Some Related Minima Stability and Minima Infinite Divisibility of the General Multivariate Pareto Distributions

This article studies the minima stable property of the general multivariate Pareto distributions MP^(k)(I), MP^(k)(II), MP^(k)(III), MP^(k)(IV) which can be applied to characterize the MP^(k) distribution via its weighted ordered coordinates minima and marginal distribution. Also, the multivariate semi-Pareto distribution (denoted by MSP) is discerned in the class of geometric minima infinite divisible and geometric minima stable distributions. If the exponent measure is satisfied by some functional equation, then the geometric minima stable property can be used to characterize the MSP distribution. Finally, the finite sample minima infinite divisible property of the MP^(k)(I), (II), and (IV) distributions is also discussed.     

Monitoring Structural Changes in Generalized Linear Models

In this article, we introduce two monitoring schemes to (sequentially) detect structural changes in generalized linear models and develop asymptotic theories for them. The first method is based on cumulative sums (CUSUM) of weighted residuals, in which the unknown in-control parameters have been replaced by its maximum likelihood (ML) estimate from the training sample, whereas the second scheme makes use of moving sums (MOSUM) of weighted residuals. We characterize the limit distribution of the test statistic and show that these tests are consistent. Moreover, we also obtain and tabulate the asymptotic critical values of the tests. Finally, we study the speed of detection under different conditions. The methods are illustrated and compared in several simulations.     

A Class of Improved Estimators for the Scale Parameter of a Mixture Model of Exponential Distribution with Unknown Location

Under Stein's loss, a class of improved estimators for the scale parameter of a mixture of exponential distribution with unknown location is constructed. The method is analogous to Maruyama's (1998 Maruyama , Y. ( 1998 ). Minimax estimators of a normal variance . Metrika 48 : 209 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) construction for the variance of a normal distribution and also an extension of the result produced in Petropoulos and Kourouklis (2002 Petropoulos , C. , Kourouklis , S. ( 2002 ). A class of improved estimators for the scale parameter of an exponential distribution with unknown location . Commun. Statist. Theor. Meth. 31 : 325 – 335 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Also, robustness properties are considered.     

Multivariate Aging with Archimedean Dependence Structures in High Dimensions

Bivariate aging notions for a vector X of lifetimes based on stochastic comparisons between X and X _t, where X _t is the multivariate residual lifetime after time t > 0, have been studied in Pellerey (2008 Pellerey , F. ( 2008 ). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas . Kybernetika 44 : 795 – 806 .[Web of Science ®] , [Google Scholar]) under the assumption that the dependence structure in X is described by an Archimedean survival copula. Similar stochastic comparisons between X _t and X _t+s, for all t; s > 0, were considered in Mulero and Pellerey (2010 Mulero , J. , Pellerey , F. ( 2010 ). Bivariate aging properties under Archimedean dependence structures . Commun. Statist. Theor. Meth. 39 : 3108 – 3121 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, these results are generalized and extended to the multivariate case. Two illustrative examples are also provided.     

The Methods for Approximation of Principal Points for Binary Distributions on the Basis of Submodularity

Principal points for binary distributions are able to be defined based on Flury’s principal points (1990 Flury, B.A. (1990). Principal points. Biometrica 77:33–41.[Crossref], [Web of Science ®] , [Google Scholar]). However, finding principal points for binary distributions is hard in a straightforward manner. In this article, a method for approximating principal points for binary distributions is proposed by formulating it as an uncapacitated location problem. Moreover, it is shown that the problem of finding principal points can be solved with the aid of submodular functions. It leads to a solution whose value is at least (1 ? 1/e) times the optimal value.     

Combining Multi-class SVMs with Linear Ensemble Methods that Estimate the Class Posterior Probabilities

Roughly speaking, there is one main model of pattern recognition support vector machine, with several variants of lower popularity. On the contrary, among the different multi-class support vector machines which can be found in the literature, none is clearly favoured. On the one hand, they exhibit distinct statistical properties. On the other hand, multiple comparative studies between multi-class support vector machines and decomposition methods have highlighted the fact that each model has its advantages and drawbacks. These observations call for the evaluation of combinations of multi-class support vector machines. In this article, we study the combination of multi-class support vector machines with linear ensemble methods. Their sample complexity is low, which should prevent them from overfitting, and the outputs of two of them are estimates of the class posterior probabilities.     

New invariant and consistent chi-squared type goodness-of-fit tests for multivariate normality and a related comparative simulation study

ABSTRACT

New invariant and consistent goodness-of-fit tests for multivariate normality are introduced. Tests are based on the Karhunen–Loève transformation of a multidimensional sample from a population. A comparison of simulated powers of tests and other well-known tests with respect to some alternatives is given. The simulation study demonstrates that power of the proposed McCull test almost does not depend on the number of grouping cells. The test shows an advantage over other chi-squared type tests. However, averaged over all of the simulated conditions examined in this article, the Anderson–Darling type and the Cramer–von Mises type tests seem to be the best.     

An adaptive multivariate CUSUM control chart for signaling a range of location shifts

ABSTRACT

In this work, we proposed an adaptive multivariate cumulative sum (CUSUM) statistical process control chart for signaling a range of location shifts. This method was based on the multivariate CUSUM control chart proposed by Pignatiello and Runger (1990 Pignatiello, J.J., Runger, G.C. (1990). Comparisons of multivariate CUSUM charts. J. Qual. Technol. 22(3):173–186.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), but we adopted the adaptive approach similar to that discussed by Dai et al. (2011 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]), which was based on a different CUSUM method introduced by Crosier (1988 Crosier, R.B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics 30(3):291–303.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The reference value in this proposed procedure was changed adaptively in each run, with the current mean shift estimated by exponentially weighted moving average (EWMA) statistic. By specifying the minimal magnitude of the mean shift, our proposed control chart achieved a good overall performance for detecting a range of shifts rather than a single value. We compared our adaptive multivariate CUSUM method with that of Dai et al. (2001 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]) and the non adaptive versions of these two methods, by evaluating both the steady state and zero state average run length (ARL) values. The detection efficiency of our method showed improvements over the comparative methods when the location shift is unknown but falls within an expected range.     

A statistically adaptive sampling policy to the Hotelling's T2 control chart: Markov chain approach

ABSTRACT

We present an alternative sampling scheme for the Hotelling's T² control chart with variable parameters (VP T²) which allows the sampling interval h, the sample size n, and control limit k to vary between minimum and maximum values while keeping the warning line fixed over time. Our method uses only one measurement scale to overcome the difficulties of using two scales in practice. Later, we demonstrate the merits of the method in terms of its performance in detecting small-to-moderate shifts and its ease of application.     

Bayesian estimation of the dispersion matrix of a multivariate normal distribution

The estimation of the dispersion matrix of a multivariate normal distribution with zero mean on the basis of a random sample is discussed from a Bayesian view. An inverted-Wishart distribu- tion for the dispersion is taken, with its defining matrix of intraclass form. Some consistency properties are described. The posterior distribution is found and its mode investigated as a possible estimate in preference to that of maximum likelihood