Asymptotic expansion of S-estimators of location and covariance

By means of a straightforward application of empirical process theory, we show that S-estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent vector and matrix valued random elements respectively. This provides an alternative proof of asymptotic normality of S-estimators and clearly explains the limiting covariance structure. It also leads to a relatively simple proof of asymptotic normality of the length of the shortest α-fraction.     

A proof of asymptotic normality for some VARX models

Here we present a proof of the asymptotic normality of least squares estimates for stable multivariate autoregressive models excited by a deterministic second order input signal.     

Efficiency. of infinite dimensional M- estimators

It is well-known that maximum likelihood estimators are asymptotically normal with covariance equal to the inverse Fisher information in smooth, finite dimensional parametric models. Thus they are asymptotically efficient. A similar phenomenon has been observed for certain infinite dimensional parameter spaces. We give a simple proof of efficiency, starting from a theorem on asymptotic normality of infinite dimensional M -estimators. The proof avoids the explicit calculation of the Fisher information. We also address Hadamard differentiability of the corresponding M -functionals.     

A note on a specification test of independence

In a recent paper Zheng (1997a) proposed a new specification test of independence between two random vectors by the kernel method. He showed asymptotic normality under the hypothesis and local alternatives. The present work investigates the asymptotic distribution of the corresponding test statistic under fixed alternatives. In this case asymptotic normality of a standardized statistic is still valid but with a different rate of convergence. Received: January 1999     

Asymptotics of self-weighted M-estimators for autoregressive models

In this paper, we consider a stationary autoregressive AR(p) time series \(y_t=\phi _0+\phi _1y_{t-1}+\cdots +\phi _{p}y_{t-p}+u_t\). A self-weighted M-estimator for the AR(p) model is proposed. The asymptotic normality of this estimator is established, which includes the asymptotic properties under the innovations with finite or infinite variance. The result generalizes and improves the known one in the literature.     

Estimation and model selection of semiparametric multivariate survival functions under general censorship

We study estimation and model selection of semiparametric models of multivariate survival functions for censored data, which are characterized by possibly misspecified parametric copulas and nonparametric marginal survivals. We obtain the consistency and root-n$n$ asymptotic normality of a two-step copula estimator to the pseudo-true copula parameter value according to KLIC, and provide a simple consistent estimator of its asymptotic variance, allowing for a first-step nonparametric estimation of the marginal survivals. We establish the asymptotic distribution of the penalized pseudo-likelihood ratio statistic for comparing multiple semiparametric multivariate survival functions subject to copula misspecification and general censorship. An empirical application is provided.     

Asymptotics for panel quantile regression models with individual effects

This paper studies panel quantile regression models with individual fixed effects. We formally establish sufficient conditions for consistency and asymptotic normality of the quantile regression estimator when the number of individuals, n$n$, and the number of time periods, T$T$, jointly go to infinity. The estimator is shown to be consistent under similar conditions to those found in the nonlinear panel data literature. Nevertheless, due to the non-smoothness of the objective function, we had to impose a more restrictive condition on T$T$ to prove asymptotic normality than that usually found in the literature. The finite sample performance of the estimator is evaluated by Monte Carlo simulations.     

A goodness-of-fit test in the multinomial legit model based on weighted squared residuals

The asymptotic (normal) distribution of the sum of weighted squared residuals in the multinomial logit model is derived. The performance of a chi-squared test based on the asymptotic normality is discussed using some empirical examples.     

The asymptotic properties of maximum likelihood estimators for marked poisson processes with a cyclic intensity measure

In this paper we are concerned with the large sample behavior of the MLE for a class of marked Poisson processes arising in hydrology. We establish strong consistency, asymptotic normality and asymptotic efficiency of the MLE. As an application we present the asymptotic distribution of the design discharge of a river flow.     

ON THE FIRST–ORDER EFFICIENCY AND ASYMPTOTIC NORMALITY OF MAXIMUM LIKELIHOOD ESTIMATORS OBTAINED FROM DEPENDENT OBSERVATIONS

Abstract. In this paper we study the first–order efficiency and asymptotic normality of the maximum likelihood estimator obtained from dependent observations. Our conditions are weaker than usual, in that we do not require convergences in probability to be uniform or third–order derivatives to exist.
The paper builds on Witting and Nolle's result concerning the asymptotic normality of the maximum likelihood estimator obtained from independent and identically distributed observations, and on a martingale theorem by McLeish.     

Spatial local M-estimation under association

We investigate the asymptotic behavior of a robust version of local linear regression estimators with variable bandwidth for spatial associated processes. The weak consistency of the proposed estimators is given under appropriate conditions. Furthermore, we establish the asymptotic normality of the estimators, from which expressions for the asymptotic bias and variance can be derived.     

