A generalized inverse regression estimator in multi-univariate linear calibration

A multivariate linear calibration problem, in which response variable is multivariate and explanatory variable is univariate, is considered. In this paper a class of generalized inverse regression estimators is proposed in multi-univariate linear calibration. It includes the classical estimator and the inverse regression one (or Krutchkoff estimator). For the proposed estimator we derive the expressions of bias and mean square error (MSE). Furthermore the behavior of these characteristics is investigated through an analytical method. In addition through a numerical study we confirm the existence of a generalized inverse regression estimator to improve both the classical and the inverse regression estimators on the MSE criterion.     

Some small-sample properties of some recently proposed multivariate outlier detection techniques

Recently, several new robust multivariate estimators of location and scatter have been proposed that provide new and improved methods for detecting multivariate outliers. But for small sample sizes, there are no results on how these new multivariate outlier detection techniques compare in terms of p _n , their outside rate per observation (the expected proportion of points declared outliers) under normality. And there are no results comparing their ability to detect truly unusual points based on the model that generated the data. Moreover, there are no results comparing these methods to two fairly new techniques that do not rely on some robust covariance matrix. It is found that for an approach based on the orthogonal Gnanadesikan–Kettenring estimator, p _n can be very unsatisfactory with small sample sizes, but a simple modification gives much more satisfactory results. Similar problems were found when using the median ball algorithm, but a modification proved to be unsatisfactory. The translated-biweights (TBS) estimator generally performs well with a sample size of n≥20 and when dealing with p-variate data where p≤5. But with p=8 it can be unsatisfactory, even with n=200. A projection method as well the minimum generalized variance method generally perform best, but with p≤5 conditions where the TBS method is preferable are described. In terms of detecting truly unusual points, the methods can differ substantially depending on where the outliers happen to be, the number of outliers present, and the correlations among the variables.     

Generalized order statistics from two parameter uniform distribution

In this paper some distributional properties of the generalized order statistics from uniform distribution are given. The minimum variance linear unbiased as well best ( in the sense of minimum mean squared error) invariant estimators of the parameters of the two parameter uniform distribution based on the first m generalized order statistics are presented.     

Sufficient conditions for the admissibility under the LINEX loss function in non-regular case

Consider an estimation problem under the LINEX loss function in one-parameter non-regular distributions where the endpoint of the support depends on an unknown parameter. The purpose of this paper is to give sufficient conditions for a generalized Bayes estimator of a parametric function to be admissible. Also, it is shown that the main result in this paper is an extension of the quadratic loss case. Some examples are given.     

A new approach to use multivariate auxiliary information in sample surveys

A new approach to form multivariate difference estimator is suggested which does not require the knowledge of unknown population parameters as such. It gives minimum variance among the class of multivariate difference estimators. The performance of this estimator with respect to Des Raj's (J. Amer. Statist. Assoc. 60 (1965), 270–277) multivariate difference estimator is illustrated. Using the information on two auxiliary variates, the robustness of Des Raj's estimator y_d is studied empirically. Two new estimators to estimate population mean/total are developed on the same lines as that of y_d. The performance of these estimators is studied for a wide variety of populations.     

Outliers detection with the minimum covariance determinant estimator in practice

Robust statistics have slowly become familiar to all practitioners. Books entirely devoted to the subject (e.g. [R.A. Maronna, R.D. Martin, V.J. Yohai, Robust Statistics: Theory and Methods. John Wiley &; Sons, New York, NY, USA, 2006; P.J. Rousseeuw, A.M. Leroy, Robust Regression and Outlier Detection, John Wiley &; Sons, New York, NY, USA, 1987], …) are without any doubt responsible for the increased practice of robust statistics in all fields of applications. Even classical books often have at least one chapter (or parts of chapters) which develops robust methodology. The improvement of computing power has also contributed to the development of a wider and wider range of available robust procedures. However, this success story is now menacing to get backwards: non-specialists interested in the application of robust methodology are faced with a large set of (assumed equivalent) methods and with over-sophistication of some of them. Which method should one use? How should the (numerous) parameters be optimally tuned? These questions are not so easy to answer for non-specialists! One could then argue that default procedures are available in most statistical software (Splus, R, SAS, Matlab, …). However, using as illustration the detection of outliers in multivariate data, it is shown that, on one hand, it is not obvious that one would feel confident with the output of default procedures, and that, on the other hand, trying to understand thoroughly the tuning parameters involved in the procedures might require some extensive research. This is not conceivable when trying to compete with the classical methodology which (while clearly unreliable) is so straightforward. The aim of the paper is to help the practitioners willing to detect in a reliable way outliers in a multivariate data set. The chosen methodology is the Minimum Covariance Determinant estimator being widely available and intuitively appealing.     

