Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.     

Percentile estimators in location-scale parameter families under absolute loss

Estimators of percentiles of location-scale parameter families are optimized based on median unbiasedness and absolute risk. Median unbiased estimators and minimum absolute risk estimators are shown to exist within a class of equivariant estimators and depend upon medians of two completely specified distributions. This work extends earlier findings to a larger class of equivariant estimators. These estimators are illustrated in the normal and exponential distributions.     

On universal admissibility of scale parameter estimators

A characterization result of Kushary (1998) regarding universal admissibility of equivariant estimators in the one parameter gamma distribution is generalized to a scale family of distributions with monotone likelihood ratio. New examples are given, among them the F-distribution with a scale parameter. In particular, universal admissibility is characterized within the class of location-scale equivariant estimators of the ratio of the variances of two normal distributions with unknown means. In this context the maximum likelihood estimator is shown to be universally inadmissible by virtue of a general sufficient condition for universal inadmissibility of a scale equivariant estimator. Received: January 2000     

On the existence of locally mvu estimators

It is proved that there exists an unbiased estimator for some real parameter of a class of distributions, which has minimal variance for some fixed distribution among all corresponding unbiased estimators, if and. only if the corresponding minimal variances for all related unbiased estimation problems concerning finite subsets of the underlying family of distributions are bounded. As an application it is shown that there does not exist some unbiased estimator for θ^k+c(ε≥0) with minimal variance for θ =0 among all corresponding unbiased estimators on the base of k i.i.d. random variables with a Cauchy-distribution, where θ denotes some location parameter.     

Estimators for the common principal components model based on reweighting: influence functions and Monte Carlo study

The common principal components model for several groups of multivariate observations is a useful parsimonious model for the scatter structure which assumes equal principal axes but different variances along those axes for each group. Due to the lack of resistance of the classical maximum likelihood estimators for the parameters of this model, several robust estimators have been proposed in the literature: plug-in estimators and projection-pursuit (PP) type estimators. In this paper, we show that it is possible to improve the low efficiency of the projection-pursuit estimators by applying a reweighting step. More precisely, we consider plug-in estimators obtained by plugging a reweighted estimator of the scatter matrices into the maximum likelihood equations defining the principal axes. The weights considered penalize observations with large values of the influence measures defined by Boente et al. (2002). The new estimators are studied in terms of theoretical properties (influence functions and asymptotic variances) and are compared with other existing estimators in a simulation study.     

Simultaneous estimation of hazard rates of several exponential populations

Suppose independent random samples are drawn from k (2) populations with a common location parameter and unequal scale parameters. We consider the problem of estimating simultaneously the hazard rates of these populations. The analogues of the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU) and the best scale equivariant (BSE) estimators for the one population case are improved using Rao‐Blackwellization. The improved version of the BSE estimator is shown to be the best among these estimators. Finally, a class of estimators that dominates this improved estimator is obtained using the differential inequality approach.     

Estimation of the order restricted scale parameters for two populations from the Lomax distribution

The usual methods of estimating the unknown parameters of a distribution, use only the information given from the sample data. In many cases, there is, also, another important information for estimating the unknown parameters of our model, such as the order of these parameters, and this last information improves the quality of estimation. In this paper, we deal with the problem of estimating the ordered scale parameters from two populations of the multivariate Lomax distribution, with unknown location parameters. It is proved that the best equivariant estimators of the scale parameters (in the unrestricted case) are not admissible and we construct estimators that improve upon the usual ones (when these parameters are known to be ordered).     

Least squares, preliminary test and Stein-type estimation in general vector AR(p) models

In the general vector autoregressive process AR ( p ), multivariate least square estimation (LSE)/maximum likelihood estimation (MLE) of a subset of the parameters is considered when the complementary subset is suspected to be redundant. This may be viewed as a special case of linear constraints of autoregressive parameters. We incorporate this nonsample information in the estimation process and propose preliminary test and Stein-type estimators for the target subset of parameters. Under local alternatives their asymptotic properties are investigated and compared with those of unrestricted and restricted LSE. The dominance picture of the estimators is presented.     

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.     

A new look at variance estimation based on low,high and closing prices taking into account the drift

The joint distribution of low, high and closing prices of the arithmetic Brownian motion is used to evaluate the properties of the most popular estimators of the variance constructed on the basis of high, low and closing prices. The expected values and mean square errors of the Parkinson, Garman–Klass and Rogers–Satchell estimators for the process with a zero drift and a non‐zero drift are derived. Moreover, new volatility estimators, more efficient in the majority of financial applications than the Rogers–Satchell estimator, are proposed. The considered estimators are applied to the estimation of the volatility of the Polish stock index WIG20.     

