Asymptotic Normality of the Minimum Non-Hilbertian Distance Estimators for a Diffusion Process with Small Noise

A one-dimensional diffusion type process with small noise is observed up to the time T. It depends on an unknown real parameter. Some minimum distance estimators of this parameter are considered. These estimators are defined using the L ^p-metric or the uniform metric. The limiting distribution of the normalizing minimum distance estimators (as the noise vanishing) is known to be the distribution of a random variable. The distribution of this random variable is studied as the time T goes to the infinity. We will prove under some conditions that it has a limiting Gaussian law. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic properties in semiparametric partially linear regression models for functional data

We consider the semiparametric partially linear regression models with mean function XTβ + g(z), where X and z are functional data. The new estimators of β and g(z) are presented and some asymptotic results are given. The strong convergence rates of the proposed estimators are obtained. In our estimation, the observation number of each subject will be completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.     

On an estimator in T3-class of linear estimators in sampling with varying probabilities from a finite population

Summary Horvitz and Thompson [4] introduced three classes of linear estimators for estimation of population characteristics on the basis of a sample drawn with varying probabilities and without replacement. TheirT ₃-class of estimators does not admit a best unbiased estimator. In this paper, the variance and an unbiased estimate of variance for an estimator in T₃-class, which is proved to have several good properties by Godambe [2], [3], are derived for sampling with varying probabilities with or without replacement.     

Nonparametric estimation of the spectral measure,and associated dependence measures,for multivariate extreme values using a limiting conditional representation

The traditional approach to multivariate extreme values has been through the multivariate extreme value distribution G, characterised by its spectral measure H and associated Pickands’ dependence function A. More generally, for all asymptotically dependent variables, H determines the probability of all multivariate extreme events. When the variables are asymptotically dependent and under the assumption of unit Fréchet margins, several methods exist for the estimation of G, H and A which use variables with radial component exceeding some high threshold. For each of these characteristics, we propose new asymptotically consistent nonparametric estimators which arise from Heffernan and Tawn’s approach to multivariate extremes that conditions on variables with marginal values exceeding some high marginal threshold. The proposed estimators improve on existing estimators in three ways. First, under asymptotic dependence, they give self-consistent estimators of G, H and A; existing estimators are not self-consistent. Second, these existing estimators focus on the bivariate case, whereas our estimators extend easily to describe dependence in the multivariate case. Finally, for asymptotically independent cases, our estimators can model the level of asymptotic independence; whereas existing estimators for the spectral measure treat the variables as either being independent, or asymptotically dependent. For asymptotically dependent bivariate random variables, the new estimators are found to compare favourably with existing estimators, particularly for weak dependence. The method is illustrated with an application to finance data.     

Recursive reconstruction on periodic trees

A periodic tree T_n consists of full n-level copies of a finite tree T. The tree T_n is labeled by random bits. The root label is chosen randomly, and the probability of two adjacent vertices to have the same label is 1−ϵ. This model simulates noisy propagation of a bit from the root, and has significance both in communication theory and in biology. Our aim is to find an algorithm which decides for every set of values of the boundary bits of T, if the root is more probable to be 0 or 1. We want to use this algorithm recursively to reconstruct the value of the root of T_n with a probability bounded away from ½ for all n. In this paper we find for all T, the values of ϵ for which such a reconstruction is possible. We then compare the ϵ values for recursive and nonrecursive algorithms. Finally, we discuss some problems concerning generalizations of this model. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 81–97, 1998     

Estimation of Time Varying Linear Systems

Based on kernel and wavelet estimators of the evolutionary spectrum and cross-spectrum we propose nonlinear wavelet estimators of the time varying coefficients of a linear system, whose input and output are locally stationary processes, in the sense of Dahlhaus (1997). We obtain large sample properties of these estimators, present some simulated examples and derive results on the L ₂-risk for the wavelet threshold estimators, assuming that the coefficients belong to some smoothness class. This revised version was published online in June 2006 with corrections to the Cover Date.     

