Controlling the size of multivariate outlier tests with the MCD estimator of scatter

Multivariate outlier detection requires computation of robust distances to be compared with appropriate cut-off points. In this paper we propose a new calibration method for obtaining reliable cut-off points of distances derived from the MCD estimator of scatter. These cut-off points are based on a more accurate estimate of the extreme tail of the distribution of robust distances. We show that our procedure gives reliable tests of outlyingness in almost all situations of practical interest, provided that the sample size is not much smaller than 50. Therefore, it is a considerable improvement over all the available MCD procedures, which are unable to provide good control over the size of multiple outlier tests for the data structures considered in this paper.     

Bayesian optimal blocking of factorial designs

The presence of block effects makes the optimal selection of fractional factorial designs a difficult task. The existing frequentist methods try to combine treatment and block wordlength patterns and apply minimum aberration criterion to find the optimal design. However, ambiguities exist in combining the two wordlength patterns and therefore, the optimality of such designs can be challenged. Here we propose a Bayesian approach to overcome this problem. The main technique is to postulate a model and a prior distribution to satisfy the common assumptions in blocking and then, to develop an optimal design criterion for the efficient estimation of treatment effects. We apply our method to develop regular, nonregular, and mixed-level blocked designs. Several examples are presented to illustrate the advantages of the proposed method.     

Robust centroid based classification with minimum error rates for high dimension,low sample size data

A new method of statistical classification (discrimination) is proposed. The method is most effective for high dimension, low sample size data. It uses a robust mean difference as the direction vector and locates the classification boundary by minimizing the error rates. Asymptotic results for assessment and comparison to several popular methods are obtained by using a type of asymptotics of finite sample size and infinite dimensions. The value of the proposed approach is demonstrated by simulations. Real data examples are used to illustrate the performance of different classification methods.     

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.     

(M,S)-optimality in selecting factorial designs

Use of the (M,S) criterion to select and classify factorial designs is proposed and studied. The criterion is easy to deal with computationally and it is independent of the choice of treatment contrasts. It can be applied to two-level designs as well as multi-level symmetrical and asymmetrical designs. An important connection between the (M,S) and minimum aberration criteria is derived for regular fractional factorial designs. Relations between the (M,S) criterion and generalized minimum aberration criteria on nonregular designs are also discussed. The (M,S) criterion is then applied to study the projective properties of some nonregular designs.     

Acceptance Sampling Plans from Truncated Life Tests Based on the Generalized Birnbaum–Saunders Distribution

In this article, we develop acceptance sampling plans when the life test is truncated at a pre-fixed time. The minimum sample size necessary to ensure the specified median life is obtained by assuming that the lifetimes of the test units follow a generalized Birnbaum–Saunders distribution. The operating characteristic values of the sampling plans as well as producer's risk are presented. Two examples are also given to illustrate the procedure developed here, with one of them being based on a real data from software reliability.     

A Bivariate Generalization of the Noncentral Negative Binomial Distribution

This article proposes a bivariate generalization of the noncentral negative binomial distribution which arises as a model in photon and neural counting. This bivariate generalization is derived as a mixed shifted bivariate negative binomial distribution. Various properties and parameter estimation, especially by a minimum distance method based on the probability generating function, are considered. To show the practical usefulness of the bivariate distribution proposed, an application to model low-flux astronomical images is discussed and a real data set has been analyzed.     

Modeling the Dependence of Conditional Correlations on Market Volatility

Several models have been developed to capture the dynamics of the conditional correlations between time series of financial returns and several studies have shown that the market volatility is a major determinant of the correlations. We extend some models to include explicitly the dependence of the correlations on the market volatility. The models differ by the way—linear or nonlinear, direct or indirect—in which the volatility influences the correlations. Using a wide set of models with two measures of market volatility on two datasets, we find that for some models, the empirical results support to some extent the statistical significance and the economic significance of the volatility effect on the correlations, but the presence of the volatility effect does not improve the forecasting performance of the extended models. Supplementary materials for this article are available online.     

Estimation of a multiple-threshold AR(p) model

The present paper deals with the multiple-threshold p-order autoregressive model which has been introduced by Tong and Lim [H. Tong, K.S. Lim, Threshold autoregression, limit cycles and cyclical data, J. R. Stat. Soc. Ser. B 42 (1980) 245–292] in nonlinear system modelling. Under some conditions on the coefficients of the model which ensure the stationarity, the existence of moments and the strong mixing property of this process and under other mild assumptions, we establish the asymptotic properties (consistency and asymptotic normality) of the minimum Hellinger distance estimates of the autoregressive coefficients of the model.     

Questions & Answers

Several methods are available for finding minimum variance unbiased estimators for functions of distribution parameters. This paper concentrates on two which are rarely used but simple when applicable. The first, previously discussed by Davis (1951) and Tate (1959), yields estimators by differentiation when the range of nonzero probability for a continuous random variable depends on an unknown parameter. The second, which has wider applicability, permits estimators for some rather complicated functions to be found by using some well-known results from distribution theory. A number of examples are presented, many of which are suitable for classroom exercises.