concomitant-based estimators,and asymptotics

Annals of the Institute of Statistical Mathematics - Various members of the class of weighted insurance premiums and risk capital allocation rules have been researched from a number of...     

The asymptotic efficiency,in the sense of Bahadur,of estimators of a multidimensional parameter of the spectral density

Institute of Mathematics and Informatics, Lithuanian Academy of Sciences. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), pp. 455–474, July–September, 1991.     

On rates of convergence for sample average approximations in the almost sure sense and in mean

Mathematical Programming - We study the rates at which optimal estimators in the sample average approximation approach converge to their deterministic counterparts in the almost sure sense and in...     

Optimal convergence rates and asymptotic efficiency of point estimators under truncated distribution families

For regular and irregular truncated distribution families, the optimal convergence rates of consistent point estimators have been found and the corresponding asymptotic efficiencies established. Also, it has been justified that commonly used estimators are all efficient. The efficiencies here are compared to the efficiencies of asymptotically median unbiased estimators, providing a lot of counter estimator examples such that those estimators are efficient in the former sense, but not in the latter.     

Self-similar groups in the sense of an iterated function system and their properties

The notion of self-similarity in the sense of iterated function system (IFS) for compact topological groups is given by ?. Koçak in Definition 3. In this work, first we give the definition of strong self-similar group in the sense of IFS. Then, we investigate the main properties of these groups. We also obtain the relations between profinite groups and strong self-similar groups in the sense of IFS. Finally, we construct some examples of these groups.     

Localized and efficient estimators for obstacle problems in the context of standard residual estimators

In order to localize the estimator contributions with respect to the active set of constraints special error measures and residual-type estimators have been derived for different variational inequalities [2–4]. Considering the classical obstacle problem, we summarize the basic ideas and draw a relation to estimators for linear elliptic problems without constraints. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)