Consider a probit regression problem in which Y1,..., Y-n are independent Bernoulli random variables such that Pr(Y-j = 1) = Phi(x(i)(T) beta) where x(i) is a p-dimensional vector of known covariates that are associat...
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Consider a probit regression problem in which Y1,..., Y-n are independent Bernoulli random variables such that Pr(Y-j = 1) = Phi(x(i)(T) beta) where x(i) is a p-dimensional vector of known covariates that are associated with Yj, 0 is' a p-dimensional vector of unknown regression coefficients and Phi(.) denotes the standard normal distribution function. We study Markov chain Monte Carlo algorithms for exploring the intractable posterior density that results when the probit regression likelihood is combined with a flat prior on beta. We prove that Albert and Chib's data augmentation algorithm and Liu and Wu's PX-DA algorithm both converge at a geometric rate, which ensures the existence of central limit theorems for ergodic averages under a second-moment condition. Although these two algorithms are essentially equivalent in terms of computational complexity, results of Hobert and Marchev imply that the PX-DA algorithm is theoretically more efficient in the sense that the asymptotic variance in the central limit theorem under the PX-DA algorithm is no larger than that under Albert and Chib's algorithm. We also construct minorization conditions that allow us to exploit regenerative simulation techniques for the consistent estimation of asymptotic variances. As an illustration, we apply our results to van Dyk and Meng's lupus data. This example demonstrates that huge gains in efficiency are possible by using the PX-DA algorithm instead of Albert and Chib's algorithm.
Many statistical problems can be formulated as discrete missing data problems (MDPs). Examples include change-point problems, capture and recapture models, sample survey with non-response, zero-inflated Poisson models...
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Many statistical problems can be formulated as discrete missing data problems (MDPs). Examples include change-point problems, capture and recapture models, sample survey with non-response, zero-inflated Poisson models, medical screening/diagnostic tests and bioassay. This paper proposes an exact non-iterative sampling algorithm to obtain independently and identically distributed (i.i.d.) samples from posterior distribution in discrete MDPs. The new algorithm is essentially a conditional sampling, thus completely avoiding problems of convergence and slow convergence in iterative algorithms such as Markov chain Monte Carlo. Different from the general inverse Bayes formulae (IBF) sampler of Tan, Tian and Ng (Statistica Sinica, 13, 2003, 625), the implementation of the new algorithm requires neither the expectation maximization nor the sampling importance resampling algorithms. The key idea is to first utilize the sampling-wise IBF to derive the conditional distribution of the missing data given the observed data, and then to draw i.i.d. samples from the complete-data posterior distribution. We first illustrate the method with a performing example and then apply the method to contingency tables with one supplemental margin for an human immunodeficiency virus study.
Reproductive scientists and couples attempting pregnancy are interested in identifying predictors of the day-specific probabilities of conception in relation to the timing of a single intercourse act. Because most men...
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Reproductive scientists and couples attempting pregnancy are interested in identifying predictors of the day-specific probabilities of conception in relation to the timing of a single intercourse act. Because most menstrual cycles have multiple days of intercourse, the occurrence of conception represents the aggregation across Bernoulli trials for each intercourse day. Because of this data structure and dependency among the multiple cycles from a woman, implementing analyses has proven challenging. This article proposes a Bayesian approach based on a generalization of the Barrett and Marshall model to incorporate a woman-specific frailty and day-specific covariates. The model results in a simple closed form expression for the marginal probability of conception, and has an auxiliary variables formulation that facilitates efficient posterior computation. Although motivated by fecundability studies, the approach can be used for efficient variable selection and model averaging in general applications with categorical or discrete event time data.
This article provides a simple method to accelerate Markov chain Monte Carlo sampling algorithms, such as the data augmentation algorithm and the Gibbs sampler, via alternating subspace-spanning resampling (ASSR). The...
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This article provides a simple method to accelerate Markov chain Monte Carlo sampling algorithms, such as the data augmentation algorithm and the Gibbs sampler, via alternating subspace-spanning resampling (ASSR). The ASSR algorithm often shares the simplicity of its parent sampler but has dramatically improved efficiency. The methodology is illustrated with Bayesian estimation for analysis of censored data from fractionated experiments. The relationships between ASSR and existing methods are also discussed.
A Gaussian random field with an unknown linear trend for the mean is considered. Methods for obtaining the distribution of the trend coefficients given exact data and inequality constraints are established, Moreover, ...
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A Gaussian random field with an unknown linear trend for the mean is considered. Methods for obtaining the distribution of the trend coefficients given exact data and inequality constraints are established, Moreover, the conditional distribution for the random field at any location is calculated so that predictions using e.g. the expectation, the mode, or the median can be evaluated and prediction error estimates using quantiles or variance can be obtained. Conditional simulation techniques are also provided.
Simplex constraints, such as monotonicity and convexity or concavity on the probabilities of a set of discrete distributions, are useful for modeling and analyzing discrete data. This article considers both maximum li...
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Simplex constraints, such as monotonicity and convexity or concavity on the probabilities of a set of discrete distributions, are useful for modeling and analyzing discrete data. This article considers both maximum likelihood estimation and Bayesian estimation of discrete distribution with a class of simplex constraints using the Expectation-Maximization (EM) algorithm and the dataaugmentation (DA) algorithm. The formulation and implementation of EM and DA for binomial, Poisson, hierarchical Poisson-binomial, multinomial, and hierarchical multinomial distributions are considered in detail and illustrated with examples.
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