Topic
Multivariate normal distribution
About: Multivariate normal distribution is a research topic. Over the lifetime, 8304 publications have been published within this topic receiving 324203 citations. The topic is also known as: multivariate Gaussian distribution & joint normal distribution.
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TL;DR: For rectangular confidence regions for the mean values of multivariate normal distributions, this paper proved that a confidence region constructed for independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates.
Abstract: For rectangular confidence regions for the mean values of multivariate normal distributions the following conjecture of 0. J. Dunn [3], [4] is proved: Such a confidence region constructed for the case of independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates. This result is based on an inequality for the probabilities of rectangles in normal distributions, which permits one to factor out the probability for any single coordinate.
2,413 citations
TL;DR: In this article, a modification of the Bonferroni procedure for testing multiple hypotheses is presented, based on the ordered p-values of the individual tests, which is less conservative than the classical BFP but is still simple to apply.
Abstract: SUMMARY A modification of the Bonferroni procedure for testing multiple hypotheses is presented. The method, based on the ordered p-values of the individual tests, is less conservative than the classical Bonferroni procedure but is still simple to apply. A simulation study shows that the probability of a type I error of the procedure does not exceed the nominal significance level, a, for a variety of multivariate normal and multivariate gamma test statistics. For independent tests the procedure has type I error probability equal to a. The method appears particularly advantageous over the classical Bonferroni procedure when several highly-correlated test statistics are involved.
2,220 citations
TL;DR: For small datasets, FAST-MCD typically finds the exact MCD, whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders.
Abstract: The minimum covariance determinant (MCD) method of Rousseeuw is a highly robust estimator of multivariate location and scatter. Its objective is to find h observations (out of n) whose covariance matrix has the lowest determinant. Until now, applications of the MCD were hampered by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size, one about a production process at Philips with n = 677 objects and p = 9 variables, and a dataset from astronomy with n = 137,256 objects and p = 27 variables. To deal with such problems we have developed a new algorithm for the MCD, called FAST-MCD. The basic ideas are an inequality involving order statistics and determinants, and techniques which we call “selective iteration” and “nested extensions.” For small datasets, FAST-MCD typically finds the exact MCD, whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders...
2,073 citations
Book•
01 Jan 1980
TL;DR: In this article, the multivariate normal distribution is used for principal component analysis and multivariate analysis of covariance and related topics, as well as multi-dimensional scaling and cluster analysis.
Abstract: Part One. Multivariate distributions. Preliminary data analysis. Part Two: Finding new underlying variables. Principal component analysis. Factor analysis. Part Three: Procedures based on the multivariate normal distribution. The multivariate normal distribution. Procedures based on normal distribution theory. The multivariate analysis of variance. The multivariate analysis of covariance and related topics. Part Four: Multi-dimensional scaling and cluster analysis. Multi-dimensional scaling. Cluster analysis.
1,856 citations
TL;DR: This work uses the pair-copula decomposition of a general multivariate distribution and proposes a method for performing inference, which represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-Copula models, that also can be applied to huge data sets automatically.
Abstract: Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method for performing inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. Our approach represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.
1,744 citations