Estimating the ratio of two scale parameters: a simple approach

We describe a simple approach for estimating the ratio ρ = σ ₂/σ ₁ of the scale parameters of two populations from a decision theoretic point of view. We show that if the loss function satisfies a certain condition, then the estimation of ρ reduces to separately estimating σ ₂ and 1/σ ₁. This implies that the standard estimator of ρ can be improved by just employing an improved estimator of σ ₂ or 1/σ ₁. Moreover, in the case where the loss function is convex in some function of its argument, we prove that such improved estimators of ρ are further dominated by corresponding ones that use all the available data. Using this result, we construct new classes of double-adjustment improved estimators for several well-known convex as well as non-convex loss functions. In particular, Strawderman-type estimators of ρ in general models are given whereas Shinozaki-type estimators of the ratio of two normal variances are briefly treated.     

On Adjusted Viterbi Training

The EM algorithm is a principal tool for parameter estimation in the hidden Markov models, where its efficient implementation is known as the Baum–Welch algorithm. This paper is however motivated by applications where EM is replaced by Viterbi training, or extraction (VT), also known as the Baum–Viterbi algorithm. VT is computationally less intensive and more stable, and has more of an intuitive appeal. However, VT estimators are also biased and inconsistent. Recently, we have proposed elsewhere the adjusted Viterbi training (VA), a new method to alleviate the above imprecision of the VT estimators while preserving the computational advantages of the baseline VT algorithm. The key difference between VA and VT is that asymptotically, the true parameter values are a fixed point of VA (and EM), but not of VT. We have previously studied VA for a special case of Gaussian mixtures, including simulations to illustrate its improved performance. The present work proves the asymptotic fixed point property of VA for general hidden Markov models. J. Lember is supported by Estonian Science Foundation Grant 5694.     

Numerical comparison of nonlinear subgridscale models via adaptive mesh refinement

In this paper, we provide a method of evaluating the efficacy of nonlinear subgridscale models for use in the large eddy simulation of incompressible viscous flow problems. We compare subgridscale “artificial” viscosity models using a posteriori error estimation and adaptive mesh refinement. Specifically, we compare α-Laplacian based subgridscale models and discuss the benefits and limitations of different values of α for some standard benchmark problems for the Navier–Stokes equations.     

Estimation of Cusp Location by Poisson Observations

We consider an inhomogeneous Poisson process X on [0, T]. The intensity function of X is supposed to be regular on [0, T] except at the point , in which it has a singularity (a cusp) of order p. We suppose that we know the shape of the intensity function, but not the location (given by the parameter ) of the point of cusp. We consider the problem of estimation of this location (shift) parameter based on n observations of the process X. We study the maximum likelihood estimator and the Bayesian estimators. We show that these estimators are consistent, their rate of convergence is n ^1/(2p+1), they have different limit distributions, and the Bayesian estimators are asymptotically efficient.     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

IPO estimation of heaviness of the distribution beyond regularly varying tails

Abstract

We introduce a completely novel method for estimation of the parameter which governs the tail behavior of the cumulative distribution function of the observed random variable. We call it Inverse Probabilities for p-Outside values (IPO) estimation method. We show that this approach is applicable for wider class of distributions than the one with regularly varying tails. We demonstrate that IPO method is a valuable competitor to regularly varying tails based estimation methods. Some of the properties of the estimators are derived. The results are illustrated by a convenient simulation study.     

Statistical analysis of stochastic resonance with ergodic diffusion noise

A subthreshold signal is transmitted through a channel and may be detected when some noise--with known structure and proportional to some level--is added to the data. There is an optimal noise level, called stochastic resonance that corresponds to the highest Fisher information in the problem of estimation of the signal. As noise we consider an ergodic diffusion process and the asymptotic is considered as time goes to infinity. We propose consistent estimators of the subthreshold signal and we solve further a problem of hypotheses testing. We also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples.     

