Multivariate Chebyshev inequality in generalized measure space based on Choquet integral

The multivariate Chebyshev inequality for a random vector on probability measure space has been studied by numerous authors. One thing that seems missing is the multivariate version of Chebyshev inequality in non additive cases. In this article, we show that this inequality still works in generalized probability theory based on Choquet integral.     

Matrix Euclidean norm Wielandt inequalities and their applications to statistics

In this paper we shall establish a new matrix inequality which will fill the gap that there has not been any matrix Euclidean norm version of the Wielandt inequality in the literature yet. This inequality can be used to present an upper bound of a new measure of association which plays a very important role in statistics, especially in multivariate analysis. A new alternative based on Euclidean norm for relative gain of the covariance adjusted estimator of parameters is provided.     

Some results on dependent random variables and a connection with the multivariate s-increasing convex order

In this paper, some new concepts of dependence are introduced that generalize the concepts of positive and negative association. The new concepts of dependence are linked to the multivariate s-increasing convex order. Furthermore, a Kolmogorov-type inequality and a Hàjek–Rènyi inequality are proven that lead to an asymptotic result for these new random objects.     

Critical values and powers for tests of uniformity of directions under multivariate normality

The Rayleigh, Ajne, Giné and two new tests of uniformity of directions are investigated as tests for multivariate normality when the population mean vector and covariance matrix are assumed to be unknown. The new tests include one which is designed especially to detect for bimodal alternatives and one which is designed to perform well under a wide variety of alternatives. Simulated percentile points are obtained under the assumption that the variates constitute a random sample from a multivariate normal distribution. Powers of the five tests are compared under alternatives in the bivariate as well as higher dimensional settings.     

Multivariate Change Point Control Chart Based on Data Depth for Phase I Analysis

A multivariate change point control chart based on data depth (CPDP) is considered for detecting shifts in either the mean vector, the covariance matrix, or both of the processes for Phase I. The proposed chart is preferable from a robustness point of view, has attractive detection performance, and can be especially useful in Phase I analysis setting, where there is limited information about the underlying process. Comparison results and an illustrative example show that our CPDP chart has great potential for Phase I analysis of multivariate individual observations. The application of CPDP chart is illustrated in a real data example.     

Bayesian multivariate normal analysis under the extended reflected normal loss function

This paper considers the Bayesian analysis of the multivariate normal distribution under a new and bounded loss function, based on a reflection of the multivariate normal density function. The Bayes estimators of the mean vector can be derived for an arbitrary prior distribution of [d]. When the covariance matrix has an inverted Wishart prior density, a Bayes estimator of[d] is obtained under a bounded loss function, based on the entropy loss. Finally the admissibility of all linear estimators c[d]+ d for the mean vector is considered     

Spatial analysis of auto-multivariate lattice data

Most problems related to environmental studies are innately multivariate. In fact, in each spatial location more than one variable is usually measured. In geostatistics multivariate data analysis, where we intend to predict the value of a random vector in a new site, which has no data, cokriging method is used as the best linear unbiased prediction. In lattice data analysis, where almost exclusively the probability modeling of data is of concern, only auto-Gaussian model has been used for continuous multivariate data. For discrete multivariate data little work has been carried out. In this paper, an auto-multinomial model is suggested for analyzing multivariate lattice discrete data. The proposed method is illustrated by a real example of air pollution in Tehran, Iran.     

An extension of the multivariate Chebyshev's inequality to a random vector with a singular covariance matrix

ABSTRACT

We extend Chebyshev's inequality to a random vector with a singular covariance matrix. Then we consider the case of a multivariate normal distribution for this generalization.     

A Multivariate Adaptive Exponentially Weighted Moving Average Control Chart

A multivariate extension of the adaptive exponentially weighted moving average (AEWMA) control chart is proposed. The new multivariate scheme can detect small and large shifts in the process mean vector effectively. The proposed scheme can be viewed as a smooth combination of a multivariate exponentially weighted moving average (MEWMA) chart and a Shewhart χ²-chart. The optimal design of the proposed chart is given according to a pre-specified in-control average run length and two shift sizes; a small and large shift each measured in terms of the non centrality parameter. The signal resistance of the newly proposed multivariate chart is also given. Comparisons among the new chart, the MEWMA chart, and the combined Shewhart-MEWMA (S-MEWMA) chart in terms of the standard and worst-case average run length profiles are presented. In addition, the three charts are compared with respect to their worst-case signal resistance values. The proposed chart gives somewhat better worst-case ARL and signal resistance values than the competing charts. It also gives better standard ARL performance especially for moderate and large shifts. The effectiveness of our proposed chart is illustrated through an example with simulated data set.     

Multivariate analysis in genstat

Genstat is a general statistical language for data analysis. The facilities for multivariate and cluster analysis within the language are described as well as the many vector and matrix operations which can be used to form multivariate analysis programs. The contents of the standard macro library relevant to multivariate analysis are also discussed.     

Robust estimation of the mean vector for high-dimensional data set using robust clustering

The first step in statistical analysis is the parameter estimation. In multivariate analysis, one of the parameters of interest to be estimated is the mean vector. In multivariate statistical analysis, it is usually assumed that the data come from a multivariate normal distribution. In this situation, the maximum likelihood estimator (MLE), that is, the sample mean vector, is the best estimator. However, when outliers exist in the data, the use of sample mean vector will result in poor estimation. So, other estimators which are robust to the existence of outliers should be used. The most popular robust multivariate estimator for estimating the mean vector is S-estimator with desirable properties. However, computing this estimator requires the use of a robust estimate of mean vector as a starting point. Usually minimum volume ellipsoid (MVE) is used as a starting point in computing S-estimator. For high-dimensional data computing, the MVE takes too much time. In some cases, this time is so large that the existing computers cannot perform the computation. In addition to the computation time, for high-dimensional data set the MVE method is not precise. In this paper, a robust starting point for S-estimator based on robust clustering is proposed which could be used for estimating the mean vector of the high-dimensional data. The performance of the proposed estimator in the presence of outliers is studied and the results indicate that the proposed estimator performs precisely and much better than some of the existing robust estimators for high-dimensional data.     

