On piecewise linear density estimators

We study piecewise linear density estimators from the L ₁ point of view: the frequency polygons investigated by S cott (1985) and J ones et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L ₁ strongly consistent. We derive large deviation inequalities. For twice differentiable densities with compact support their expected L ₁ error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.     

Regular estimators for subclasses of linear estimators

Summary The concept of regular estimator is due toRoy/Chakravarti. For its application they confined to the most general class of linear estimators. The present paper considers some subclasses of linear 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.     

Linear programming-based estimators in simple linear regression

In this paper we introduce a linear programming estimator (LPE) for the slope parameter in a constrained linear regression model with a single regressor. The LPE is interesting because it can be superconsistent in the presence of an endogenous regressor and, hence, preferable to the ordinary least squares estimator (LSE). Two different cases are considered as we investigate the statistical properties of the LPE. In the first case, the regressor is assumed to be fixed in repeated samples. In the second, the regressor is stochastic and potentially endogenous. For both cases the strong consistency and exact finite-sample distribution of the LPE is established. Conditions under which the LPE is consistent in the presence of serially correlated, heteroskedastic errors are also given. Finally, we describe how the LPE can be extended to the case with multiple regressors and conjecture that the extended estimator is consistent under conditions analogous to the ones given herein. Finite-sample properties of the LPE and extended LPE in comparison to the LSE and instrumental variable estimator (IVE) are investigated in a simulation study. One advantage of the LPE is that it does not require an instrument.     

A characterization of admissible linear estimators of fixed and random effects in linear models

In the paper the problem of simultaneous linear estimation of fixed and random effects in the mixed linear model is considered. A necessary and sufficient conditions for a linear estimator of a linear function of fixed and random effects in balanced nested and crossed classification models to be admissible are given.     

Kernel-weighted GMM estimators for linear time series models

This paper analyzes the higher-order asymptotic properties of generalized method of moments (GMM) estimators for linear time series models using many lags as instruments. A data-dependent moment selection method based on minimizing the approximate mean squared error is developed. In addition, a new version of the GMM estimator based on kernel-weighted moment conditions is proposed. It is shown that kernel-weighted GMM estimators can reduce the asymptotic bias compared to standard GMM estimators. Kernel weighting also helps to simplify the problem of selecting the optimal number of instruments. A feasible procedure similar to optimal bandwidth selection is proposed for the kernel-weighted GMM estimator.     

Necessary conditions for admissibility of matrix linear estimators in a multivariate linear model

This article provides necessary conditions for the admissibility of matrix linear estimators of an estimable parameter matrix linear function under two kinds of quadratic matrix loss functions in a multivariate linear model following a family of matrix normal distributions, where the covariance matrix associated is completely unknown. Further it is demonstrated that if a more concrete condition supplied for one of the subdivided conditions is satisfied, then the special condition concerning the Stein problem is necessary for the admissibility of the kind of estimators under each of the loss functions.     

Contamination in linear regression models and its influence on estimators

The consequences of the omission of possibly contaminated observations in a linear regression model for the performance of the ordinary least squares ( LS- ) estimator are discussed. We compare the ordinary L Sestimator with the corresponding 'never pooled' LS -estimator with respect to the matrix-valued mean squared error. Necessary and sufficient conditions are derived for the superiority of an estimator to another one and tests are proposed to check these conditions. Finally the resulting preliminary-test-estimators are investigated.     

Exponential convergence properties of linear estimators under exponential and uniform distribution

Summary Certain measures of asymptotic efficiency of test statistics are based on exponential convergence properties of the underlying error probabilities (Bahadur-, Hodges-Lehmann-efficiency). From a general large deviation theorem, that is specified to weighted sums of independent identically distributed (i.i.d.) random variables, such exponential convergence properties are derived for test statistics which are linear functions of order statistics of i.i.d. random variables under exponential and uniform distribution. For that purpose some smoothness-conditions for the weights have to be established. In a series of examples it is shown that these conditions are fulfilled for certain robust linear estimators of location or scale parameters. With the help of some numerical results two of them, namely Winsorized and trimmed mean, are compared with regard to the asymptotic relative efficiency against each other.

