Minimax invariant estimator of a continuous distribution function under a general loss function

The problem of invariant estimation of a continuous distribution function is considered under a general loss function. Minimaxity of the minimum risk invariant estimator of a continuous distribution function is proved for any sample size n ≥ 2.     

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     

On the strong uniform consistency of a new kernel density estimator

Summary A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ _n andV _n, respectively, whereμ _n is an unspecified estimator of the mean vector andV _n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively. Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.     

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     

Another look at the naive estimator in a regression model

We consider the linear regression model where only a particular linear function of the dependent variables is observed, Stahlecker and Schmidt (1987) proposed a naive least squares (LS) estimator for regression coefficients in such a case. In this note we represent their estimator as a general ridge estimator. This observation leads to a view different from the previous work and provides an easy way of obtaining many important properties of the naive LS estimator. Our approach also gives some insight into the relationship between the naive LS estimator and the generalized least squares estimator.     

The depth function of a population distribution

Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an arbitrary probability distribution on Â^p, and even of a non-probability measure. The halfspace depth of any point / in Â^p is the smallest measure of a closed halfspace that contains /. We review the properties of halfspace depth, enriched with some new results. For various measures, uniform as well as non-uniform, we derive an expression for the depth function. We also compute the Tukey median, which is the / in which the depth function attains its maximal value. Various interesting phenomena occur. For the uniform distribution on a triangle, a square or any regular polygon, the depth function has ridges that correspond to an 'inversion' of depth contours. And for a product of Cauchy distributions, the depth contours are squares. We also consider an application of the depth function to voting theory.     

Locally minimum variance unbiased estimator in a discontinuous density function

The exact forms of the locally minimum variance unbiased estimators and their variances are given in the case of a discontinuous density function.     

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).     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

The conditional maximum likelihood estimator of the shape parameter in the gamma distribution

Summary The conditional maximum likelihood estimator of the shape parameter in the gamma distribution is studied for a finite sample size in comparison with the (unconditional) maximum likelihood estimator. The former estimator is concluded to be strictly superior to the latter. The reasons for the conclusion include the undesirable behavior of the residual likelihood, the consistency and relatively less bias of the conditional maximum likelihood estimator. Simulation studies for risk comparisons also support the conclusion.     

A property of Stein's estimator for the mean of a multivariate normal distribution

A uniform bound on the risk (under squared error loss) of Stein's estimator Ψ₁ for the mean of the multivariate normal distribution is given. Using the bound, the asymptotic behaviour of the risk of Ψ₁ under a Bayesian assumption is obtained.     

Semiparametric time series regression modeling with a diverging number of parameters

Variable selection and error structure determination of a partially linear model with time series errors are important issues. In this paper, we investigate the regression coefficient and autoregressive order shrinkage and selection via the smoothly clipped absolute deviation penalty for a partially linear model with a divergent number of covariates and finite order autoregressive time series errors. Both consistency and asymptotic normality of the proposed penalized estimators are derived. The oracle property of the resultant estimators is proved. Simulation studies are carried out to assess the finite‐sample performance of the proposed procedure. A real data analysis is made to illustrate the usefulness of the proposed procedure as well.     

MSE-improvement of the least squares estimator by dropping variables

It is well known that dropping variables in regression analysis decreases the variance of the least squares (LS) estimator of the remaining parameters. However, after elimination estimates of these parameters are biased, if the full model is correct. In his recent paper, Boscher (1991) showed that the LS-estimator in the special case of a mean shift model (cf. Cook and Weisberg, 1982) which assumes no “outliers” can be considered in the framework of a linear regression model where some variables are deleted. He derived conditions under which this estimator outperforms the LS-estimator of the full model in terms of the mean squared error (MSE)-matrix criterion. We demonstrate that this approach can be extended to the general set-up of dropping variables. Necessary and sufficient conditions for the MSE-matrix superiority of the LS-estimator in the reduced model over that in the full model are derived. We also provide a uniformly most powerful F-statistic for testing the MSE-improvement.     

GARCH models,tail indexes and error distributions: An empirical investigation

We perform a large simulation study to examine the extent to which various generalized autoregressive conditional heteroskedasticity (GARCH) models capture extreme events in stock market returns. We estimate Hill's tail indexes for individual S&P 500 stock market returns and compare these to the tail indexes produced by simulating GARCH models. Our results suggest that actual and simulated values differ greatly for GARCH models with normal conditional distributions, which underestimate the tail risk. By contrast, the GARCH models with Student's t conditional distributions capture the tail shape more accurately, with GARCH and GJR-GARCH being the top performers.     