Classification and Discrimination Problems with Applications, Part IIa

Abstract  In Part I exact results for univariate (" p = 1") two-group ("k = 2") classification problems were derived assuming normality and equality of the variances. In Part IIa asymptotic results for multivariate (" p > I") two-group classification and discrimination problems are based on the corresponding assumptions of multivariate normality and equality of the covariance matrices. The results (4.6.5), (4.6.6) and (4.6.7) are believed to be new.
The asymptotic results in Section 4.6, together with results presented elsewhere in the literature, constitute the basis of various detailed proposals to deal with problems from actual statistical practice. Most of these proposals are modifications or specifications of existing ones. We shall pay some attention to (I) testing whether differences exist. But we are mainly interested in: (II) constructing a discriminant function, (III) assigning the individual under classification, and in (IV) constructing a confidence interval for "the" posterior probability that the individual under classification belongs to Population 2.
An important part in our theory is played by various techniques for selecting variables in discriminant analysis. The need for such techniques follows from Section 4.10. The consequences of building-in a selection technique are discussed in Section 4.12. One of our proposals motivates the theory presented in Chapter 3 and is mentioned here for that reason: employ a large part of the data, say 70%, in order to construct a discriminant function (via a selection of variables); by applying this function to the rest of the data, the exact univariate theory of Part I becomes of application. Part IIb will contain a chapter on applications.     

Nonparametric fixed effects model for panel data with locally stationary regressors

We develop methods for inference in nonparametric time-varying fixed effects panel data models that allow for locally stationary regressors and for the time series length $T$ and cross-section size $N$ both being large. We first develop a pooled nonparametric profile least squares dummy variable approach to estimate the nonparametric function, and establish the optimal convergence rate and asymptotic normality of the resultant estimator. We then propose a test statistic to check whether the bivariate nonparametric function is time-varying or the time effect is separable, and derive the asymptotic distribution of the proposed test statistic. We present several simulated examples and two real data analyses to illustrate the finite sample performance of the proposed methods.     

Consistency and asymptotic normality of maximum likelihood estimation for Gaussian Markov processes from discrete observations

In this paper we prove the weak consistency and the asymptotic normality of the maximum likelihood estimation based on discrete observations ofn independent Gaussian Markov processes. The Ornstein Uhlenbeck process is a special Gaussian Markov process. We derive asymptotic simultaneous confidence regions for the parameters of the Ornstein Uhlenbeck process as an application.     

Combining Significance of Correlated Statistics with Application to Panel Data*

The inverse normal method, which is used to combine P‐values from a series of statistical tests, requires independence of single test statistics in order to obtain asymptotic normality of the joint test statistic. The paper discusses the modification by Hartung (1999, Biometrical Journal, Vol. 41, pp. 849–855) , which is designed to allow for a certain correlation matrix of the transformed P‐values. First, the modified inverse normal method is shown here to be valid with more general correlation matrices. Secondly, a necessary and sufficient condition for (asymptotic) normality is provided, using the copula approach. Thirdly, applications to panels of cross‐correlated time series, stationary as well as integrated, are considered. The behaviour of the modified inverse normal method is quantified by means of Monte Carlo experiments.     

A Central Limit Theorem for M-estimators by the von Mises Method

Asymptotic normality of M- or maximum likelihood type estimators was established in a classic paper by Huber (1967). Reeds (1976) argued that this could have been obtained simply as an application of the delta-method, using the tool of compactly differentiating von Mises functionals with respect to the empirical distribution function F_n. His proof however contains some errors and has been largely ignored. A corrected version of the proof is given.     

Asymptotics of SIMEX-based variance estimation

In this paper, we study the asymptotic properties of simulation extrapolation (SIMEX) based variance estimation that was proposed by Wang et al. (J R Stat Soc Series B 71:425–445, 2009). We first investigate the asymptotic normality of the parameter estimator in general parametric variance function and the local linear estimator for nonparametric variance function when permutation SIMEX (PSIMEX) is used. The asymptotic optimal bandwidth selection with respect to approximate mean integrated squared error (AMISE) for nonparametric estimator is also studied. We finally discuss constructing confidence intervals/bands of the parameter/function of interest. Other than applying the asymptotic results so that normal approximation can be used, we recommend a nonparametric Monte Carlo algorithm to avoid estimating the asymptotic variance of estimator. Simulation studies are carried out for illustration.     

Some large-concentration-parameter asymptotics for the k-class estimators

A sufficient condition is derived in this paper for the consistency and asymptotic normality of the k-class estimators (k-stochastic or nonstochastic) as the concentration parameter increases indefinitely, with the sample size either staying fixed or also increasing. It is further shown that the limited-information maximum likelihood estimator satisfies this condition. Since large sample size implies a large concentration parameter, but not vice versa, the usual conditions for consistency and asymptotic normality of the k-class estimators as the sample size increases can be inferred from the results given in this paper. But more importantly, the results in this paper shed further light on the small-sample properties of the stochastic k-class estimators and can serve as a starting point for the derivation of asymptotic approximations for these estimators as the concentration parameter goes to infinity, while the sample size either stays fixed or also goes to infinity.     

An invariance principle forU-statistics in simple random sampling without replacement

Summary Supplementing the well-known invariance principle forU-statistics based on i.i.d. observations (Miller and Sen 1972) we establish an invariance principle forU-statistics in case of simple random sampling from a sequence of finite populations. This generalizes the asymptotic normality-result of Nandi and Sen (1963), and permits e.g. to prove asymptotic normality ofU-statistics in the presence of random non-response.     

Asymptotic properties of the maximum likelihood estimator in dichotomous logit models

The existence and strong consistency of the maximum likelihood estimator are analyzed in the context of dichotomous logit models. Sufficient conditions are given for the asymptotic normality of this estimator.