Effects of the estimation of covariance matrix parameters in the generalized multivariate linear model

We Consider the generalized multivariate linear model and assume the covariance matrix of the p x 1 vector of responses on a given individual can be represented in the general linear structure form described by Anderson (1973). The effects of the use of estimates of the parameters of the covariance matrix on the generalized least squares estimator of the regression coefficients and on the prediction of a portion of a future vector, when only the first portion of the vector has been observed, are investigated. Approximations are derived for the covariance matrix of the generalized least squares estimator and for the mean square error matrix of the usual predictor, for the practical case where estimated parameters are used.     

Phi-Divergence Statistics for Testing Linear Hypotheses in Logistic Regression Models

In this paper we introduce and study two new families of statistics for the problem of testing linear combinations of the parameters in logistic regression models. These families are based on the phi-divergence measures. One of them includes the classical likelihood ratio statistic and the other the classical Pearson's statistic for this problem. It is interesting to note that the vector of unknown parameters, in the two new families of phi-divergence statistics considered in this paper, is estimated using the minimum phi-divergence estimator instead of the maximum likelihood estimator. Minimum phi-divergence estimators are a natural extension of the maximum likelihood estimator.     

A balanced multi-level rotation sampling design and its efficient composite estimators

We present a multi-level rotation sampling design which includes most of the existing rotation designs as special cases. When an estimator is defined under this sampling design, its variance and bias remain the same over survey months, but it is not so under other existing rotation designs. Using the properties of this multi-level rotation design, we derive the mean squared error (MSE) of the generalized composite estimator (GCE), incorporating the two types of correlations arising from rotating sample units. We show that the MSEs of other existing composite estimators currently used can be expressed as special cases of the GCE. Furthermore, since the coefficients of the GCE are unknown and difficult to determine, we present the minimum risk window estimator (MRWE) as an alternative estimator. This MRWE has the smallest MSE under this rotation design and yet, it is easy to calculate. The MRWE is unbiased for monthly and yearly changes and preserves the internal consistency in total. Our numerical study shows that the MRWE is as efficient as GCE and more efficient than the existing composite estimators and does not suffer from the drift problem [Fuller W.A., Rao J.N.K., 2001. A regression composite estimator with application to the Canadian Labour Force Survey. Surv. Methodol. 27 (2001) 45–51] unlike the regression composite estimators.     

Estimating powers of the generalized variance under the Pitman closeness criterion

For estimating powers of the generalized variance under a multivariate normal distribution with an unknown mean, the inadmissibility of the closest affine equivariant estimator is shown for the Pitman closeness criterion.     

Some Improvements in Numerical Evaluation of Symmetric Stable Density and Its Derivatives

A new multivariate inverse Polya distribution of order k, type I, is derived by means of a generalized urn scheme and by compounding the multivariate negative binomial distribution of order k, type I, of Philippou, Antzoulakos and Tripsiannis (1988) with the Dirichlet distribution. It is noted that this new distribution includes as special cases a new multivariate inverse hypergeometric distribution of order k and a new multivariate negative inverse one of the same order. The mean and variance-covariance of the multivariate inverse Polya distribution of order k, type I, are derived, and two known distributions of the same order are shown to be limiting cases of it.     

Estimation of parameters of a pareto distribution by generalized order statistics

In this paper we address the problem of estimating the parameters of Pareto II distribution based on generalized order statistics. The estimators based on order statistics and record values are shown to be special cases of these estimators.     

