Asymptotic consequences of neglected dynamics in individual effects models*

We assess the asymptotic consequences of estimating static models based on cross-section or panel data, when in reality the data are generated by a dynamic relationship, involving lagged dependent and current and lagged exogenous variables as well as individual effects. If the exogenous variable follows a stationary process, then the static estimators usually underestimate its long-run effect. This inconsistency is less severe, the higher the autocorrelation of the exogenous variable. If the exogenous variable follows a random walk with or without individual-specific drift, then the estimators are found to be consistent for the long-run effect.     

Comparison of tail index estimators

We compare various estimators for the index of distribution functions with regularly varying tails by calculating their asymptotic mean squared errors after choosing the optimal number of upper order statistics involved (which is different for different estimators).     

Asymptotic linearity of minimax estimators

The celebrated local asymptotic minimax (LAM) theorem due to HÁjek (1972) also includes the statement that a LAM estimator Is necessarily asymptotically linear. A similar result. is true for semi-parametric models, but Hájek's result doesn't apply to this case as the efficient influence function is often not contained in the (proper) tangent space. This note gives a simple, elementary proof of both the LAM theorem and the necessity of asymptotic linearity of a LAM estimator sequence.     

Limiting distribution of the collision resolution interval

We use the theory of probability metrics to study the asymptotic normality of the collison resolution intervals in the CTM multi-access protocol under general conditions on the number of retransmitted messages and of new arrivals during the collision slots. Our main result establishes stability of the central limit theorem for the CTM algorithm. We provide extensive simulation results investigating the extent to which the mean of the collision resolution interval eventually becomes unstable for increasing values of n , the number of users who initially collide. The normal fit is numerically investigated and is shown to be quite satisfactory and stable with respect to moderate perturbations and n ≥50.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

A consistent estimator in general functional errors-in-variables models

The problem of estimation in nonlinear functional errors-in-variables model is considered. A modified least squares estimator is studied, its consistency and asymptotic normality is established. Simulation results are also presented showing the performance of the estimator in comparison with the naive ordinary least squares estimator. Received: June 1999     

A new sequential design based on the robbins-monro procedure

Summary A sequential estimation procedure based on the Robbins-Monro process is proposed. Through observations of a random variableY(x), estimates of a regression functionM(x) are obtained, whereM(x)=E[Y(x)]. The estimates of the roots ofM(x) are shown to be strongly convergent and asymptotically normal with minimum variance.     

Consistent estimation in the bilinear multivariate errors-in-variables model

A bilinear multivariate errors-in-variables model is considered. It corresponds to an overdetermined set of linear equations AXB=C, A∈ℝ^m×n, B∈ℝ^p×q, in which the data A, B, C are perturbed by errors. The total least squares estimator is inconsistent in this case.  An adjusted least squares estimator is constructed, which converges to the true value X, as m →∞, q →∞. A small sample modification of the estimator is presented, which is more stable for small m and q and is asymptotically equivalent to the adjusted least squares estimator. The theoretical results are confirmed by a simulation study. Acknowledgements. We thank two anonymous reviewers for their suggestions and corrections.? A. Kukush is supported by a postdoctoral research fellowship of the Belgian office for Scientific, Technical and Cultural Affairs, promoting Scientific and Technical Collaboration with Central and Eastern Europe.? S. Van Huffel is a full professor with the Katholieke Universiteit Leuven.? I. Markovsky is a research assistant with the Katholieke Universiteit Leuven.? This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction (IUAP V-22), initiated by the Belgian State, Prime Minister's Office – Federal Office for Scientific, Technical and Cultural Affairs of the Concerted Research Action (GOA) projects of the Flemish Government MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology), of the IDO/99/03 project (K.U. Leuven) “Predictive computer models for medical classification problems using patient data and expert knowledge”, of the FWO projects G.0078.01, G.0200.00, and G0.0270.02.? The scientific responsibility is assumed by its authors.     

A consistent estimator for nonlinear regression models

In this paper an estimator for the general (nonlinear) regression model with random regressors is studied which is based on the Fourier transform of a certain weight function. Consistency and asymptotic normality of the estimator are established and simulation results are presented to illustrate the theoretical ones.Supported by the Hungarian National Science Foundation OTKA under Grants No. F 032060/2000 and F 046061/2004 and by the Bolyai Grant of the Hungarian Academy of Sciences.Received October 2003     

One-step R-estimation in linear models with stable errors

Classical estimation techniques for linear models either are inconsistent, or perform rather poorly, under α$\alpha$-stable error densities; most of them are not even rate-optimal. In this paper, we propose an original one-step R-estimation method and investigate its asymptotic performances under stable densities. Contrary to traditional least squares, the proposed R-estimators remain root-n$n$ consistent (the optimal rate) under the whole family of stable distributions, irrespective of their asymmetry and tail index. While parametric stable-likelihood estimation, due to the absence of a closed form for stable densities, is quite cumbersome, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values (_α0,_b0)$({\alpha }_0,b_0)$ of the tail index α$\alpha$ and skewness parameter b$b$, while preserving root-n$n$ consistency under any (α,b)$(\alpha ,b)$ as well as under usual light-tailed densities. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.     

THE ASYMPTOTIC BEHAVIOR OF SIMPLE LINEAR RANK STATISTICS

The concept of a simple linear rank statistic is essential in nonparametric theory. These statistics may be used for testing the null-hypothesis that all observations are independent and identically distributed against classes of alternatives indicated by the choice of regression constants and scores. Special cases are the two-sample Linear rank statistics and Spearman's rank correlation coefficient rho under the null-hypothesis of independence.
In this paper we shall give a survey of the asympotic distribution theory of simple linear rank statistics.