97,085 research outputs found
Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit
This paper presents the R package AdMit which provides flexible functions to approximate a certain target distribution and to efficiently generate a sample of random draws from it, given only a kernel of the target density function. The core algorithm consists of the function AdMit which fits an adaptive mixture of Student-t distributions to the density of interest. Then, importance sampling or the independence chain Metropolis-Hastings algorithm is used to obtain quantities of interest for the target density, using the fitted mixture as the importance or candidate density. The estimation procedure is fully automatic and thus avoids the time-consuming and difficult task of tuning a sampling algorithm. The relevance of the package is shown in two examples. The first aims at illustrating in detail the use of the functions provided by the package in a bivariate bimodal distribution. The second shows the relevance of the adaptive mixture procedure through the Bayesian estimation of a mixture of ARCH model fitted to foreign exchange log-returns data. The methodology is compared to standard cases of importance sampling and the Metropolis-Hastings algorithm using a naive candidate and with the Griddy-Gibbs approach.
Adaptive Mixture of Student-t distributions as a Flexible Candidate Distribution for Efficient Simulation
This paper presents the R package AdMit which provides functions to approximate and sample from a certain target distribution given only a kernel of the target density function. The core algorithm consists in the function AdMit which fits an adaptive mixture of Student-t distributions to the density of interest via its kernel function. Then, importance sampling or the independence chain Metropolis- Hastings algorithm are used to obtain quantities of interest for the target density, using the fitted mixture as the importance or candidate density. The estimation procedure is fully automatic and thus avoids the time-consuming and difficult task of tuning a sampling algorithm. The relevance of the package is shown in two examples. The first aims at illustrating in detail the use of the functions provided by the package in a bivariate bimodal distribution. The second shows the relevance of the adaptive mixture procedure through the Bayesian estimation of a mixture of ARCH model fitted to foreign exchange log-returns data. The methodology is compared to standard cases of importance sampling and the Metropolis-Hastings algorithm using a naive candidate and with the Griddy-Gibbs approach
Adaptive mixture of Student-t distributions as a flexible candidate distribution for efficient simulation: the R package AdMit
textabstractThis paper presents the R package AdMit which provides flexible functions to approximate a certain target distribution and to efficiently generate a sample of random draws from it, given only a kernel of the target density function. The core algorithm consists of the function AdMit which fits an adaptive mixture of Student-t distributions to the density of interest. Then, importance sampling or the independence chain Metropolis-Hastings algorithm is used to obtain quantities of interest for the target density, using the fitted mixture as the importance or candidate density. The estimation procedure is fully automatic and thus avoids the time-consuming and difficult task of tuning a sampling algorithm. The relevance of the package is shown in two examples. The first aims at illustrating in detail the use of the functions provided by the package in a bivariate bimodal distribution. The second shows the relevance of the adaptive mixture procedure through the Bayesian estimation of a mixture of ARCH model fitted to foreign exchange log-returns data. The methodology is compared to standard cases of importance sampling and the Metropolis-Hastings algorithm using a naive candidate and with the Griddy-Gibbs approach
Convergence rates for Bayesian density estimation of infinite-dimensional exponential families
We study the rate of convergence of posterior distributions in density
estimation problems for log-densities in periodic Sobolev classes characterized
by a smoothness parameter p. The posterior expected density provides a
nonparametric estimation procedure attaining the optimal minimax rate of
convergence under Hellinger loss if the posterior distribution achieves the
optimal rate over certain uniformity classes. A prior on the density class of
interest is induced by a prior on the coefficients of the trigonometric series
expansion of the log-density. We show that when p is known, the posterior
distribution of a Gaussian prior achieves the optimal rate provided the prior
variances die off sufficiently rapidly. For a mixture of normal distributions,
the mixing weights on the dimension of the exponential family are assumed to be
bounded below by an exponentially decreasing sequence. To avoid the use of
infinite bases, we develop priors that cut off the series at a
sample-size-dependent truncation point. When the degree of smoothness is
unknown, a finite mixture of normal priors indexed by the smoothness parameter,
which is also assigned a prior, produces the best rate. A rate-adaptive
estimator is derived.Comment: Published at http://dx.doi.org/10.1214/009053606000000911 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
SPADES and mixture models
This paper studies sparse density estimation via penalization
(SPADES). We focus on estimation in high-dimensional mixture models and
nonparametric adaptive density estimation. We show, respectively, that SPADES
can recover, with high probability, the unknown components of a mixture of
probability densities and that it yields minimax adaptive density estimates.
These results are based on a general sparsity oracle inequality that the SPADES
estimates satisfy. We offer a data driven method for the choice of the tuning
parameter used in the construction of SPADES. The method uses the generalized
bisection method first introduced in \citebb09. The suggested procedure
bypasses the need for a grid search and offers substantial computational
savings. We complement our theoretical results with a simulation study that
employs this method for approximations of one and two-dimensional densities
with mixtures. The numerical results strongly support our theoretical findings.Comment: Published in at http://dx.doi.org/10.1214/09-AOS790 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Adaptive Smoothing Parameter in Kernel Density Estimation and Parameter Estimation in Normal Mixture Distributions
Kernel density estimation is a widely used tool in nonparametric density estimation procedures. Choice of a kernel function and a smoothing parameter are two important issues in implementing kernel density estimation procedures. In this paper, four different kernel functions are considered in implementing an adaptive selection procedure in choosing the smoothing parameter. In simulation, a skewed bimodal density which is a mixture of two normal distributions is considered along with the standard normal and the standard exponential densities. In skewed bimodal data, parameter estimation is also explored in the context of the parameter estimation in mixtures of normal distributions. Maximum likelihood estimation procedure is implemented in parameter estimation in mixtures of normal distributions
- …