For multivariate non-Gaussian involving copulas, likelihood inference is dominated by the data in the middle, and fitted models might not be very good for joint tail inference, such as assessing the ...strength of tail dependence. When preliminary data and likelihood analysis suggest asymmetric tail dependence, a method is proposed to improve extreme value inferences based on the joint lower and upper tails. A prior that uses previous information on tail dependence can be used in combination with the likelihood. With the combination of the prior and the likelihood (which in practice has some degree of misspecification) to obtain a tilted log-likelihood, inferences with suitably transformed parameters can be based on Bayesian computing methods or with numerical optimization of the tilted log-likelihood to obtain the posterior mode and Hessian at this mode.
For multivariate copula-based models for which maximum likelihood is computationally difficult, a two-stage estimation procedure has been proposed previously; the first stage involves maximum ...likelihood from univariate margins, and the second stage involves maximum likelihood of the dependence parameters with the univariate parameters held fixed from the first stage. Using the theory of inference functions, a partitioned matrix in a form amenable to analysis is obtained for the asymptotic covariance matrix of the two-stage estimator. The asymptotic relative efficiency of the two-stage estimation procedure compared with maximum likelihood estimation is studied. Analysis of the limiting cases of the independence copula and Fréchet upper bound help to determine common patterns in the efficiency as the dependence in the model increases. For the Fréchet upper bound, the two-stage estimation procedure can sometimes be equivalent to maximum likelihood estimation for the univariate parameters. Numerical results are shown for some models, including multivariate ordinal probit and bivariate extreme value distributions, to indicate the typical level of asymptotic efficiency for discrete and continuous data.
A
d-dimensional positive definite correlation matrix
R
=
(
ρ
ij
)
can be parametrized in terms of the correlations
ρ
i
,
i
+
1
for
i
=
1
,
…
,
d
-
1
, and the partial correlations
ρ
ij
|
i
+
1
,
…
j
...-
1
for
j
-
i
⩾
2
. These
d
2
parameters can independently take values in the interval
(
-
1
,
1
)
. Hence we can generate a random positive definite correlation matrix by choosing independent distributions
F
ij
,
1
⩽
i
<
j
⩽
d
, for these
d
2
parameters. We obtain conditions on the
F
ij
so that the joint density of
(
ρ
ij
)
is proportional to a power of
det
(
R
)
and hence independent of the order of indices defining the sequence of partial correlations. As a special case, we have a simple construction for generating
R that is uniform over the space of positive definite correlation matrices. As a byproduct, we determine the volume of the set of correlation matrices in
d
2
-dimensional space. To prove our results, we obtain a simple remarkable identity which expresses
det
(
R
)
as a function of
ρ
i
,
i
+
1
for
i
=
1
,
…
,
d
-
1
, and
ρ
ij
|
i
+
1
,
…
j
-
1
for
j
-
i
⩾
2
.
Multivariate models with parsimonious dependence have been used for a large number of variables, and have mainly been developed for multivariate Gaussian. Graphical dependence model representations ...include Bayesian networks, conditional independence graphs, and truncated vines. The class of Gaussian truncated vines is a subset of Gaussian Bayesian networks and Gaussian conditional independence graphs, but has an extension to non-Gaussian dependence with (i) combinations of continuous and discrete random variables with arbitrary univariate margins, and (ii) accommodation of latent variables. To illustrate the importance of graphical models with latent variables that do not rely on the Gaussian assumption, the combined factor-vine structure is presented and applied to a data set of stock returns.