Hierarchical clustering of variables: a comparison among strategies of analysis

In this paper some hierarchical methods for identifying groups of variables are illustrated and compared. It is shown that the use of multivariate association measures between two sets of variables can overcome the drawbacks of the usually employed bivariate correlation coefficient, but the resulting methods are generally not monotonic. Thus a new multivariate association measure is proposed, based on the links existing between canonical correlation analysis and principal component analysis, which can be more suitably used for the purpose at hand. The hierarchical method based on the suggested measure is illustrated and compared with other possible solutions by analysing simulated and real data sets. Finally an extension of the suggested method to the more general situation of mixed (qualitative and quantitative) variables is proposed and theoretically discussed.     

Some properties of a linear combination CUSUM statistic for controlling a multivariate mean vector

We develop a class of new multivariate procedures for monitoring quality by detecting a change in the level of a multivariate process. Following the ideas of S.N. Roy, we first consider a linear combination statistic which results from projecting the multivariate observations onto a unit vector and then maximizing a selected univariate statistic over all directions.     

A new family of multivariate skew slash distribution

In this article, the new family of multivariate skew slash distribution is defined. According to the definition, a stochastic representation of the multivariate skew slash distribution is derived. The first four moments and measures of skewness and kurtosis of a random vector with the multivariate skew slash distribution are obtained. The distribution of quadratic forms for the multivariate skew slash distribution and the non central skew slash χ² distribution are studied. Maximum likelihood inference and real data illustration are discussed. In the end, the potential extension of multivariate skew slash distribution is discussed.     

Estimating the structural dimension of regressions via parametric inverse regression

A new estimation method for the dimension of a regression at the outset of an analysis is proposed. A linear subspace spanned by projections of the regressor vector X , which contains part or all of the modelling information for the regression of a vector Y on X , and its dimension are estimated via the means of parametric inverse regression. Smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model. No restrictions are placed on the distribution of the regressors. The estimate of the dimension of the regression is based on optimal estimation procedures. A simulation study shows the method to be more powerful than sliced inverse regression in some situations.     

New control charts for simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles

Simultaneous monitoring of the mean vector and covariance matrix in multivariate processes allows practitioners to avoid the inflated false alarm rate that results from using two independent control charts. In this paper, we extend exponentially weighted moving average semicircle and generally weighted moving average semicircle control charts to monitor the mean vector and covariance matrix of multivariate multiple linear regression profiles in Phase II simultaneously. These new control charts are compared with the existing control charts in the literature in terms of the average run length criterion. Finally, a case is considered to show the application of the proposed charts.     

Hayter and Tsui's test with double sampling for the vector mean of multivariate normal population

Abstract

In this paper, we introduce a version of Hayter and Tsui's statistical test with double sampling for the vector mean of a population under multivariate normal assumption. A study showed that this new test was more or as efficient than the well-known Hotelling's T² with double sampling. Some nice features of Hayter and Tsui's test are its simplicity of implementation and its capability of identifying the errant variables when the null hypothesis is rejected. Taking that into consideration, a new control chart called HTDS is also introduced as a tool to monitor multivariate process vector mean when using double sampling.     

Estimates of regression coefficients based on the sign covariance matrix

Summary. A new estimator of the regression parameters is introduced in a multivariate multiple-regression model in which both the vector of explanatory variables and the vector of response variables are assumed to be random. The affine equivariant estimate matrix is constructed using the sign covariance matrix (SCM) where the sign concept is based on Oja's criterion function. The influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the SCM regression estimate. The estimate is shown to be consistent with a limiting multinormal distribution. The influence function, as a function of the length of the contamination vector, is shown to be linear in elliptic cases; for the least squares (LS) estimate it is quadratic. The asymptotic relative efficiencies with respect to the LS estimate are given in the multivariate normal as well as the t -distribution cases. The SCM regression estimate is highly efficient in the multivariate normal case and, for heavy-tailed distributions, it performs better than the LS estimate. Simulations are used to consider finite sample efficiencies with similar results. The theory is illustrated with an example.     

Nonparametric estimation of copula-based measures of multivariate association from contingency tables

Nonparametric estimation of copula-based measures of multivariate association in a continuous random vector X=(X₁, …, X_d) is usually based on complete continuous data. In many practical applications, however, these types of data are not readily available; instead aggregated ordinal observations are given, for example, ordinal ratings based on a latent continuous scale. This article introduces a purely nonparametric and data-driven estimator of the unknown copula density and the corresponding copula based on multivariate contingency tables. Estimators for multivariate Spearman's rho and Kendall's tau are based thereon. The properties of these estimators in samples of medium and large size are evaluated in a simulation study. An increasing bias can be observed along with an increasing degree of association between the components. As it is to be expected, the bias is severely influenced by the amount of information available. Additionally, the influence of sample size is only marginal. We further give an empirical illustration based on daily returns of five German stocks.     

some results on cononical correlations and measures of multivariate association

The canonical correlations and several orher measures of multivariate association between two sets of variables (x and y) are considered when the covariance matrices are singular. A useful inequality for the canonical correlations when new vari- ables are brought into x or y is obtained for both the nonsingular and singular cases. It is also shown that, under a simple condition, measures of multivariate association equal one if and only if there exists a linear relationship between the sets of variables.