Some model-restricted shrinkage estimators for contingency tables with missing data

For contingency tables with extensive missing data, the unrestricted MLE under the saturated model, computed by the EM algorithm, is generally unsatisfactory. In this case, it may be better to fit a simpler model by imposing some restrictions on the parameter space. Perlman and Wu (1999) propose lattice conditional independence (LCI) models for contingency tables with arbitrary missing data patterns. When this LCI model fits well, the restricted MLE under the LCI model is more accurate than the unrestricted MLE under the saturated model, but not in general. Here we propose certain empirical Bayes (EB) estimators that adaptively combine the best features of the restricted and unrestricted MLEs. These EB estimators appear to be especially useful when the observed data is sparse, even in cases where the suitability of the LCI model is uncertain. We also study a restricted EM algorithm (called the ER algorithm) with similar desirable features. Received: July 1999     

Some strange properties of panel data estimators

We study the biases that are likely to arise in practice with panel data when parameters vary across individuals, but this is not allowed for in estimation. We consider both stationary and non-stationary regressors. We find that biases can be severe for relatively small parameter variation, and that this problem is hard to detect. We study in some detail by Monte-Carlo the performance of the Anderson-Hsiao estimator in the presence of this particular mis-specification.     

Some results concerningHorvitz-Thompson's T 1-class of estimators

Summary The question of availability of a unique best estimator inHorvitz-Thompson's T ₁-class of estimators is dealt with in general and a simple sampling scheme yielding such an estimator is found out; the efficiency of the same is studied and a method of improving on it is also discussed.     

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.     

The graph of the k-class estimator: An algebraic and statistical interpretation

This paper presents an algebraic analysis of the graphs of the k-class estimator, its asymptotic standard error and asymptotic t-ratio as functions of k for a single structural equation containing one or more endogenous explanatory variables. These results are illustrated by the corresponding graphs of the second and fifth equations of the Girshick-Haavelmo (1947) Demand for Food Model.Tests of the rank condition for identification are also developed. They are found to involve the values of k which explode the k-class estimator.     

Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties

We examine several modified versions of the heteroskedasticity-consistent covariance matrix estimator of Hinkley (1977) and White (1980). On the basis of sampling experiments which compare the performance of quasi t-statistics, we find that one estimator, based on the jackknife, performs better in small samples than the rest. We also examine the finite-sample properties of using modified critical values based on Edgeworth approximations, as proposed by Rothenberg (1984). In addition, we compare the power of several tests for heteroskedasticity, and find that it may be wise to employ the jackknife heteroskedasticity-consistent covariance matrix even in the absence of detected heteroskedasticity.     

Recursions for the two-stage least-squares estimators

This paper presents recursion formulae for the two-stage least-squares estimators of the structural coefficients in a simultaneous equation model and for the residual sum of squares used in estimating the asymptotic covariance matrix. Included are formulae for updating estimates when a new set of observations is obtained and for revising estimates when a set of observations is discarded. The recursion formulae should prove to be of both practical and theoretical interest to econometricians.     

Price asymptotics

This paper examines the pricing decisions of a seller facing an unknown demand function. It is assumed that partial information, in the form of an independent random sample of values, is available. The optimal price for the inferred demand satisfies a consistency property—as the size of the sample increases, the maximum profit and price approach the values for the case where demand is known. The main results deduced here are asymptotics for prices. Prices converge at a rate of O _p (n ^−1/3) with a limit that can be expressed as a functional of a Gaussian process. Implications for the comparison of mechanisms are discussed.      

Some small sample properties of estimators and test statistics in the multivariate logit model

In this paper we compare three estimators for the multivariate logit model: two asymptotically efficient methods and a consistent method. The most interesting result is that at sample sizes of more than one hundred, the simple consistent estimator performs almost as well as the asymptotically efficient estimators.     

Dominating estimators for minimum-variance portfolios

In this paper, we derive two shrinkage estimators for minimum-variance portfolios that dominate the traditional estimator with respect to the out-of-sample variance of the portfolio return. The presented results hold for any number of assets d≥4$d\ge 4$ and number of observations n≥d+2$n\ge d+2$. The small-sample properties of the shrinkage estimators as well as their large-sample properties for fixed d$d$ but n→∞$n\to \infty$ and n,d→∞$n,d\to \infty$ but n/d→q≤∞$n/d\to q\le \infty$ are investigated. Furthermore, we present a small-sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification.     