Des modèles multivariés à dépendance éparse ont été utilisés avec un nombre élevé de variables, mais ils ont surtout été développés dans un contexte multivarié gaussien. La représentation graphique de modèles de dépendance inclut les réseaux bayésiens, les graphes d’indépendance conditionnelle, et les vignes tronquées. La classe de vignes gaussiennes tronquées est un sous-ensemble des réseaux bayésiens gaussiens et des graphes d’indépendance conditionnelle dont une extension non-gaussienne peut être obtenue, permettant (i) des combinaisons de variables aléatoires continues et discrètes avec des marges univariées arbitraires, et (ii) la présence de variables latentes. L’auteur présente la structure combinée de vigne factorielle afin d’illustrer l’importance de disposer de modèles graphiques comportant des variables latentes mais n’étant pas basés sur l’hypothèse de normalité. Il applique la méthode à des données réelles de rendement boursier.
Count Time Series: A Methodological Review Davis, Richard A.; Fokianos, Konstantinos; Holan, Scott H. ...
Journal of the American Statistical Association,
05/2021, Volume:
116, Issue:
535
Journal Article
Peer reviewed
A growing interest in non-Gaussian time series, particularly in series comprised of nonnegative integers (counts), is taking place in today's statistics literature. Count series naturally arise in ...fields, such as agriculture, economics, epidemiology, finance, geology, meteorology, and sports. Unlike stationary Gaussian series where autoregressive moving-averages are the primary modeling vehicle, no single class of models dominates the count landscape. As such, the literature has evolved somewhat ad-hocly, with different model classes being developed to tackle specific situations. This article is an attempt to summarize the current state of count time series modeling. The article first reviews models having prescribed marginal distributions, including some recent developments. This is followed by a discussion of state-space approaches. Multivariate extensions of the methods are then studied and Bayesian approaches to the problem are considered. The intent is to inform researchers and practitioners about the various types of count time series models arising in the modern literature. While estimation issues are not pursued in detail, reference to this literature is made.
We extend and improve two existing methods of generating random correlation matrices, the onion method of Ghosh and Henderson S. Ghosh, S.G. Henderson, Behavior of the norta method for correlated ...random vector generation as the dimension increases, ACM Transactions on Modeling and Computer Simulation (TOMACS) 13 (3) (2003) 276–294 and the recently proposed method of Joe H. Joe, Generating random correlation matrices based on partial correlations, Journal of Multivariate Analysis 97 (2006) 2177–2189 based on partial correlations. The latter is based on the so-called
D
-vine. We extend the methodology to any regular vine and study the relationship between the multiple correlation and partial correlations on a regular vine. We explain the onion method in terms of elliptical distributions and extend it to allow generating random correlation matrices from the same joint distribution as the vine method. The methods are compared in terms of time necessary to generate 5000 random correlation matrices of given dimensions.
We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to ...order "r." They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when "r" is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson's Xsuperscript 2. For this model in nonsparse situations, the proposed statistics are also more powerful than Xsuperscript 2. In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.
For multivariate data from an observational study, inferences of interest can include conditional probabilities or quantiles for one variable given other variables. For statistical modeling, one ...could fit a parametric multivariate model, such as a vine copula, to the data and then use the model-based conditional distributions for further inference. Some results are derived for properties of conditional distributions under different positive dependence assumptions for some copula-based models. The multivariate version of the stochastically increasing ordering of conditional distributions is introduced for this purpose. Results are explained in the context of multivariate Gaussian distributions, as properties for Gaussian distributions can help to understand the properties of copula extensions based on vines.
For modeling count time series data, one class of models is generalized integer autoregressive of order p based on thinning operators. It is shown how numerical maximum likelihood estimation is ...possible by inverting the probability generating function of the conditional distribution of an observation given the past p observations. Two data examples are included and show that thinning operators based on compounding can substantially improve the model fit compared with the commonly used binomial thinning operator.
So-called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep ...them tractable for inference and model selection, the simplifying assumption, that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on, is popular.
We show that the only Archimedean copulas in dimension d≥3 which are of the simplified type are those based on the Gamma Laplace transform or its extension, while the Student-t copulas are the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on, and demonstrate a technique to assess the distance of a multivariate distribution from a nearby distribution that satisfies the simplifying assumption.
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