On the asymptotlc normality for l2-error of wavelei density estimator with application

The l₂ error of linear wavelet estimator of a density from a random sample is shown to be asymptotically normal without imposing smoothing conditions on the density. As an application, the goodness-of-fit test and the approximation of its power for closeness alternatives are considered.     

ON A STRONGLY CONSISTENT WAVELET DENSITY ESTIMATOR FOR THE DECONVOLUTION PROBLEM

ABSTRACT

The problem of wavelet density estimation is studied when the sample observations are contaminated with random noise. In this paper a linear wavelet estimator based on Meyer-type wavelets is shown to be strongly consistent when Fourier transform of random noise has polynomial descent or exponential descent.     

Rates of strong consistencies of the regression function estimator for functional stationary ergodic data

The aim of this paper is to study both the pointwise and uniform consistencies of the kernel regression estimate and to derive also rates of convergence whenever functional stationary ergodic data are considered. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric separable abstract space. While estimating the regression function using the well-known Nadaraya-Watson estimator, we establish the strong pointwise and uniform consistencies with rates. Depending on the Vapnik-Chervonenkis size of the class over which uniformity is considered, the pointwise rate of convergence may be reached in the uniform case. Notice, finally, that the ergodic data framework extends the dependence setting to cases that are not covered by the usual mixing structures.     

Image estimators based on marked bins

The problem of approximating an ‘image’ S?? ^d from a random sample of points is considered. If S is included in a grid of square bins, a plausible estimator of S is defined as the union of the ‘marked’ bins (those containing a sample point). We obtain convergence rates for this estimator and study its performance in the approximation of the border of S. The practical aspects of implementation are discussed, including some technical improvements on the estimator, whose performance is checked through a real data example.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

Wavelet estimation of density for censored data with censoring indicator missing at random

In this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.     

Non-parametric Estimation for the Location of a Change-point in an Otherwise Smooth Hazard Function under Random Censoring

A non-parametric wavelet based estimator is proposed for the location of a change-point in an otherwise smooth hazard function under non-informative random right censoring. The proposed estimator is based on wavelet coefficients differences via an appropriate parametrization of the time-frequency plane. The study of the estimator is facilitated by the strong representation theorem for the Kaplan–Meier estimator established by Lo and Singh (1986). The performance of the estimator is checked via simulations and two real examples conclude the paper.     

Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)^?λ] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.     

Edgeworth expansions for nonparametric distribution estimation with applications

In this paper, we will investigate the nonparametric estimation of the distribution function F of an absolutely continuous random variable. Two methods are analyzed: the first one based on the empirical distribution function, expressed in terms of i.i.d. lattice random variables and, secondly, the kernel method, which involves nonlattice random vectors dependent on the sample size n; this latter procedure produces a smooth distribution estimator that will be explicitly corrected to reduce the effect of bias or variance. For both methods, the non-Studentized and Studentized statistics are considered as well as their bootstrap counterparts and asymptotic expansions are constructed to approximate their distribution functions via the Edgeworth expansion techniques. On this basis, we will obtain confidence intervals for F(x) and state the coverage error order achieved in each case.     

Wavelet estimators for the derivatives of the density function from data contaminated with heteroscedastic measurement errors

Measurement errors occur in many real data applications. In this paper, the linear and the non linear wavelet estimators of the derivatives of the density function are constructed in the case of data contaminated with heteroscedastic measurement errors. We establish L_p risk performance of the estimators and show that they achieve fast convergence rates under quite general conditions.     

Nonparametric Regression with Sample Design Following a Random Process

Nonparametric regression is considered where the sample point placement is not fixed and equispaced, but generated by a random process with rate n. Conditions are found for the random processes that result in optimal rates of convergence for nonparametric regression when using a block thresholded wavelet estimator. Previous results on nonparametric regression via wavelets on both fixed and random sample point placement are shown to be special cases of the general result given here. The estimator is adaptive over a large range of Hölder function spaces and the convergence rate exhibited is an improvement over term-by-term wavelet estimators. Threshold selection is implemented in a data-adaptive fashion, rather than using a fixed threshold as is usually done in block thresholding. This estimator, BlockSure, is compared against fixed-threshold block estimators and the more traditional term-by-term threshold wavelet estimators on several random design schemes via simulations.     

Nonparametric estimation of the derivatives of a density by the method of wavelet for mixing sequences

The problem of estimation of the derivative of a probability density f is considered, using wavelet orthogonal bases. We consider an important kind of dependent random variables, the so-called mixing random variables and investigate the precise asymptotic expression for the mean integrated error of the wavelet estimators. We show that the mean integrated error of the proposed estimator attains the same rate as when the observations are independent, under certain week dependence conditions imposed to the {X _i }, defined in {Ω, N, P}.     

