Gamma-minimax estimation of a multivariate normal mean

Summary The mean vector of a multivariate normal distribution is to be estimated. A class Γ of priors is considered which consists of all priors whose vector of first moments and matrix of second moments satisfy some given restrictions. The Γ-minimax estimator under arbitrary squared error loss is characterized. The characterization follows from an application of a result of Browder and Karamardian published in Ichiishi (1983) which is a special version of a minimax inequality due to Ky Fan (1972). In particular, it is shown that within the set of all estimators a linear estimator is Γ-minimax. The authors would like to thank the Deutsche Forschungsgemeinschaft for financial support.     

Rough-and-ready assessment of the degree and importance of smoothing in functional estimation

In nonparametric estimation of functionals of a distribution, it may or may not be desirable, or indeed necessary, to introduce a degree of smoothing into this estimation. In this article, I describe a method for assessing, with just a little thought about the functional of interest, (i) whether smoothing is likely to prove worthwhile, and (ii) if so, roughly how much smoothing is appropriate (in order-of-magnitude terms). This rule-of-thumb is not guaranteed to be accurate nor does it give a complete answer to the smoothing problem. However, I have found it very useful over a number of years; many examples of its use, and limitations, are given.     

Iterative Density Estimation from Contaminated Observations

We introduce an iterative procedure for estimating the unknown density of a random variable X from n independent copies of Y=X+ɛ, where ɛ is normally distributed measurement error independent of X. Mean integrated squared error convergence rates are studied over function classes arising from Fourier conditions. Minimax rates are derived for these classes. It is found that the sequence of estimators defined by the iterative procedure attains the optimal rates. In addition, it is shown that the sequence of estimators converges exponentially fast to an estimator within the class of deconvoluting kernel density estimators. The iterative scheme shows how, in practice, density estimation from indirect observations may be performed by simply correcting an appropriate ordinary density estimator. This allows to assess the effect that the perturbation due to contamination by ɛ has on the density to be estimated. We also suggest a method to select the smoothing parameter required by the iterative approach and, utilizing this method, perform a simulation study.     

ON THE ESTIMATION OF POPULATION MEANS AND RATIOS USING SUPPLEMENTARY INFORMATION

In this paper we discuss the problem of estimating population means and ratios of population means using supplementary information on an auxiliary variable. Two classes of estimators are proposed, depending on two parameters. The bias and mean square error of each of the involved estimators is obtained to the first order of approximation. It is shown that, with a proper choice of the values for the parameters, the estimators are more efficient than the conventional estimators. Numerical examples are provided.     

Estimation of ratio of population means in survey sampling when some observations are missing

This paper considers the estimation of the ratio of population means when some observations are missing. Four estimators are presented and their bias and mean square error properties are studied. Received: June 1998     

Some theoretical results on forecast combinations

This paper proposes a framework for the analysis of the theoretical properties of forecast combination, with the forecast performance being measured in terms of mean squared forecast errors (MSFE). Such a framework is useful for deriving all existing results with ease. In addition, it also provides insights into two forecast combination puzzles. Specifically, it investigates why a simple average of forecasts often outperforms forecasts from single models in terms of MSFEs, and why a more complicated weighting scheme does not always perform better than a simple average. In addition, this paper presents two new findings that are particularly relevant in practice. First, the MSFE of a forecast combination decreases as the number of models increases. Second, the conventional approach to the selection of optimal models, based on a simple comparison of MSFEs without further statistical testing, leads to a biased selection.     

On optimal allocations for estimating the surface of a random field

In general, the construction of optimal designs is apparently a difficult task for the approximation of a random field indexed by more than one dimension. Besides the rate of convergence of the minimum achievable error hardly anything is known until now. However, if there is an immanent structure present in the random field, then, taking this structure into account, improved estimates can be obtained. For this situation we present adequate designs which show, at least, a nearly optimal performance. work supported by 313/ARC/VII/93/151 of the DAAD work supported by Ku719/2-1 of the DFG     

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).     

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.     

动迁安置用房价格对动迁成本的影响分析

本文以CQ公司作为典型案例,运用方差和回归分析的方法,对动迁行业中安置用房价格给动迁 成本造成的影响程度进行了分析,为专业动迁公司的动迁成本测算提供参考。     

Frequentist model averaging for multinomial and ordered logit models

Multinomial and ordered Logit models are quantitative techniques which are used in a range of disciplines nowadays. When applying these techniques, practitioners usually select a single model using either information-based criteria or pretesting. In this paper, we consider the alternative strategy of combining models rather than selecting a single model. Our strategy of weight choice for the candidate models is based on the minimization of a plug-in estimator of the asymptotic squared error risk of the model average estimator. Theoretical justifications of this model averaging strategy are provided, and a Monte Carlo study shows that the forecasts produced by the proposed strategy are often more accurate than those produced by other common model selection and model averaging strategies, especially when the regressors are only mildly to moderately correlated and the true model contains few zero coefficients. An empirical example based on credit rating data is used to illustrate the proposed method. To reduce the computational burden, we also consider a model screening step that eliminates some of the very poor models before averaging.     

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.     

Boosting nonlinear predictability of macroeconomic time series

We apply the boosting estimation method in order to investigate to what extent and at what horizons macroeconomic time series have nonlinear predictability that comes from their own history. Our results indicate that the U.S. macroeconomic time series have more exploitable nonlinear predictability than previous studies have found. On average, the most favorable out-of-sample performance is obtained via a two-stage procedure, where a conventional linear prediction model is fitted first and the boosting technique is applied to build a nonlinear model for its residuals.     

Choosing instrumental variables in conditional moment restriction models

Properties of GMM estimators are sensitive to the choice of instrument. Using many instruments leads to high asymptotic asymptotic efficiency but can cause high bias and/or variance in small samples. In this paper we develop and implement asymptotic mean square error (MSE) based criteria for instrument selection in estimation of conditional moment restriction models. The models we consider include various nonlinear simultaneous equations models with unknown heteroskedasticity. We develop moment selection criteria for the familiar two-step optimal GMM estimator (GMM), a bias corrected version, and generalized empirical likelihood estimators (GEL), that include the continuous updating estimator (CUE) as a special case. We also find that the CUE has lower higher-order variance than the bias-corrected GMM estimator, and that the higher-order efficiency of other GEL estimators depends on conditional kurtosis of the moments.