Data from One Multivariate Distribution

2004 ◽  
pp. 89-150
Author(s):  
Anthony C. Atkinson ◽  
Marco Riani ◽  
Andrea Cerioli
Keyword(s):  
Multivariate Distribution

A methodology to evaluate and reduce reservoir uncertainties using multivariate distribution

2015 ◽  
Vol 128 ◽  
pp. 1-14 ◽  
Author(s):  
Andre Carlos Bertolini ◽  
Célio Maschio ◽  
Denis Jose Schiozer
Keyword(s):  
Multivariate Distribution

scholarly journals Testing Group Symmetry of a Multivariate Distribution

Symmetry ◽  
2009 ◽  
Vol 1 (2) ◽  
pp. 180-200 ◽  
Author(s):  
Lyudmila Sakhanenko
Keyword(s):  
Multivariate Distribution ◽  
Group Symmetry ◽  
Testing Group

scholarly journals Application of Copula functions in statistics

2007 ◽  
Author(s):  
Αριστείδης Νικολουλόπουλος
Keyword(s):  
Discrete Data ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Multivariate Distribution ◽  
Dependence Structure ◽  
Copula Models ◽  
Cumulative Distribution Functions ◽  
Multivariate Discrete Distributions ◽  
Dependence Properties ◽  
Multivariate Copula

Studying associations among multivariate outcomes is an interesting problem in statistical science. The dependence between random variables is completely described by their multivariate distribution. When the multivariate distribution has a simple form, standard methods can be used to make inference. On the other hand one may create multivariate distributions based on particular assumptions, limiting thus their use. Unfortunately, these limitations occur very often when working with multivariate discrete distributions. Some multivariate discrete distributions used in practice can have only certain properties, as for example they allow only for positive dependence or they can have marginal distributions of a given form. To solve this problem copulas seem to be a promising solution. Copulas are a currently fashionable way to model multivariate data as they account for the dependence structure and provide a flexible representation of the multivariate distribution. Furthermore, for copulas the dependence properties can be separated from their marginal properties and multivariate models with marginal densities of arbitrary form can be constructed, allowing a wide range of possible association structures. In fact they allow for flexible dependence modelling, different from assuming simple linear correlation structures. However, in the application of copulas to discrete data marginal parameters affect dependence structure, too, and, hence the dependence properties are not fully separated from the marginal properties. Introducing covariates to describe the dependence by modelling the copula parameters is of special interest in this thesis. Thus, covariate information can describe the dependence either indirectly through the marginalparameters or directly through the parameters of the copula . We examine the case when the covariates are used both in marginal and/or copula parameters aiming at creating a highly flexible model producing very elegant dependence structures. Furthermore, the literature contains many theoretical results and families of copulas with several properties but there are few papers that compare the copula families and discuss model selection issues among candidate copula models rendering the question of which copulas are appropriate and whether we are able, from real data, to select the true copula that generated the data, among a series of candidates with, perhaps, very similar dependence properties. We examined a large set of candidate copula families taking intoaccount properties like concordance and tail dependence. The comparison is made theoretically using Kullback-Leibler distances between them. We have selected this distance because it has a nice relationship with log-likelihood and thus it can provide interesting insight on the likelihood based procedures used in practice. Furthermore a goodness of fit test based on Mahalanobisdistance, which is computed through parametric bootstrap, will be provided. Moreover we adopt a model averaging approach on copula modelling, based on the non-parametric bootstrap. Our intention is not to underestimate variability but add some additional variability induced by model selection making the precision of the estimate unconditional on the selected model. Moreover our estimates are synthesize from several different candidate copula models and thus they can have a flexible dependence structure. Taking under consideration the extended literature of copula for multivariate continuous data we concentrated our interest on fitting copulas on multivariate discrete data. The applications of multivariate copula models for discrete data are limited. Usually we have to trade off between models with limited dependence (e.g. only positive association) and models with flexible dependence but computational intractabilities. For example, the elliptical copulas provide a wide range of flexible dependence, but do not have closed form cumulative distribution functions. Thus one needs to evaluate the multivariate copula and, hence, a multivariate integral repeatedly for a large number of times. This can be time consuming but also, because of the numerical approach used to evaluate a multivariate integral, it may produce roundoff errors. On the other hand, multivariate Archimedean copulas, partially-symmetric m-variate copulas with m − 1 dependence parameters and copulas that are mixtures of max-infinitely divisible bivariate copulas have closed form cumulative distribution functions and thus computations are easy, but allow only positive dependence among the random variables. The bridge of the two above-mentioned problems might be the definition of a copula family which has simple form for its distribution function while allowing for negative dependence among the variables. We define such a multivariate copula family exploiting the use of finite mixture of simple uncorrelated normal distributions. Since the correlation vanishes, the cumulative distribution is simply the product of univariate normal cumulative distribution functions. The mixing operation introduces dependence. Hence we obtain a kind of flexible dependence, and allow for negative dependence.

Error bounds on multivariate Normal approximations for word count statistics

2002 ◽  
Vol 34 (03) ◽  
pp. 559-586 ◽  
Author(s):  
Haiyan Huang
Keyword(s):  
Covariance Matrix ◽  
Error Bounds ◽  
Multivariate Distribution ◽  
Finite Alphabet ◽  
Multivariate Normal ◽  
Word Count ◽  
First Order ◽  
Normal Approximations ◽  
Necessary And Sufficient ◽  
The Given

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w 1), …, N(S,w d )), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O ((log n) / √n) in the i.i.d. case and O ((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

Modeling multivariate distribution of multiple soil parameters using vine copula model

2020 ◽  
Vol 118 ◽  
pp. 103340 ◽  
Author(s):  
Tian-Jian Lü ◽  
Xiao-Song Tang ◽  
Dian-Qing Li ◽  
Xiao-Hui Qi
Keyword(s):  
Multivariate Distribution ◽  
Soil Parameters ◽  
Copula Model ◽  
Vine Copula

Multivariate distribution of returns in financial time series

2007 ◽  
Vol 6 (5-6) ◽  
pp. 315-324
Author(s):  
M.I. Krivoruchenko ◽  
E. Alessio ◽  
V. Frappietro ◽  
L.J. Streckert
Keyword(s):  
Time Series ◽  
Financial Time Series ◽  
Multivariate Distribution ◽  
Financial Time
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