Estimating a density on the positive half line by the method of orthogonal series

The kernel method of density estimation is not so attractive when the density has its support confined to (0, ∞), particularly when the density is unsmooth at the origin. In this situation the method of orthogonal series is competitive. We consider three essentially different orthogonal series—those based on the even and odd Hermite functions, respectively, and that based on Laguerre functions—and compare them from the point of view of mean integrated square error.     

Strong consistency of density estimation by orthogonal series methods for dependent variables with applications

Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables. The results obtained are then applied to two estimates of the functional Δ(f)=∫f ² (x)dx were strong consistency is established. One of the suggested two estimates of Δ(f) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where they established consistency in the second mean of the estimate. Research supported in part by the National Research Council of Canada and in part by McMaster University Research Board. Now at Memphis State University, Memphis, Tennessee 38152, U.S.A.     

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

Large sample properties of Jaeckel's adaptive trimmed mean

Summary A critical examination of Jaeckel's (1971,Ann. Math. Statist.,42, 1540–1552) study of his adaptive trimmed mean reveals that the theory is not applicable in many important cases, such as when the optimal trimming proportion is close to 0 or 1/2. This region includes the normal and double exponential distributions, among others, which have received considerable attention in the study of other adaptive location estimates. In this paper we obtain results which justify the use of Jaeckel's trimmed mean for a very large class of distributions. By restricting this class we obtain weak and strong rates of convergence which are much faster than those given by Jaeckel.     

Sharp adaptive estimation of quadratic functionals

Estimation of a quadratic functional of a function observed in the Gaussian white noise model is considered. A data-dependent method for choosing the amount of smoothing is given. The method is based on comparing certain quadratic estimators with each other. It is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the “regular” and “irregular” region. We consider l_p bodies and construct bounds for the risk of the estimator which show that for p=4 the estimator is exactly optimal and for example when p ∈[3,100], then the upper bound is at most 1.055 times larger than the lower bound. We show the connection of the estimator to the theory of optimal recovery. The estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators. Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/6-2, CIES, France, and Jenny and AnttiWihuri Foundation.     

Asymptotic properties of nonparametric curve estimates

We consider a class of nonparametric estimators for the regression functionm(t) in the model:y _i =m(t _i ) + _i , 1 i n, t _i [0, 1], which are linear in the observationsy _i . Several limit theorems concerning local and global stochastic and a.s. convergence and limit distributions are given.     

Asymptotic normality of the sample mean and covariances of evanescent fields in noise

We consider the asymptotic properties of the sample mean and the sample covariance sequence of a field composed of the sum of a purely indeterministic and evanescent components. The asymptotic normality of the sample mean and sample covariances is established. A Bartlett-type formula for the asymptotic covariance matrix of the sample covariances of this field, is derived.     

Bootstrap,wild bootstrap,and asymptotic normality

Summary We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.This work has been supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 Stochastische Mathematische Modelle     

Asymptotic normality and Berry-Esseen results for conditional density estimator with censored and dependent data

In this paper we derive the asymptotic normality and a Berry-Esseen type bound for the kernel conditional density estimator proposed in Ould-Saïd and Cai (2005) [26] when the censored observations with multivariate covariates form a stationary α-mixing sequence.     

Asymptotic normality of test statistics under alternative hypotheses

The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics under alternative hypotheses. Second order asymptotics, and their implications for bias corrections, are also discussed in a somewhat informal manner. As an example, asymptotics of test statistics in the analysis of covariance structures are discussed in detail.     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

Integrated mean square properties of density estimation by orthogonal series methods for dependent variables

Summary The rates at which integrated mean square and mean squre errors of nonparametric density estimation by orthogonal series method for sequences of strictly stationary strong mixing random variables are obtained. These rates are better than those known to hold for the independent case and they are shown to hold for Markov processes. In fact our results when specialized to the independent case are improvements over previously known results of Schwartz (1967,Ann. Math. Statist.,38, 1262–1265). An extension of the results to estimation of the bivariate density is also given. Research supported by a faculty summer research grant MS-STAT-42 from the University of Petroleum and Minerals.     

A note on a quadratic measure of deviation of density estimates

Summary The results of Rosenblatt on quadratic measure of deviations of density estimates have been generalized to a wider class of weight functions. It is pointed out that the proof of Theorem 1 of Rosenblatt is incorrect. A corrected version of the proof is also provided.     

High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus

We prove a multivariate CLT for skewness and kurtosis of the wavelets coefficients of a stationary field on the torus. The results are in the framework of the fixed-domain asymptotics, i.e. we refer to observations of a single field which is sampled at higher and higher frequencies. We consider also studentized statistics for the case of an unknown correlation structure. The results are motivated by the analysis of high-frequency financial data or cosmological data sets, with a particular interest towards testing for Gaussianity and isotropy.     

Comparison of two orthogonal series methods of estimating a density and its derivatives on an interval

We compare the merits of two orthogonal series methods of estimating a density and its derivatives on a compact interval—those based on Legendre polynomials, and on trigonometric functions. By examining the rates of convergence of their mean square errors we show that the Legendre polynomial estimators are superior in many respects. However, Legendre polynomial series can be more difficult to construct than trigonometric series, and to overcome this difficulty we show how to modify trigonometric series estimators to make them more competitive.     

On the Haezendonck-Goovaerts risk measure for extreme risks

In this paper, we are interested in the calculation of the Haezendonck-Goovaerts risk measure, which is defined via a convex Young function and a parameter q∈(0,1) representing the confidence level. We mainly focus on the case in which the risk variable follows a distribution function from a max-domain of attraction. For this case, we restrict the Young function to be a power function and we derive exact asymptotics for the Haezendonck-Goovaerts risk measure as q↑1. As a subsidiary, we also consider the case with an exponentially distributed risk variable and a general Young function, and we obtain an analytical expression for the Haezendonck-Goovaerts risk measure.     

Strong laws for density estimators of bernstein type

Starting from the classical theorem of Weierstrass (and its modifications) on approximation of continuous functions by means of Bernstein polynomials a smoothed histogram type estimator is developed for estimating probability densities and its derivatives. Consistency results are obtained in form of various strong laws. In particular, one gets estimates for the rates for pointwise and uniform strong convergence of estimators for the derivatives. Moreover, for approximating the density itself the exact order of consistency is established. This is done by a law of iterated logarithm for pointwise approximation and by a law of logarithm in case of uniform approximation.This paper contains parts of the author's Habilitationsschrift written at the Department of Mathematics of the University of Ulm.     

A bivariate histogram density estimator: Consistency and asymptotic normality

This paper presents a nonparametric histogram density estimator based on the spacings of order statistics. This estimator generalizes to the bivariate case the univariate histogram estimator proposed by Van Ryzin (1973). The first of the two theorems in this paper gives conditions under which the estimator is pointwise strongly consistent. The second theorem provides conditions for the asymptotic normality of the estimator for points at which the density function possesses continuous partial derivatives of second order.