Asymptotic behaviour of regression pre-test estimators with minimal Bayes risk

Pre-test estimators (PTE) are considered which are optimal under a Bayes risk among PTE with general measurable sets as “regions of significance” for the test statistic t associated with the estimate of a given regression coefficient. Asymptotic and some finite sample results are stated and numerical experiments are commented on.     

Slow convergence of the number of near-maxima for Burr XII distributions

A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number K _n (a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that for all Burr XII distributions K _n (a) converges almost surely to unity, but this convergence property may not become clear under certain cases even for very large n. We explore the reason of such slow convergence by studying a distributional continuity between Burr XII and Weibull distributions. We have also given a theoretical explanation of slow convergence of K _n (a) for the Burr XII distributions by showing that the rate of convergence in terms of P{K _n (a) > 1} tending to zero changes very little with the sample size n. Illustrations of the limiting behaviour K _n (a) for the Burr XII and the Weibull distributions are given by simulations and real data. The study also raises an important issue that although the Burr XII provides overall better fit to a given data set than the Weibull distribution, cautions should be taken for the extrapolation of the upper tail behaviour in the case of slow convergence.      

Asymptotic normality

Summary The object of this paper is to show that — under certain regularity conditions — a dominated family of probability measures with Euclidean parameter space behaves approximately like a family of normal distributions if each probability measure is the independent product of a great number of identical components. The paper was written while this author was employed by a grant of the Deutsche Forschungsgemeinschaft.     

Asymptotic distributions of smoothed histograms

As an estimator for an unknown probability density functionf, concentrated on a known intervalI, one can use a histogram smoothed by a suitable family of lattice distributions. For such an estimator a uniform weak consistency result and a central limit theorem with an error bound are given. Further for the global deviation of fromf the asymptotic distribution is developed.Partially supported by the Natural Sciences and Engineering Research Council of Canada, grant A 2983, A4806, and A3988.     

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.     

Asymptotic seat bias formulas

Apportionment methods are used to round the vote proportions of parties in a proportional representation system to integer numbers of seats in the parliament. Seat biases quantify by how much on average a particular apportionment method favors larger (or smaller) parties. In this paper, we prove a previous conjecture on asymptotic seat biases of stationary divisor methods and the quota method of greatest remainders, as the size of the parliament tends to infinity.     

Asymptotic theory and econometric practice

The classical paradigm of asymptotic theory employed in econometrics presumes that model dimensionality, p, is fixed as sample size, n, tends to inifinity. Is this a plausible meta-model of econometric model building? To investigate this question empirically, several meta-models of cross- sectional wage equation models are estimated and it is concluded that in the wage-equation literature at least that p increases with n roughly like n ^l/4, while that hypothesis of fixed model dimensionality of the classical asymptotic paradigm is decisively rejected. The recent theoretical literature on ‘large-p’ asymptotics is then very briefly surveyed, and it is argued that a new paradigm for asymptotic theory has already emerged which explicitly permits p to grow with n. These results offer some guidance to econometric model builders in assessing the validity of standard asymptotic confidence regions and test statistics, and may eventually yield useful correction factors to conventional test procedures when p is non-negligible relative to n.     

Asymptotic inference from multi-stage samples

I develop a GMM-based framework for asymptotic inference to analyze data from surveys whose designs involve stratification and clustering. I set up the estimation problem, derive the appropriate asymptotic distribution theory as the number of clusters per stratum tends to infinity and compute asymptotic standard errors that are robust to sample-design effects. The analysis is then extended to nonparametric regression and to semiparametric estimation based on U-processes. Empirical illustrations are provided using consumption expenditure data from the complexly designed Indian national household survey.     

Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE

In generalized autoregressive conditional heteroskedastic (GARCH) models, the standard identifiability assumption that the variance of the iid process is equal to 1 can be replaced by an alternative moment assumption. We show that, for estimating the original specification based on the standard identifiability assumption, efficiency gains can be expected from using a quasi-maximum likelihood (QML) estimator based on a non Gaussian density and a reparameterization based on an alternative identifiability assumption. A test allowing to determine whether a reparameterization is needed, that is, whether the more efficient QMLE is obtained with a non Gaussian density, is proposed.     

Local Asymptotic Power of Breitung's Test*

In this article, we derive the local asymptotic power function of the unit root test proposed by Breitung [Journal of Econometrics (2002) Vol. 108, pp. 343–363]. Breitung's test is a non‐parametric test and is free of nuisance parameters. We compare the local power curve of the Breitungs’ test with that of the Dickey–Fuller test. This comparison is in fact a quantification of the loss of power that one has to accept when applying a non‐parametric test.     