Transforms for finding probabilities and variate values of a distribution function in tables of a related distribution function

Het verband tussen enige bedkende verdelingen.
In dit artikel worden enkele schema's gegeven voor het verband tussen een aantal van de meest bekende verdelingen. Bovendien wordt een aantal andere relaties besproken. He doel van dit artikel is drieledig. Ten eerste kunnen statistische afdelingen, die in het bezit zijn van een uitgebreide tabellenverzamelign de gegeven transformaties gebruiken om bestaande tabellen uit andere tabellen te vervolledigen ten dienste van bepaalde praktische toepassingen. Ten tweede kunnen statistici, die niet in het bezit van volledige tabellenverzamelingen zijn, ontbreakende tabellen vervanden tussen deze verdelingen leiden tot meer inzicht in bepaalde toetsings-methoden. Dit laatste punt wordt niet uitvoerig uitgewerkt.
Par. 1. geeft een inleiding en overzicht; par. 2 vermeldt de gebruikte symbolen en verwijst naar bekende tabellenverzamelingen; par. 3 en 4 geven schematisch de transformaties weer; par. 5 geeft een schematisch overzicht van de relaties; par. 6 geeft een aantal numerieke voorbeelden; in de appendix worden de wiskundige formules van de verdelingen vermeld.
Afleidingen worden niet gegeven.     

K-matrix-class estimators and the full-information maximum-likelihood estimator as a special case

This article presents a unified treatment of simultaneous system estimation. A general class of full-information estimators is proposed, called K-matrix-class (KMC). It is shown that the K-matrix-class includes both full-information maximum-likelihood and three-stage least- squares estimators as special cases and that the k-class can be regarded as a subclass of the K-matrix-class. Conditions under which KMC estimators are consistent (similar to those of the k-class estimators) are given. Furthermore, as a full information-generalization of the double k-class estimators, the double K-matrix-class estimators (DKMC) are proposed.     

The historical construction of correlation as a conceptual and operative instrument for empirical research

This article is meant to reconstruct—from the standpoint of sociology and history of science—the development of the concept and the operative instruments of statistical correlation. The starting point is the discussion of some key mathematical aspects of the Error Theory, including a detailed analysis of the various positions regarding its contributions, if any, to the theory of correlation. Then proceeds to examine how the concept (and its relative instruments) emerged in its modern sense, by the late Nineteenth century, thanks to the work of Francis Galton. Finally, it considers the numerous contributions that rendered possible the formalisation and generalisation of both Galton’s concept and methodological tools, in particular those of Karl Pearson, but also those of Walter Weldon, Francis Ysidro Edgeworth, George Udny Yule and Charles Spearman.

“Co-relation or correlation of structure” is a phrase much used in biology, and not least in that branch of it which refers to heredity, and the idea is even more frequently present than the phrase; but I am not aware of any previous attempt to define it clearly, to trace its mode of action in detail, or to show how to measure its degree. Francis Galton (1888, 135)

     

Monotone empirical Bayes estimators for the continuous one-parameter exponential family

Abstract  A class of empirical Bayes estimators (EBE's) is proposed for estimating the natural parameter of a one-parameter exponential family. In contrast to related EBE's proposed and investigated until now, the EBE's presented in this paper possess the nice property of being monotone by construction. Based on an arbitrary reasonable estimator of the underlying marginal density, a simple algorithm is given to construct a monotone EBE. Two representations of these EBE's are given, one of which serves as a tool in establishing asymptotic results, while the other one, related with isotonic regression, proves useful in the actual computation.     

Strong representation of the presmoothed quantile function estimator for censored data

We consider lifetime data subject to right random censorship. In this context, this paper deals with the topic of estimating the distribution function of the lifetime and the corresponding quantile function. As it has been shown that the classical Kaplan–Meier estimator of the distribution function can be improved by means of presmoothing ideas, we introduce a quantile function estimator via the presmoothed distribution function estimator studied by Cao et al. [Journal of Nonparametric statistics, Vol. 17 (2005) pp. 31–56.] The main result of this paper is an almost sure representation of this presmoothed estimator. As a consequence, its strong consistency and asymptotic normality are established. The performance of this new quantile estimator is analyzed in a simulation study and applied to a real data example.