228 research outputs found
On approximating copulas by finite mixtures
Copulas are now frequently used to approximate or estimate multivariate
distributions because of their ability to take into account the multivariate
dependence of the variables while controlling the approximation properties of
the marginal densities. Copula based multivariate models can often also be more
parsimonious than fitting a flexible multivariate model, such as a mixture of
normals model, directly to the data. However, to be effective, it is imperative
that the family of copula models considered is sufficiently flexible. Although
finite mixtures of copulas have been used to construct flexible families of
copulas, their approximation properties are not well understood and we show
that natural candidates such as mixtures of elliptical copulas and mixtures of
Archimedean copulas cannot approximate a general copula arbitrarily well. Our
article develops fundamental tools for approximating a general copula
arbitrarily well by a mixture and proposes a family of finite mixtures that can
do so. We illustrate empirically on a financial data set that our approach for
estimating a copula can be much more parsimonious and results in a better fit
than approximating the copula by a mixture of normal copulas.Comment: 26 pages and 1 figure and 2 table
Approximate Bayesian inference in semiparametric copula models
We describe a simple method for making inference on a functional of a
multivariate distribution. The method is based on a copula representation of
the multivariate distribution and it is based on the properties of an
Approximate Bayesian Monte Carlo algorithm, where the proposed values of the
functional of interest are weighed in terms of their empirical likelihood. This
method is particularly useful when the "true" likelihood function associated
with the working model is too costly to evaluate or when the working model is
only partially specified.Comment: 27 pages, 18 figure
Estimating Discrete Markov Models From Various Incomplete Data Schemes
The parameters of a discrete stationary Markov model are transition
probabilities between states. Traditionally, data consist in sequences of
observed states for a given number of individuals over the whole observation
period. In such a case, the estimation of transition probabilities is
straightforwardly made by counting one-step moves from a given state to
another. In many real-life problems, however, the inference is much more
difficult as state sequences are not fully observed, namely the state of each
individual is known only for some given values of the time variable. A review
of the problem is given, focusing on Monte Carlo Markov Chain (MCMC) algorithms
to perform Bayesian inference and evaluate posterior distributions of the
transition probabilities in this missing-data framework. Leaning on the
dependence between the rows of the transition matrix, an adaptive MCMC
mechanism accelerating the classical Metropolis-Hastings algorithm is then
proposed and empirically studied.Comment: 26 pages - preprint accepted in 20th February 2012 for publication in
Computational Statistics and Data Analysis (please cite the journal's paper
Bayesian Semiparametric Covariate Informed Multivariate Density Deconvolution
Estimating the marginal and joint densities of the long-term average intakes
of different dietary components is an important problem in nutritional
epidemiology. Since these variables cannot be directly measured, data are
usually collected in the form of 24-hour recalls of the intakes. The problem of
estimating the density of the latent long-term average intakes from their
observed but error contaminated recalls then becomes a problem of multivariate
deconvolution of densities. The underlying densities could potentially vary
with the subjects' demographic characteristics such as sex, ethnicity, age,
etc. The problem of density deconvolution in the presence of associated
precisely measured covariates has, however, never been considered before, not
even in the univariate setting. We present a flexible Bayesian semiparametric
approach to covariate informed multivariate deconvolution. Building on recent
advances in copula deconvolution and conditional tensor factorization
techniques, our proposed method not only allows the joint and the marginal
densities to vary flexibly with the associated predictors but also allows
automatic selection of the most influential predictors. Importantly, the method
also allows the density of interest and the density of the measurement errors
to vary with potentially different sets of predictors. We design Markov chain
Monte Carlo algorithms that enable efficient posterior inference, appropriately
accommodating uncertainty in all aspects of our analysis. The empirical
efficacy of the proposed method is illustrated through simulation experiments.
Its practical utility is demonstrated in the afore-described nutritional
epidemiology applications in estimating covariate-adjusted long term intakes of
different dietary components. Supplementary materials include substantive
additional details and R codes are also available online.Comment: arXiv admin note: text overlap with arXiv:1912.0508
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