Upper Bound for p-Value of the Test of Multivariate Normal Ordered Mean Vectors Against all Alternatives

Consider the problem of testing the isotonic of several p-variate normal mean vectors against all alternatives. It is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown. In the present paper, a test statistic is proposed for this problem of testing. A reformulation of the test statistic is given based on the orthogonal projections on the closed convex cones and then the upper bound for p-value of the test statistic is computed.     

Optimal Simultaneous Confidence Bands in Multiple Linear Regression with Predictor Variables Constrained in an Ellipsoidal Region

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model function, just as a confidence interval gives the plausible range of an unknown parameter. For a multiple linear regression model, confidence bands of different shapes, such as the hyperbolic band and the constant width band, can be constructed and the predictor variable region over which a confidence band is constructed can take various forms. One interesting but unsolved problem is to find the optimal (shape) confidence band over an ellipsoidal region χ_E under the Minimum Volume Confidence Set (MVCS) criterion of Liu and Hayter (2007 Liu, W., Hayter, A.J. (2007). Minimum area confidence set optimality for confidence bands in simple linear regression. J. Amer. Statist. Assoc. 102:181–190.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Liu et al. (2009 Liu, W., Bretz, F., Hayter, A.J., Wynn, H.P. (2009). Assessing non-superiority, non-inferiority or equivalence when comparing two regression models over a restricted covariate region. Biometrics 65:1279–1287.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). This problem is challenging as it involves optimization over an unknown function that determines the shape of the confidence band over χ_E. As a step towards solving this difficult problem, in this paper, we introduce a family of confidence bands over χ_E, called the inner-hyperbolic bands, which includes the hyperbolic and constant-width bands as special cases. We then search for the optimal confidence band within this family under the MVCS criterion. The conclusion from this study is that the hyperbolic band is not optimal even within this family of inner-hyperbolic bands and so cannot be the overall optimal band. On the other hand, the constant width band can be optimal within the family of inner-hyperbolic bands when the region χ_E is small and so might be the overall optimal band.     

On an ad hoc test for order restricted multivariate normal means

Abstract

For several normal mean vectors restricted by a simple ordering with respect to a multivariate order, this article derives sufficient and necessary conditions for the restricted MLEs for both mean vectors and covariance matrix, and develops an ad hoc test. It establishes conditions for the bounds of the p-values. One example of such bound is given with some comments.     

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.     

Sufficient reduction methods for multivariate time-dependent health surveillance data

ABSTRACT

Early detection with a low false alarm rate (FAR) is the main aim of outbreak detection as used in public health surveillance or in regard to bioterrorism. Multivariate surveillance is preferable to univariate surveillance since correlation between series (CBS) is recognized and incorporated. Sufficient reduction has proved a promising method for handling CBS, but has not previously been used when correlation within series (CWS) is present. Here we develop sufficient reduction methods for reducing a p-dimensional multivariate series to a univariate series of statistics shown to be sufficient to monitor a sudden, but persistent, shift in the multivariate series mean. Correlation both within and between series is taken into account, as public health data typically exhibit both forms of association. Simultaneous and lagged changes and different shift sizes are investigated. A one-sided exponentially weighted moving average chart is used as a tool for detection of a change. The performance of the proposed method is compared with existing sufficient reduction methods, the parallel univariate method and both VarR and Z charts. A simulation study using bivariate normal autoregressive data shows that the new method gives shorter delays and a lower FAR than other methods, which have high FARs when CWS is clearly present.     

Sample-size requirements for confidence bounds of prescribed precision on multivariate normal quantiles

The problem of setting confidence bounds on a central multivariate normal quantile is considered. It is shown that for the setting of exact confidence bounds of specified closeness to the quantile,the required minimum size of a normal sample is large and rises rapidly with the number of variates considered.     

Vector correlation for elliptical distributions

A measure of multivariate correlation between two sets of vectors is considered when the underlying joint distribution is a member of the class of elliptical distributions. Its asymptotic distribution is derived under different situations and these results are used to test hypotheses on vector correlation when the underlying joint distribution is non-normal.     

Stability conditions for a bivariate arch system which is cointegrated in mean

This note gives necessary and sufficient conditions for covariance stationarity in a bivariate system of individually first-order integrated (i.e. difference-stationary) processes. The two processes are linked by a cointegrating relation and by a special ARCH model that describes their time-changing volatility. The problem of deriving necessary conditions for strict stationarity in this system is demonstrated using Monte Carlo simulation. The conditions of the theorem are highlighted using empirical examples.     

A Generalization of Hotelling's T 2

ABSTRACT

We derive an asymptotic version of Hotelling's T ² for the multivariate proper dispersion models of Jøtrgensen and Lauritzen (2000 Jørgensen , B. , Lauritzen , S. L. ( 2000 ). Multivariate dispersion models . J. Multivariate Anal. 74 : 267 – 281 . [CSA] [CROSSREF]  [Google Scholar]), the main tool being the saddlepoint approximation. Multivariate dispersion models are suitable for positive, directional, compositional, and other non normal data. We illustrate the results by a multivariate gamma model.