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Zusammenfassung Bestimmte asymptotische Effizienzbegriffe für Tests basieren auf einem exponentiellen Konvergenzverhalten der zugrundeliegenden Fehlerwahrscheinlichkeiten (Bahadur-, Hodges-Lehmann-Effizienz). Mit Hilfe eines allgemeinen Satzes üver Wahrscheinlichkeiten großer Abweichungen, der spezialisiert wird auf gewichtete Summen unabhängiger, identisch verteilter (i.i.d.) Zufallsvariablen mit momenterzeugenden Funktionen, wird ein solches exponentielles Konvergenzverhalten nachgewiesen für Linearkombinationen von order statistics von i.i.d. Zufallsvariablen unter Exponential- und Rechteckverteilung. Dazu sind bestimmte Bedingungen an die Gewichte zu stellen. In einigen Beispielen wird gezeigt, daß solche Gewichtsbedingungen für eine Reihe von robusten Schätzern erfüllt sind. Zwei spezielle, nämlich das Winsorisierte und getrimmte Mittel, werden mit Hilfe einiger numerischer Ergebnisse hinsichtlich ihrer asymptotischen Effizienz miteinander verglichen.

     

On construction of admissible quadratic estimators

A random linear model for spatially located sensors measured intensity of a source of signals in discrete instants of time is considered. A basis of a quadratic subspace useful in quadratic estimation of a function of model parameters is given. Received: December 1999     

A new class of limited-information estimators for simultaneous equations systems

The TSLS and LIML estimators are evaluated by means of a new class of limited-information estimators, the so-called Ω-class estimators. Under certain assumptions the Ω-class estimator is a maximun-likelihood estimator. These assumptions are superfluous, however, if we view the Ω-class as a class of minimun-distance estimators; all the members are shown to be consistent under general conditions. Besides the TSLS and the LIML estimators some other interesting members are introduced, and it is shown that, under certain conditions, the Ω-class estimators are weighted averages of different TSLS estimators. The use of TSLS in small samples is criticized; an alternative estimator is proposed.     

Examining bias in estimators of linear rational expectations models under misspecification

Most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value. We investigate the asymptotic bias of generalized method of moment (GMM) and maximum likelihood (ML) estimators in such models under misspecification. We consider several misspecifications, and focus more specifically on the case of omitted dynamics in the dependent variable. In a stylized DGP, we derive analytically the asymptotic biases of these estimators. We establish that in many cases of interest the two estimators of the degree of forward-lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter. We also propose a quasi-Hausman test of misspecification based on the difference between the GMM and ML estimators. Using Monte-Carlo simulations, we show that the ordering and direction of the estimators still hold in a more realistic New Keynesian macroeconomic model. In this set-up, misspecification is in general found to be more harmful to GMM than to ML estimators.     

Moment consistency of estimators in partially linear models under NA samples

Consider the heteroscedastic regression model Y ^(j)(x _in , t _in ) = t _in β + g(x _in ) + σ _in e ^(j)(x _in ), 1 ≤ j ≤ m, 1 ≤ i ≤ n, where s_in²=f(u_in){\sigma_{in}^{2}=f(u_{in})}, (x _in , t _in , u _in ) are fixed design points, β is an unknown parameter, g(·) and f(·) are unknown functions, and the errors {e ^(j)(x _in )} are mean zero NA random variables. The moment consistency for least-squares estimators and weighted least-squares estimators of β is studied. In addition, the moment consistency for estimators of g(·) and f(·) is investigated.     

Kantorovich inequalities and efficiency comparisons for several classes of estimators in linear models

New matrix, determinant and trace versions of the Kantorovich inequality (KI) involving two positive definite matrices are presented. Some of these are used to study the efficiencies of minimum-distance (MD) estimators, generalized method-of-moments (GMM) estimators and several estimators specific to longitudinal or panel-data analysis. They are also used to give upper bounds for the determinant and trace of the asymptotic variance matrix of a weighted least-squares (WLS) estimator in the generalized linear model.