Asymptotic interpretations for equilibria of nonatomic games

We show that a mixed equilibrium of a semi-anonymous nonatomic game can be used to generate pure-strategy profiles for finite games randomly generated from the type distribution of the nonatomic game. As the numbers of players involved in the finite games increase, the generated profiles will be asymptotically equilibrium. The converse of this result is also true, i.e., a mixed-strategy profile that is not an equilibrium for the nonatomic game will not be able to achieve the above asymptotic rationality for large finite games. The combined finding can be specialized to situations where the nonatomic game is anonymous and where the given equilibrium is pure. Besides their practical values, these results offer yet one more justification for the study of nonatomic games. They also suggest that efforts may be better spent on searching for mixed rather than pure equilibria of nonatomic games.     

Asymptotic theory for heterogeneous dynamic pseudo-panels

Dynamic pseudo-panel models with inter-cohort parameter heterogeneity are studied. The population is divided into cohorts and the cohort sample means are used as a heterogeneous panel. Least squares and instrumental variables estimators are considered. Multidimensional limits are analyzed as the cross-sectional and temporal dimensions of the data pass to infinity, allowing for both stationary and nonstationary cases. Monte-Carlo simulations on the finite-sample performance of the estimators in these two dimensions are conducted and these, in conjunction with the asymptotic results, are used to make recommendations for practical implementation of the techniques. An empirical illustration finds heterogeneity in consumption growth rates among Taiwanese birth cohorts.     

On Some Characteristics of Gaussian Covariance Functions

The concepts of isotropy/anisotropy and separability/non‐separability of a covariance function are strictly related. If a covariance function is separable, it cannot be isotropic or geometrically anisotropic, except for the Gaussian covariance function, which is the only model both separable and isotropic. In this paper, some interesting results concerning the Gaussian covariance model and its properties related to isotropy and separability are given, and moreover, some examples are provided. Finally, a discussion on asymmetric models, with Gaussian marginals, is furnished and the strictly positive definiteness condition is discussed.     

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.     

Locally best tests for Gaussian processes

A continuous-time stochastic process occuring in a representation of the likelihood ratio process for Gaussian processes with common covariance kernel is shown to be a Wiener process with respect to a certain family of -algebras. This is applied to the problem of sequentially testing one-sided hypotheses for Gaussian processes, and it is proved that certain continuous-time SPRT's are locally best sequential tests under some restrictions.     

Fisher information under Gaussian quadrature models

This paper develops formulae to compute the Fisher information matrix for the regression parameters of generalized linear models with Gaussian random effects. The Fisher information matrix relies on the estimation of the response variance under the model assumptions. We propose two approaches to estimate the response variance: the first is based on an analytic formula (or a Taylor expansion for cases where we cannot obtain the closed form), and the second is an empirical approximation using the model estimates via the expectation–maximization process. Further, simulations under several response distributions and a real data application involving a factorial experiment are presented and discussed. In terms of standard errors and coverage probabilities for model parameters, the proposed methods turn out to behave more reliably than does the ‘disparity rule’ or direct extraction of results from the generalized linear model fitted in the last expectation–maximization iteration.     

Asymptotic distribution for products of sums under dependence

The product of partial sums of linear positive (negative) quadrant dependent, positive random variables is asymptotically lognormal. This extends the earlier work on independent, positive random variables [see Rempala and Wegolowski Elect Comm Probab 7:47–54, 2002]. Research supported by National Natural Science Foundation of China (No. 10471126); Research supported by Science Foundation of Zhejiang Provincial Education (No. 20060122).     

Asymptotic least-squares estimation efficiency considerations and applications

This paper is concerned with the large sample efficiency of the asymptotic least-squares (ALS) estimators introduced by Gouriéroux, Monfort, and Trognon (1982, 1985) and Chamberlain (1982, 1984). We show how the efficiency of these estimators is affected when additional information is incorporated into the estimation procedure. The relationship between ALS and maximum likelihood is discussed. It is shown that ALS can be used to obtain asymptotically efficient estimates for a large range of econometric models. Many results from the literature on estimation are special cases of the framework adopted in this paper. An application of ALS to a dynamic rational expections factor demand model in the manufacturing sector in The Netherlands demonstrates the potential of the method in the estimation of the parameters in models which are subject to nonlinear cross-equation restrictions.     

Asymptotic normality in vacancies on a line

Abstract  Covering a row of hooks by hats (one hat needs two adjacent hooks) by choosing for the next hat a pair of free hooks with equal probability from the free hooks, we encounter x_n , the number of isolated hooks remaining uncovered. We prove that x_n is asymptotically normal.