Improved Liu estimator in a linear regression model

In this paper, we mainly aim to introduce the notion of improved Liu estimator (ILE) in the linear regression model y=Xβ+e. The selection of the biasing parameters is investigated under the PRESS criterion and the optimal selection is successfully derived. We make a simulation study to show the performance of ILE compared to the ordinary least squares estimator and the Liu estimator. Finally, the main results are applied to the Hald data.     

Small-area estimation by combining time-series and cross-sectional data

A model involving autocorrelated random effects and sampling errors is proposed for small-area estimation, using both time-series and cross-sectional data. The sampling errors are assumed to have a known block-diagonal covariance matrix. This model is an extension of a well-known model, due to Fay and Herriot (1979), for cross-sectional data. A two-stage estimator of a small-area mean for the current period is obtained under the proposed model with known autocorrelation, by first deriving the best linear unbiased prediction estimator assuming known variance components, and then replacing them with their consistent estimators. Extending the approach of Prasad and Rao (1986, 1990) for the Fay-Herriot model, an estimator of mean squared error (MSE) of the two-stage estimator, correct to a second-order approximation for a small or moderate number of time points, T, and a large number of small areas, m, is obtained. The case of unknown autocorrelation is also considered. Limited simulation results on the efficiency of two-stage estimators and the accuracy of the proposed estimator of MSE are présentés.     

Adaptive wavelet estimation of a function from an m-dependent process with possibly unbounded m

The estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.     

Testing for random effects and serial correlation in spatial autoregressive models

This paper constructs and evaluates tests for random effects and serial correlation in spatial autoregressive panel data models. In these models, ignoring the presence of random effects not only produces misleading inference but inconsistent estimation of the regression coefficients. Two different estimation methods are considered: maximum likelihood and instrumental variables. For each estimator, optimal tests are constructed: Lagrange multiplier in the first case; Neyman's C(α)$C(\alpha )$ in the second. In addition, locally size-robust tests, for individual hypotheses under local misspecification of the unconsidered parameter, are constructed. Extensive Monte Carlo evidence is presented.     

A simulation study: estimation of population mean using two auxiliary variables in stratified random sampling

ABSTRACT

This paper deals with the problem of estimating the finite population mean in stratified random sampling by using two auxiliary variables. This paper proposed a ratio-cum-product exponential type estimator of population mean under different situations: (i) when there is presence of non-response and measurement errors on the study as well as auxiliary variables; (ii) when there is non-response on the study and auxiliary variables but with no measurement error; (iii) when there is complete response on study variable but there is presence of non-response and measurement error on the auxiliary variables and (iv) when there are complete response and measurement error on study as well as auxiliary variables. The expressions of the bias and mean square error of the proposed estimator have been obtained up to the first degree of approximation. The proposed estimator has been compared with usual unbiased estimator, ratio estimator and other existing estimators and the conditions obtained to show the efficacy of the proposed estimator over other considered estimators. Simulation study is carried out to support the theoretical findings.     

Convolution power kernels for density estimation

We propose a new type of non-parametric density estimators fitted to random variables with lower or upper-bounded support. To illustrate the method, we focus on nonnegative random variables. The estimators are constructed using kernels which are densities of empirical means of m i.i.d. nonnegative random variables with expectation 1. The exponent m   plays the role of the bandwidth. We study the pointwise mean square error and propose a pointwise adaptive estimator. The risk of the adaptive estimator satisfies an almost oracle inequality. A noteworthy result is that the adaptive rate is in correspondence with the smoothness properties of the unknown density as a function on (0,+∞)$(0,+\infty )$. The adaptive estimators are illustrated on simulated data. We compare our approach with the classical kernel estimators.     

A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times

Abstract. In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a non‐parametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and an application to biological data are also provided.     

A consistent simulation-based estimator in generalized linear mixed models

We propose a strongly root-n consistent simulation-based estimator for the generalized linear mixed models. This estimator is constructed based on the first two marginal moments of the response variables, and it allows the random effects to have any parametric distribution (not necessarily normal). Consistency and asymptotic normality for the proposed estimator are derived under fairly general regularity conditions. We also demonstrate that this estimator has a bounded influence function and that it is robust against data outliers. A bias correction technique is proposed to reduce the finite sample bias in the estimation of variance components. The methodology is illustrated through an application to the famed seizure count data and some simulation studies.