Confidence Bounds for Multiplicative Comparisons

Statements that are inherently multiplicative have historically been justified using ratios of random variables. Although recent work on ratios has extended the classical theory to produce confidence bounds conditioned on a positive denominator, this current article offers a novel perspective that eliminates the need for such a condition. Although seemingly trivial, this new perspective leads to improved lower confidence bounds to support multiplicative statements. This perspective is also more satisfying as it allows comparisons that are inherently multiplicative in nature to be properly analyzed as such.     

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.     

An Alpha Half Normal Slash Distribution for Analyzing Non Negative Data

In this paper, we study a new class of slash distribution. We define the distribution through means of a stochastic representation as the mixture of an alpha half normal random variable with respect to the power of a uniform random variable. Properties involving moments and moment generating function are derived. The usefulness and flexibility of the proposed distribution is illustrated through a real application by maximum likelihood procedure.     

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.     

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.     

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.     

Bimodal skew-symmetric normal distribution

ABSTRACT

We introduce a new parsimonious bimodal distribution, referred to as the bimodal skew-symmetric Normal (BSSN) distribution, which is potentially effective in capturing bimodality, excess kurtosis, and skewness. Explicit expressions for the moment-generating function, mean, variance, skewness, and excess kurtosis were derived. The shape properties of the proposed distribution were investigated in regard to skewness, kurtosis, and bimodality. Maximum likelihood estimation was considered and an expression for the observed information matrix was provided. Illustrative examples using medical and financial data as well as simulated data from a mixture of normal distributions were worked.     

Use of the 21% rulf in experimental design

When the probabilities underlying a contingency table are assumed to be symmetric, it is advantageous to combine the frequencies of equiprobable cells (Haber, 1981, 1982).The present work introduces models, as well as exact and asymptotic methods, for analyzing such ‘intraclass’ contingency tables in several dimensions.