Continuous Multivariate Distributions

Kotz, Samuel; Balakrishnan, N.; Johnson, Norman L.

doi:10.1002/0471722065

Cited by 1,255 publications

(418 citation statements)

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“…Assuming that the function µ 2 (X 1 ) is any continuous function in X 1 , one obtains a wide and interesting extension of the class of bivariate normal densities. We called this class FF-normal (previously named "pseudonormal", see [2] and also [7]). In this case the parameter σ 2 can as well become a continuous function of the stress X 1 (X 1 may have a "stress" interpretation in a very wide sense).…”

Section: The Ff-normal (Pseudonormal) Extensionmentioning

confidence: 99%

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Parameter Dependence in Stochastic Modeling—Multivariate Distributions

Filus¹

2014

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We start with analyzing stochastic dependence in a classic bivariate normal density framework. We focus on the way the conditional density of one of the random variables depends on realizations of the other. In the bivariate normal case this dependence takes the form of a parameter (here the "expected value") of one probability density depending continuously (here linearly) on realizations of the other random variable. The point is, that such a pattern does not need to be restricted to that classical case of the bivariate normal. We show that this paradigm can be generalized and viewed in ways that allows one to extend it far beyond the bivariate or multivariate normal probability distributions class.

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Section: The Ff-normal (Pseudonormal) Extensionmentioning

confidence: 99%

“…These distributions we constructed in our papers since 2000 up to recently (see, Filus and Filus [1]- [6], also see Kotz, Balakrishnan and Johnson [7] pp. 217-218).…”

Section: Introductionmentioning

confidence: 99%

Parameter Dependence in Stochastic Modeling—Multivariate Distributions

Filus¹

2014

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“…A generalized t-distribution has been derived by Siddiqui [18] only in the bivariate case. An approximate multivariate extension in the equicorrelated case is given by Kotz et al [19]; this approximation is exact in the bivariate case. Nevertheless, under H 0 the test statistics T 1 ; .…”

Section: Testing Problemmentioning

confidence: 99%

Multiple Contrasts for Repeated Measures

Hasler

2013

The International Journal of Biostatistics

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This article addresses the problem of multiple contrast tests for repeated measures. Three procedures are described and compared by simulations concerning the familywise error type I. The procedure based on sandwich estimators seems to be most robust, except for all-pair comparisons.Keywords: repeated measures, multiple contrasts, sandwich estimator, multivariate t-distribution, familywise error type I *Corresponding author: Mario Hasler, Christian-Albrechts-University, Kiel, Germany, E-mail: hasler@email.uni-kiel.de IntroductionMultiple contrast tests (MCTs) and related simultaneous confidence intervals (SCIs) are well-known methods for testing and estimating linear functions of means called contrasts. A broad class of testing problems can be handled in specifying suitable contrast coefficients. The many-to-one comparison of Dunnett [1] is one of the most frequently applied and cited testing procedures today and it represents a simple example. Several treatments are compared with one control and tested for deviation. The all-pair comparison of Tukey [2], comparing all treatments against each other, is another very well-known example. Bretz [3] has formulated the trend test of Williams [4] as an approximate MCT. Here, the contrast coefficients depend in addition on the sample sizes of the treatment groups. Moreover, other problem-specific contrasts can be defined (see Nelson [5], Westfall [6] or Bretz et al. [7] for example). Furthermore, MCTs and SCIs can also be formulated for ratios of means (see Dilba et al. [8]) if conclusions about ratios -rather than differencesof means are of interest. This applies when relative changes are to be analyzed. Because correlations between the contrasts are involved by a joint distribution, MCTs exactly maintain the familywise error type I (FWE) over all contrasts. No further multiplicity adjustment is needed. SCIs (and adjusted p-values) are obtained for all hypotheses to be tested, complying with the guideline of the ICH [9]. Stepwise or gatekeeping procedures, for example, do not allow informative SCIs; see Strassburger and Bretz [10] and Guilbaud [11].MCTs and related SCIs are usually confined to normally distributed, homoscedastic and independent data with one primary endpoint. For the heteroscedastic case, Hasler and Hothorn [12] described a solution based on Games and Howell [13]. Konietschke et al. [14] presented a non-parametric version. Hasler and Hothorn [15,16] gave extensions for the case of multiple correlated endpoints. Unlike conventional MCTs, these approaches represent approximate solutions. They focus on situations where the usual assumptions are not fulfilled. Repeated measures also represent a situation where such an assumption is not met. For example, the means of several groups have to be compared simultaneously but the measurement values for these groups come from the same measurement objects (patients, plants, etc.), respectively. The groups are mostly according to time points but they can also represent different parts of the body, for ...

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“…In general the multivariate gamma distributions belong to MVME whenever the shape parameters are integer. The main reference is Kotz, Balakrishnan & Johnson (2000). In this reference the distributions are classified as bivariate and multivariate exponential respectively gamma distributions.…”

Section: Mph * Representations For Multivariate Exponential and Gammamentioning

confidence: 99%