Sampling from a truncated multivariate normal distribution （TMVND） constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data au...Sampling from a truncated multivariate normal distribution （TMVND） constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation （DA） algorithm and a non-iterative inverse Bayes formulae （IBF） sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA Mgorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and elimi- nares problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.展开更多
et X=(X1,...,Xn )' have a multivariate normal distribution with mean μ and covariance matrix Σ. In the case μ=0, Karlin and Rinott[6] obtained a necessary and sufficient condition on Σ for |X|=(|X1|,...,|Xn|)&...et X=(X1,...,Xn )' have a multivariate normal distribution with mean μ and covariance matrix Σ. In the case μ=0, Karlin and Rinott[6] obtained a necessary and sufficient condition on Σ for |X|=(|X1|,...,|Xn|)' to be MTP2. In this paper we consider the case μ≠0, and give some conditions under which |X| is MTP2. A necessary and sufficient condition is given for |X| to be TP2 when n=2 and μ≠0. Some results about the TP2 stochastic ordering are also given. The results are applied to obtain positive dependence and associated inequalities for multinormal and related distributions.展开更多
基金Supported by the National Social Science Foundation of China (No. 09BTJ012)Scientific Research Fund ofHunan Provincial Education Department (No. 09c390)+1 种基金supported in part by a HKUSeed Funding Program for Basic Research (Project No. 2009-1115-9042)a grant from Hong Kong ResearchGrant Council-General Research Fund (Project No. HKU779210M)
文摘Sampling from a truncated multivariate normal distribution （TMVND） constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation （DA） algorithm and a non-iterative inverse Bayes formulae （IBF） sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA Mgorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and elimi- nares problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.
文摘et X=(X1,...,Xn )' have a multivariate normal distribution with mean μ and covariance matrix Σ. In the case μ=0, Karlin and Rinott[6] obtained a necessary and sufficient condition on Σ for |X|=(|X1|,...,|Xn|)' to be MTP2. In this paper we consider the case μ≠0, and give some conditions under which |X| is MTP2. A necessary and sufficient condition is given for |X| to be TP2 when n=2 and μ≠0. Some results about the TP2 stochastic ordering are also given. The results are applied to obtain positive dependence and associated inequalities for multinormal and related distributions.