On progressively censored generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is considered in this paper. This lifetime distribution is capable of modeling various shapes of failure rates, and hence various shapes of aging criteria. The model can be considered as another useful two-parameter generalization of the IED. Maximum likelihood and Bayes estimates for two parameters of the generalized inverted exponential distribution (GIED) are obtained on the basis of a progressively type-II censored sample. We also showed the existence, uniqueness and finiteness of the maximum likelihood estimates of the parameters of GIED based on progressively type-II censored data. Bayesian estimates are obtained using squared error loss function. These Bayesian estimates are evaluated by applying the Lindley's approximation method and via importance sampling technique. The importance sampling technique is used to compute the Bayes estimates and the associated credible intervals. We further consider the Bayes prediction problem based on the observed samples, and provide the appropriate predictive intervals. Monte Carlo simulations are performed to compare the performances of the proposed methods and a data set has been analyzed for illustrative purposes.     

On the UMVU estimator for the generalized variance of the multinomial family

Abstract

The generalized variance is an important statistical indicator which appears in a number of statistical topics. It is a successful measure for multivariate data concentration. In this article, we established, in a closed form, the bias of the generalized variance maximum likelihood estimator of the Multinomial family. We also derived, with a complete proof, the uniformly minimum variance unbiased estimator (UMVU) for the generalized variance of this family. These results rely on explicit calculations, the completeness of the exponential family and the Lehmann–Scheffé theorem.     

Theoretical evaluation of prediction error in linear regression with a bivariate response variable containing missing data

Methods for linear regression with multivariate response variables are well described in statistical literature. In this study we conduct a theoretical evaluation of the expected squared prediction error in bivariate linear regression where one of the response variables contains missing data. We make the assumption of known covariance structure for the error terms. On this basis, we evaluate three well-known estimators: standard ordinary least squares, generalized least squares, and a James–Stein inspired estimator. Theoretical risk functions are worked out for all three estimators to evaluate under which circumstances it is advantageous to take the error covariance structure into account.     

Estimation of Generalized Variance Under an Asymetric Loss Function “Squared Log Error”

In this article, we consider the problem of estimating the generalized variance, when the observation follows from a non singular multivariate normal distribution with unknown mean under the squared log error loss function.     

A new hybrid estimation method for the generalized pareto distribution

ABSTRACT

The generalized Pareto distribution (GPD) is important in the analysis of extreme values, especially in modeling exceedances over thresholds. Most of the existing methods for estimating the scale and shape parameters of the GPD suffer from theoretical and/or computational problems. A new hybrid estimation method is proposed in this article, which minimizes a goodness-of-fit measure and incorporates some useful likelihood information. Compared with the maximum likelihood method and other leading methods, our new hybrid estimation method retains high efficiency, reduces the estimation bias, and is computation friendly.     

A class of Stein‐rules in multivariate regression model with structural changes

In this paper, we consider an estimation problem of the matrix of the regression coefficients in multivariate regression models with unknown change‐points. More precisely, we consider the case where the target parameter satisfies an uncertain linear restriction. Under general conditions, we propose a class of estimators that includes as special cases shrinkage estimators (SEs) and both the unrestricted and restricted estimator. We also derive a more general condition for the SEs to dominate the unrestricted estimator. To this end, we extend some results underlying the multidimensional version of the mixingale central limit theorem as well as some important identities for deriving the risk function of SEs. Finally, we present some simulation studies that corroborate the theoretical findings.     

Multivariate boundary kernels and a continuous least squares principle

Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.     

Inferences on a linear combination of K multivariate normal mean vectors

In this paper, the hypothesis testing and confidence region construction for a linear combination of mean vectors for K independent multivariate normal populations are considered. A new generalized pivotal quantity and a new generalized test variable are derived based on the concepts of generalized p-values and generalized confidence regions. When only two populations are considered, our results are equivalent to those proposed by Gamage et al. [Generalized p-values and confidence regions for the multivariate Behrens–Fisher problem and MANOVA, J. Multivariate Aanal. 88 (2004), pp. 117–189] in the bivariate case, which is also known as the bivariate Behrens–Fisher problem. However, in some higher dimension cases, these two results are quite different. The generalized confidence region is illustrated with two numerical examples and the merits of the proposed method are numerically compared with those of the existing methods with respect to their expected areas, coverage probabilities under different scenarios.