Performance of empirical Bayes estimators of random coefficients in multilevel analysis: Some results for the random intercept-only model

For a multilevel model with two levels and only a random intercept, the quality of different estimators of the random intercept is examined. Analytical results are given for the marginal model interpretation where negative estimates of the variance components are allowed for. Except for four or five level-2 units, the Empirical Bayes Estimator (EBE) has a lower average Bayes risk than the Ordinary Least Squares Estimator (OLSE). The EBEs based on restricted maximum likelihood (REML) estimators of the variance components have a lower Bayes risk than the EBEs based on maximum likelihood (ML) estimators. For the hierarchical model interpretation, where estimates of the variance components are restricted being positive, Monte Carlo simulations were done. In this case the EBE has a lower average Bayes risk than the OLSE, also for four or five level-2 units. For large numbers of level-1 (30) or level-2 units (100), the performances of REML-based and ML-based EBEs are comparable. For small numbers of level-1 (10) and level-2 units (25), the REML-based EBEs have a lower Bayes risk than ML-based EBEs only for high intraclass correlations (0.5).     

Model averaging estimators for the stochastic frontier model

Journal of Productivity Analysis - Model uncertainty is a prominent feature in many applied settings. This is certainty true in the efficiency analysis realm where concerns over the proper...     

Robust estimators for estimating discontinuous functions

We study the asymptotic behavior of a wide class of kernel estimators for estimating an unknown regression function. In particular we derive the asymptotic behavior at discontinuity points of the regression function. It turns out that some kernel estimators based on outlier robust estimators are consistent at jumps.     

Approximations for densities of sufficient estimators

A simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described. It is an extension of the one developed by O. Barndorff-Nielsen and D.R. Cox (1979) for exponential families. A series expansion in powers of n^?1 is derived of which the first term has an error of order n^?1 which can effectively be reduced to $\text{n}{}^{\mathrm{<mtext>-</mtext><mtext>?3</mtext><mtext>2</mtext>}}$ by renormalization. The results obtained are similar to those given by H.E. Daniels's (1954) saddlepoint method but the derivations are simpler. A brief treatment of approximations to conditional densities is given. Theorems are proved which extend the validity of the multivariate Edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors. These extensions permit the treatment of problems arising in time series analysis. The technique is used by J. Durbin (1980) to obtain approximations to the densities of partial serial correlation coefficients.     

Hermite series estimators for probability densities

Summary For Hermite series density estimators assertions about rates of convergence of MSE, MISE and about asymptotic normality are given. Moreover, we study the behaviour of these estimators if the density is not continuous. Hermite series estimators with random length are also considered. Convergence in probability and a.s. of these estimators is proved.     

On the admissibility of estimators for the finite population variance

Summary With each unti of a finite population is associated an unknown variate value. We are interested in the variance of these values and consider (1) simple random sampling without replacement. (2) quadratic loss and (3) a one parameter class of estimators. We determine all admissible elements of the class. The usual unbiased estimator for the variance which is an element of the class considered turns out to be inadmissible.     

Riesz estimators

We consider properties of estimators that can be written as vector lattice (Riesz space) operations. Using techniques widely used in economic theory and functional analysis, we study the approximation properties of these estimators paying special attention to additive models. We also provide two algorithms RIESZVAR(i-ii) for the consistent parametric estimation of continuous multivariate piecewise linear functions.     

Some sampling properties of minimum expected loss (MELO) estimators of structural coefficients

Some sampling properties of Zellner's (1978) MELO estimates of structural coefficients of linear simultaneous equation models are examined by a series of sampling experiments. The MELO estimates appear to have more pronounced biases in estimating structural coefficients than the 2SLS estimates. However, MELO is found to outperform 2SLS according to several criteria, including MSE and MAE in a wide range of situations generated by varying structural coefficients, the variance-covariance matrix of structural disturbances, and the sample size. The magnitude of absolute sampling errors, the estimation of the variance of structural disturbances, and the large-sample standard errors are also compared among OLS, 2SLS, and MELO.