Estimator selection with respect to Hellinger-type risks

We observe a random measure N and aim at estimating its intensity s. This statistical framework allows to deal simultaneously with the problems of estimating a density, the marginals of a multivariate distribution, the mean of a random vector with nonnegative components and the intensity of a Poisson process. Our estimation strategy is based on estimator selection. Given a family of estimators of s based on the observation of N, we propose a selection rule, based on N as well, in view of selecting among these. Little assumption is made on the collection of estimators and their dependency with respect to the observation N need not be known. The procedure offers the possibility to deal with various problems among which model selection, convex aggregation and construction of T-estimators as studied recently in Birgé (Ann Inst H Poincaré Probab Stat 42(3):273?C325, 2006). For illustration, we shall consider the problems of estimation, complete variable selection and selection among linear estimators in possibly non-Gaussian regression settings.     

Estimating long-range dependence in the presence of periodicity: An empirical study

Recent results in applied statistics have shown that the presence of periodicity in a time series may have an influence on the estimation of the long memory (long-range dependence) parameter H. In particular, some estimators falsely detect the presence of long-range dependence when periodicity is present. In this paper, we apply various estimation procedures to synthetic periodic time series in order to verify the performance of each estimation method and to determine which estimators should be used when periodicity may be present.     

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.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

Dimension reduction in partly linear error-in-response models with validation data

Consider partial linear models of the form Y=X^τβ+g(T)+e with Y measured with error and both p-variate explanatory X and T measured exactly. Let be the surrogate variable for Y with measurement error. Let primary data set be that containing independent observations on and the validation data set be that containing independent observations on , where the exact observations on Y may be obtained by some expensive or difficult procedures for only a small subset of subjects enrolled in the study. In this paper, without specifying any structure equations and distribution assumption of Y given , a semiparametric dimension reduction technique is employed to obtain estimators of β and g(·) based the least squared method and kernel method with the primary data and validation data. The proposed estimators of β are proved to be asymptotically normal, and the estimator for g(·) is proved to be weakly consistent with an optimal convergent rate.     

Asymptotics for estimation and testing procedures under loss of identifiability

Statistical analyses commonly make use of models that suffer from loss of identifiability. In this paper, we address important issues related to the parameter estimation and hypothesis testing in models with loss of identifiability. That is, there are multiple parameter points corresponding to the same true model. We refer the set of these parameter points to as the set of true parameter values. We consider the case where the set of true parameter values is allowed to be very large or even infinite, some parameter values may lie on the boundary of the parameter space, and the data are not necessarily independently and identically distributed. Our results are applicable to a large class of estimators and their related testing statistics derived from optimizing an objective function such as a likelihood. We examine three specific examples: (i) a finite mixture logistic regression model; (ii) stationary ARMA processes; (iii) general quadratic approximation using Hellinger distance. The applications to these examples demonstrate the applicability of our results in a broad range of difficult statistical problems.     

Shrinkage drift parameter estimation for multi‐factor Ornstein–Uhlenbeck processes

We consider some inference problems concerning the drift parameters of multi‐factors Vasicek model (or multivariate Ornstein–Uhlebeck process). For example, in modeling for interest rates, the Vasicek model asserts that the term structure of interest rate is not just a single process, but rather a superposition of several analogous processes. This motivates us to develop an improved estimation theory for the drift parameters when homogeneity of several parameters may hold. However, the information regarding the equality of these parameters may be imprecise. In this context, we consider Stein‐rule (or shrinkage) estimators that allow us to improve on the performance of the classical maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, their relative dominance is explored and assessed. We illustrate the suggested methods by analyzing interbank interest rates of three European countries. Further, a simulation study illustrates the behavior of the suggested method for observation periods of small and moderate lengths of time. Our analytical and simulation results demonstrate that shrinkage estimators (SEs) provide excellent estimation accuracy and outperform the MLE uniformly. An over‐ridding theme of this paper is that the SEs provide powerful extensions of their classical counterparts. Copyright © 2009 John Wiley & Sons, Ltd.