Improved Estimation of Parameter Matrices in a One-Sample and Two-Sample Problems

In this paper, the problems of estimating the covariance matrix in a Wishart distribution (refer as one-sample problem) and the scale matrix in a multi-variate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their harmonic mean is proposed. It is shown that the new estimator dominates the best linear estimator under two scale invariant loss functions.     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.     

Robustness Comparisons of Some Classes of Location Parameter Estimators

Asymptotic biases and variances of M-, L- and R-estimators of a location parameter are compared under ε-contamination of the known error distribution F ₀ by an unknown (and possibly asymmetric) distribution. For each ε-contamination neighborhood of F ₀, the corresponding M-, L- and R-estimators which are asymptotically efficient at the least informative distribution are compared under asymmetric ε-contamination. Three scale-invariant versions of the M-estimator are studied: (i) one using the interquartile range as a preliminary estimator of scale: (ii) another using the median absolute deviation as a preliminary estimator of scale; and (iii) simultaneous M-estimation of location and scale by Huber's Proposal 2. A question considered for each case is: when are the maximal asymptotic biases and variances under asymmetric ε-contamination attained by unit point mass contamination at ∞? Numerical results for the case of the ε-contaminated normal distribution show that the L-estimators have generally better performance (for small to moderate values of ε) than all three of the scale-invariant M-estimators studied.     

Risk and Pitman closeness properties of feasible generalized double k-class estimators in linear regression models with non-spherical disturbances under balanced loss function

In this article, a family of feasible generalized double k-class estimator in a linear regression model with non-spherical disturbances is considered. The performance of this estimator is judged with feasible generalized least-squares and feasible generalized Stein-rule estimators under balanced loss function using the criteria of quadratic risk and general Pitman closeness. A Monte-Carlo study investigates the finite sample properties of several estimators arising from the family of feasible double k-class estimators.     

On improved estimation of normal precision matrix and discriminant coefficients

The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the “unbiased” estimator of the discriminant coefficients.     

The k-step spatial sign covariance matrix

The Sign Covariance Matrix is an orthogonal equivariant estimator of multivariate scale. It is often used as an easy-to-compute and highly robust estimator. In this paper we propose a k-step version of the Sign Covariance Matrix, which improves its efficiency while keeping the maximal breakdown point. If k tends to infinity, Tyler’s M-estimator is obtained. It turns out that even for very low values of k, one gets almost the same efficiency as Tyler’s M-estimator.     

Robust Bayesian prediction and estimation under a squared log error loss function

Robust Bayesian analysis is concerned with the problem of making decisions about some future observation or an unknown parameter, when the prior distribution belongs to a class Γ instead of being specified exactly. In this paper, the problem of robust Bayesian prediction and estimation under a squared log error loss function is considered. We find the posterior regret Γ-minimax predictor and estimator in a general class of distributions. Furthermore, we construct the conditional Γ-minimax, most stable and least sensitive prediction and estimation in a gamma model. A prequential analysis is carried out by using a simulation study to compare these predictors.     

Life expectancy under random censorship

This paper deals with estimation of life expectancy used in survival analysis and competing risk study under the condition that the data are randomly censored by K independent censoring variables. The estimator constructed is based on a theorem due to Berman [2], and it involves an empirical distribution function which is related to the Kaplan-Meier estimate used in biometry. It is shown that the estimator, considered as a function of age, converges weakly to a Gaussian process. It is found that for the estimator to have finite limiting variance requires the assumption that the censoring variables be stochastically larger than the “survival” random variable under investigation.     

Almost sure convergence of a tail index estimator in the presence of censoringConvergence presque sûre d'un index de queue en présence de censure

In Beirlant and Guillou [1] an exponential regression model was introduced on the basis of scaled log-spacing between subsequent extreme order statistics from a Pareto-type distribution in the presence of censoring. From this representation, they derived an estimator for the Pareto index. In this note, we revisit this adaptation of the popular Hill [5] estimator for heavy-tailed distributions, generalizing the almost sure convergence of this estimator under very general conditions on N_r, the number of non-censored observations. To cite this article: E. Delafosse, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 375–380.     

Analysis of ϕ-divergence for loglinear models with constraints under product-multinomial sampling

The loglinear model under product-multinomial sampling with constraints is considered. The asymptotic expansion and normality of the restricted minimum φ-divergence estimator (RMφDE) which is a generalization of the maximum likelihood estimator is presented. Then various statistics based on φ-divergence and RMφDE are used to test various hypothesis test problems under the model considered. These statistics contain the classical loglikelihood ratio test statistics and Pearson chi-squared test statistics. In the last section, a simulation study is implemented.     