Estimation of the mean of the selected population under asymmetric loss function

This paper is concerned with the comparison of seven estimators of the mean of the selected population from two normal populations with unknown means and common known variance under an asymmetric loss namely the LINEX loss function. The proposed estimators are invariant under location transformation. The bias and risks of the seven estimators are computed and compared. The conclusion recommend the use of δ_P (σ) which is simple to use and it is minimax. Received: January 1999     

Large-sample estimation strategies for eigenvalues of a Wishart matrix

The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.     

Restricted maximum likelihood estimation under Eisenhart model Ill

For a balanced two-way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non-negativity requirements of the variance components by L ee and K apadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non-negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.     

A Comparative Analysis of Different IV and GMM Estimators of Dynamic Panel Data Models

It is well known that the usual procedures for estimating panel data models are inconsistent in the dynamic setting. A large number of consistent estimators however, have been proposed in the literature. This paper provides a survey of the majority of mainstream estimators, which tend to consist of IV and GMM ones. It also considers a newly proposed extension to the promising Wansbeek–Bekker estimator (Harris & Mátyás, 2000). To provide guidance to the applied researcher working on micro-datasets, the small sample performance of these estimators is evaluated using a set of Monte Carlo experiments.     

Overnight stock returns and realized volatility

The information flow in modern financial markets is continuous, but major stock exchanges are open for trading for only a limited number of hours. No consensus has yet emerged on how to deal with overnight returns when calculating and forecasting realized volatility in markets where trading does not take place 24 hours a day. Based on a recently introduced formal testing procedure, we find that for the S&P 500 index, a realized volatility estimator that optimally incorporates overnight information is more accurate in-sample. In contrast, estimators that do not incorporate overnight information are more accurate for individual stocks. We also show that accounting for overnight returns may affect the conclusions drawn in an out-of-sample horserace of forecasting models. Finally, there is considerably less variation in the selection of the best out-of-sample forecasting model when only the most accurate in-sample RV estimators are considered.     

Detecting Outliers in Frontier Models: A Simple Approach

In frontier analysis, most of the nonparametric approaches (DEA, FDH) are based on envelopment ideas which suppose that with probability one, all the observed units belong to the attainable set. In these deterministic frontier models, statistical theory is now mostly available (Simar and Wilson, 2000a). In the presence of super-efficient outliers, envelopment estimators could behave dramatically since they are very sensitive to extreme observations. Some recent results from Cazals et al. (2002) on robust nonparametric frontier estimators may be used in order to detect outliers by defining a new DEA/FDH deterministic type estimator which does not envelop all the data points and so is more robust to extreme data points. In this paper, we summarize the main results of Cazals et al. (2002) and we show how this tool can be used for detecting outliers when using the classical DEA/FDH estimators or any parametric techniques. We propose a methodology implementing the tool and we illustrate through some numerical examples with simulated and real data. The method should be used in a first step, as an exploratory data analysis, before using any frontier estimation.     

THE BIVARIATE EXPONENTIAL DISTRIBUTION; A REVIEW AND SOME NEW RESULTS

Various models have been proposed as bivariate forms of the exponential distribution. A brief but comprehensive review is presented which classifies, interrelates and contrasts the different models and outlines what is known about distributional properties, applicability and estimation and testing of parameters (particularly the association parameter). Some new results are presented for one particular model. Maximum likelihood, and moment–type, estimators of the association parameter are examined. Asymptotic variances are derived and attention is given to the relative efficiency of the estimators and to problems of their evaluation.     

Estimation procedures for a family of density functions representing various life-testing models

A family of density functions is considered which contains several life-testing models as specific cases. Uniformly minimum variance unbiased estimators are obtained for the positive and negative powers of the parameter, moments and reliability function. These general results provide the estimators for the specific models.     

Estimation of the parameters of log-gamma distribution using order statistics

In this work we propose a technique of estimating the location parameter μ and scale parameter σ of log-gamma distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.     

Uniformly most accurate equivariant prediction limit

Summary Prediction limits are widely used for reliability problems and other related problems. The determination of prediction limits has been extensively investigated, but few optimal properties of these limits have been explored. This paper introduces a concept of uniform accuracy in order to compare equivariant prediction limits and show that the prediction limits used for the normal distribution and the exponential distribution are uniformly most accurate equivariant.