Asymptotic normality for wavelet deconvolution density estimators

This current paper shows the asymptotic normality for wavelet deconvolution density estimators, when a density function belongs to some $L^p(\mathbb{R})$ $(p>2)$ and the noises are moderately ill-posed with the index β. The estimators include both the linear and non-linear wavelet ones. It turns out that the situation for $0<\beta \le 1$ is more complicated than that for $\beta >1$.     

Asymptotic normality of p-norm estimators in multiple regression

Summary The consistency and asymptotic normality of p-norm estimators (1<p<2) is established by applying some of the ideas of Huber (1973), where asymptotic normality of the so-called M-estimators for regression parameters is shown. A central role is played by a weight function . Huber assumed that , and are bounded. This is, however, not the case for p-norm estimators with 1<p<2, but some of his ideas can still be applied.     

Asymptotic normality of some kernel-type estimators of probability density

Central limit theorems are proved for some kernel-type estimators of probability density in the case where the observations form a strictly random sequence satisfying the ?-mixing condition with a certain logarithmic mixing rate.     

Asymptotic normality of H-L estimators based on dependent data

LetX ₁,X ₂, ... be a strictly stationary φ-mixing sequence of r.v.'s with a common continuous cdfF. Let θ be a location parameter ofF. We prove the asymptotic normality of a class of Hodges-Lehmann estimators of θ under various regularity conditions on the mixing number φ and the underlyingF. We also establish the asymptotic linearity of signed rank statistics in the parameter θ. Our results also enable us to study the effect of φ-dependence on the asymptotic power of signed rank tests for testingH ₀: θ=0 againstH _n :θ=θ ₀ n ^?1/2,θ ₀≠0. Finally these results are shown to remain valid for strongly mixing processes {X _i } also.     

Asymptotic normality of prediction error estimators for approximate system models

A general class of parameter estimation methods for stochastic dynamical systems is studied. The class contains the least squares method, output-error methods, the maximum likelihood method and several other techniques. It is shown that the class of estimates so obtained are asymptotically normal and expressions for the resulting asymptotic covariance matrices are given. The regularity conditions that are imposed to obtain these results, are fairly weak. It is, for example, not assumed that the true system can be described within the chosen model set, and, as a consequence, the results in this paper form a part of the so-called approximate modeling approach to system identification. It is also noteworthy that arbitrary feedback from observed system outputs to observed system inputs is allowed and stationarity is not required     

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.     

Asymptotic theory for estimators under random censorship

Summary The product limit estimator of an unknown distributionF is represented as aU-statistic plus an error of the ordero(1/n). Using this, the maximum likelihood estimator of the specific risk rate in the time interval [0,M], is shown to admit a two term Edgeworth expansion. This risk rate for a specific cause of death is defined as the ratio of the probability of death, due to that particular cause, in the time interval [0,M], to the mean life time of an individual up to that time pointM. Similar expansions for the bootstrapped statistics are used to show that the bootstrap distribution, of the studentized estimator of the risk rate, approximates the sampling distribution better than the corresponding normal distribution.Research supported in part by NSA Grant MDA 904-90-H-1001 and by NSF Grant DMS-9007717     

Asymptotic normality of the likelihood moment estimators for a stationary linear process with heavy-tailed innovations

The authors recently proved in Martig and Hüsler (2016) that the likelihood moment estimators are consistent estimators for the parameters of the Generalized Pareto distribution for the case where the underlying data arises from a (stationary) linear process with heavy-tailed innovations. In this paper we derive the bivariate asymptotic normality under some additional assumptions and give an explicit example on how to check these conditions by using asymptotic expansions. Some finite sample comparisons are presented to investigate the bias and variance behavior for some of the estimators.     

Asymptotic normality and strong consistency of maximum quasi-likelihood estimates in generalized linear models

In a generalized linear model with q x 1 responses, the bounded and fixed (or adaptive) p × q regressors Zi and the general link function, under the most general assumption on the minimum eigenvalue of ZiZ'i,the moment condition on responses as weak as possible and the other mild regular conditions, we prove that the maximum quasi-likelihood estimates for the regression parameter vector are asymptotically normal and strongly consistent.     

Asymptotic normality of wavelet density estimator under censored dependent observations

In this paper, we discuss the asymptotic normality of the wavelet estimator of the density function based on censored data, when the survival and the censoring times form a stationary ??-mixing sequence. To simulate the distribution of estimator such that it is easy to perform statistical inference for the density function, a random weighted estimator of the density function is also constructed and investigated. Finite sample behavior of the estimator is investigated via simulations too.     

Regression estimation under strong mixing data

Annals of the Institute of Statistical Mathematics - This paper studies multivariate wavelet regression estimators with errors-in-variables under strong mixing data. We firstly prove the strong...     

Asymptotic normality of spectral estimates

The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and φ-mixing sequences and linear transformations of martingale differences.     

On the strong mixing and weak Bernoulli conditions

Summary A strictly stationary random sequence is constructed which has exactly two states and satisfies the strong mixing condition but not the weak Bernoulli condition.This research was partially supported by the National Science Foundation and the Office of Naval Research     

混合误差半参数回归模型估计的相合性

研究了误差为ψ混合和ψ混合序列的半参数回归模型,综合最小二乘法和非参数权函数估计方法,分别定义了待估参数β和未知函数夕的估计量βm,n和9m,n(χ).利用混合序列的矩不等式及凸函数的性质,在较弱的条件下证明了这些估计量的强相合性与矩相合性,这些结果推广了已有的相应的研究结果.     

Asymptotic normality of cumulant spectral estimates

The paper considers higher-order cumulant spectral estimates obtained by directly Fourier transforming weighted cumulant estimates. Such estimates computationally are different from those based on the finite Fourier transform. These estimates can be looked at continuously as well as directly on submanifolds. The estimates of cumulants are based on unbiased moment estimates. Asymptotic normality is obtained for these estimates and is based on a strong mixing condition and only a finite number of cumulant summability conditions.     

Asymptotic normality of double-indexed linear permutation statistics

The paper provides sufficient conditions for the asymptotic normality of statistics of the form a _ijb_RiRj, wherea _ijandb _ijare real numbers andR _iis a random permutation.     

Tests for normality based on density estimators of convolutions

Recent results show that densities of convolutions can be estimated by local U-statistics at the root-n rate in various norms. Motivated by this and the fact that convolutions of normal densities are normal, we introduce new tests for normality which use as test statistics weighted L₁-distances between the standard normal density and local U-statistics based on standardized observations. We show that such test statistics converge at the root-n rate and determine their limit distributions as functionals of Gaussian processes. We also address a choice of bandwidth. Simulations show that our tests are competitive with other tests of normality.