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.     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

The exact and approximate distributions of linear combinations of selected order statistics from a uniform distribution

Summary The exact probability density function is given for linear combinations ofk=k(n) order statistics selected from whole order statistics based on random sample of sizen drawn from a uniform distribution. Normal approximation to the linear combinations is made with the aid of Berry-Esseen's theorem. Necessary and sufficient conditions of the asymptotic normality for the statistic are obtained, too. An exact distribution and its normal approximation of linear combination of mutually independent gamma variables with integer valued parameters are also given as associated consequences. The Institute of Statistical Mathematics     

关于线性秩统计量的渐近正态性及其收敛速度

本文讨论线性秩统计量的渐近正态性的条件及其收敛速度．推广了Hajek关于线性秩统计量收敛于正态分布的条件的重要定理，并得出了一个较易验证的充分条件．对于一般形式的计分函数，在一定条件下得出了相应线性秩统计量收敛于正态分布的速度．     

Estimation in mixed effects model with errors in variables

In this paper, we consider a linear mixed-effects model with measurement errors in both fixed and random effects and find the moment of estimators for the parameters of interest. The strong consistency and asymptotic normality of the estimators are obtained under regularity conditions. Moreover, we obtain the strong consistent estimators of the asymptotic covariance matrices involved in the limiting theory. Simulations are reported for illustration.     

Semi-parametric Estimation of the Second Order Parameter in Statistics of Extremes

We present a class of semi-parametric estimators for the second order parameter related to a probability distribution with a regularly varying tail. The second order parameter plays an important role whenever dealing with optimization problems in statistics of extreme values. Consistency and asymptotic normality are proven under appropriate conditions.     

Consistency and asymptotic normality of the maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with random regressors

This paper proposes some regularity conditions, which result in the existence, strong consistency and asymptotic normality of maximum quasi-likelihood estimator （MQLE） in quasi-likelihood nonlinear models （QLNM） with random regressors. The asymptotic results of generalized linear models （GLM） with random regressors are generalized to QLNM with random regressors.     

Asymptotic normality of the maximum likelihood estimate in the independent not identically distributed case

In this paper, we assume the existence and consistency of the maximum likelihood estimate (MLE) in the independent not identically distributed (i.n.i.d.) case and we establish its asymptotic normality. The regularity conditions employed do not involve the third order derivatives of the underlying probability density functions (p.d.f.'s). This research was supported by the National Science Foundation, Grant GR-20036, and the Office of Scientific Research and Development of the Greek Government.     

负相协平均剩余寿命函数和有效函数的一类非参数估计

本文在 NA 样本下,讨论了平均剩余寿命函数和有效函数的非参数递归型估计的相合性和渐近正态性．     

Asymptotic normality of estimating risk upon the wavelet-vaguelette decomposition of a signal function in a model with correlated noise

The problem of estimating the function when using a homogeneous linear operator in a model with correlated noise is considered. The asymptotic properties of estimating risk upon the threshold wavelet-vaguelette decomposition of a signal are studied. The conditions under which the asymptotic normality of an unbiased risk estimate holds are given.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

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.     

Consistent estimation in a linear regression problem with random errors in coefficients

We consider the linear regression model in the case when the independent variables are measured with errors, while the variances of the main observations depend on an unknown parameter. In the case of normally distributed replicated regressors we propose and study new classes of two-step estimates for the main unknown parameter. We find consistency and asymptotic normality conditions for first-step estimates and an asymptotic normality condition for second-step estimates. We discuss conditions under which these estimates have the minimal asymptotic variance.     

On the calculation of cumulants of estimators arising from a linear time series regression model

Bounds for higher-order cumulants of statistics arising from a linear time series regression model are investigated. A result given in Brillinger is proved and extended. The bounds permit derivation of asymptotic moments and asymptotic normality for estimators of parameters in the model. Two examples are given as illustrations.     

On asymptotic theory for multivariate GARCH models

The paper investigates the asymptotic theory for a multivariate GARCH model in its general vector specification proposed by Bollerslev, Engle and Wooldridge (1988) [4], known as the VEC model. This model includes as important special cases the so-called BEKK model and many versions of factor GARCH models, which are often used in practice. We provide sufficient conditions for strict stationarity and geometric ergodicity. The strong consistency of the quasi-maximum likelihood estimator (QMLE) is proved under mild regularity conditions which allow the process to be integrated. In order to obtain asymptotic normality, the existence of sixth-order moments of the process is assumed.     

The asymptotic behavior of linear placement statistics

Orban and Wolfe (1982) and Kim (1999) provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of Orban and Wolfe (1982) and Kim (1999) when the sample sizes of each group go to infinity simultaneously.     

A sharpening of the remainder term in the higher-dimensional central limit theorem for multilinear rank statistics

A new approach to the asymptotic normality of the multivariate linear rank statistics is provided along with the Berry-Esséen and the Prohorov distance estimates for the remainder term in the convergence to normality.     

Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have aninfill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber’s ψ-function and the sign functions (corresponding to the sample quantiles). Research of Lahiri is partially supported by NSF grant no. DMS-0072571. Research of Mukherjee is partially supported by the Academic Research Grant R-155-000-003-112 from the National University of Singapore.     

再论逆自相关函数自回归估计的渐近性质（Ⅱ）：渐近正态性

本文在一组相当广泛的条件下,证明了线性平稳时间序列逆自相关函数自回归估计的渐近正态性,并获得了由这一估计所得的MA(q)模型参数估计的渐近正态性和优效渐近正态性。     

A general autoregressive model with Markov switching: Estimation and consistency

In this paper, a general autoregressive model with Markov switching is considered, where the autoregression may be of an infinite order. The consistency of the maximum likelihood estimators for this model is obtained under regularity assumptions. Examples of finite and infinite order autoregressive models with Markov switching are discussed. Simulation studies with these examples illustrate the consistency and asymptotic normality of the estimators.