Asymptotic efficiency of ridge estimator in linear and semiparametric linear models

The linear model with a growing number of predictors arises in many contemporary scientific endeavor. In this article, we consider the commonly used ridge estimator in linear models. We propose analyzing the ridge estimator for a finite sample size n and a growing dimension p. The existence and asymptotic normality of the ridge estimator are established under some regularity conditions when p→∞. It also occurs that a strictly linear model is inadequate when some of the relations are believed to be of certain linear form while others are not easily parameterized, and thus a semiparametric partial linear model is considered. For these semiparametric partial linear models with p>n, we develop a procedure to estimate the linear coefficients as if the nonparametric part is not present. The asymptotic efficiency of the proposed estimator for the linear component is studied for p→∞. It is shown that the proposed estimator of the linear component asymptotically performs very well.     

Post-<Emphasis Type="Italic">J</Emphasis> test inference in non-nested linear regression models

This paper considers the post-J test inference in non-nested linear regression models. Post-J test inference means that the inference problem is considered by taking the first stage J test into account. We first propose a post-J test estimator and derive its asymptotic distribution. We then consider the test problem of the unknown parameters, and a Wald statistic based on the post-J test estimator is proposed. A simulation study shows that the proposed Wald statistic works perfectly as well as the two-stage test from the view of the empirical size and power in large-sample cases, and when the sample size is small, it is even better. As a result, the new Wald statistic can be used directly to test the hypotheses on the unknown parameters in non-nested linear regression models.     

Combining empirical likelihood and generalized method of moments estimators: Asymptotics and higher order bias

This paper proposes an estimator combining empirical likelihood (EL) and the generalized method of moments (GMM) by allowing the sample average moment vector to deviate from zero and the sample weights to deviate from n⁻¹. The new estimator may be adjusted through free parameter δ∈(0,1) with GMM behavior attained as δ?0 and EL as δ?1. When the sample size is small and the number of moment conditions is large, the parameter space under which the EL estimator is defined may be restricted at or near the population parameter value. The support of the parameter space for the new estimator may be adjusted through δ. The new estimator performs well in Monte Carlo simulations.     

Location estimation in nonparametric regression with censored data

Consider the heteroscedastic model Y=m(X)+σ(X)?, where ? and X are independent, Y is subject to right censoring, m(·) is an unknown but smooth location function (like e.g. conditional mean, median, trimmed mean…) and σ(·) an unknown but smooth scale function. In this paper we consider the estimation of m(·) under this model. The estimator we propose is a Nadaraya-Watson type estimator, for which the censored observations are replaced by ‘synthetic’ data points estimated under the above model. The estimator offers an alternative for the completely nonparametric estimator of m(·), which cannot be estimated consistently in a completely nonparametric way, whenever high quantiles of the conditional distribution of Y given X=x are involved.We obtain the asymptotic properties of the proposed estimator of m(x) and study its finite sample behaviour in a simulation study. The method is also applied to a study of quasars in astronomy.     

Estimation of the Scale Matrix and its Eigenvalues in the Wishart and the Multivariate F Distributions

In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.     

Optimum calibration points estimating distribution functions

The calibration method has been widely discussed in the recent literature on survey sampling, and calibration estimators are routinely computed by many survey organizations. The calibration technique was introduced in [12] to estimate linear parameters as mean or total. Recently, some authors have applied the calibration technique to estimate the finite distribution function and the quantiles. The computationally simpler method in [14] is built by means of constraints that require the use of a fixed value t₀. The precision of the resulting calibration estimator changes with the selected point t₀. In the present paper, we study the problem of determining the optimal value t₀ that gives the best estimation under simple random sampling without replacement. A limited simulation study shows that the improvement of this optimal calibrated estimator over possible alternatives can be substantial.     

On near neighbour estimates of a multivariate density

A recent paper by Mack and Rosenblatt (J. Multivar. Anal.9 (1979), 1–15) has shown that near neighbour estimators of a density may perform more poorly than other kernel-type estimators, particularly for x values in the tail of a distribution. In order to overcome the difficulties discovered by Mack and Rosenblatt, a generalized type of near neighbour estimator is proposed. Here the window size, or bandwidth, is chosen as a function of near neighbour distances, rather than actually equal to one of the distances. Two forms for this function are suggested and it is proved that for large samples the resulting estimator does not suffer the drawbacks of the usual near neighbour estimator.     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.     

Tail dimension reduction for extreme quantile estimation

In a regression context where a response variable Y ∈ ? is recorded with a covariate X ∈ ? ^p , two situations can occur simultaneously: (a) we are interested in the tail of the conditional distribution and not on the central part of the distribution and (b) the number p of regressors is large. To our knowledge, these two situations have only been considered separately in the literature. The aim of this paper is to propose a new dimension reduction approach adapted to the tail of the distribution in order to propose an efficient conditional extreme quantile estimator when the dimension p is large. The results are illustrated on simulated data and on a real dataset.     

Optimal generalized ridge estimator under the generalized cross-validation criterion in linear regression

In this short paper, we mainly aim to study the generalized ridge estimator in a linear regression model. Through matrix techniques including Hadamard product and derivative of a vector, the globally optimal generalized ridge estimator is derived under the generalized cross-validation criterion from the theoretical point of view. It will be seen that the notion of linearized ridge estimator plays an important role in the process. A numerical example is applied to illustrate the main results of the paper.