A non-iterative posterior sampling algorithm for linear quantile regression model

In this article, a non-iterative posterior sampling algorithm for linear quantile regression model based on the asymmetric Laplace distribution is proposed. The algorithm combines the inverse Bayes formulae, sampling/importance resampling, and the expectation maximization algorithm to obtain independently and identically distributed samples approximately from the observed posterior distribution, which eliminates the convergence problems in the iterative Gibbs sampling and overcomes the difficulty in evaluating the standard deviance in the EM algorithm. The numeric results in simulations and application to the classical Engel data show that the non-iterative sampling algorithm is more effective than the Gibbs sampling and EM algorithm.     

Hit and run ARMS: adaptive rejection Metropolis sampling with hit and run random direction

An algorithm for sampling from non-log-concave multivariate distributions is proposed, which improves the adaptive rejection Metropolis sampling (ARMS) algorithm by incorporating the hit and run sampling. It is not rare that the ARMS is trapped away from some subspace with significant probability in the support of the multivariate distribution. While the ARMS updates samples only in the directions that are parallel to dimensions, our proposed method, the hit and run ARMS (HARARMS), updates samples in arbitrary directions determined by the hit and run algorithm, which makes it almost not possible to be trapped in any isolated subspaces. The HARARMS performs the same as ARMS in a single dimension while more reliable in multidimensional spaces. Its performance is illustrated by a Bayesian free-knot spline regression example. We showed that it overcomes the well-known ‘lethargy’ property and decisively find the global optimal number and locations of the knots of the spline function.     

Mixture Model Analysis of Partially Rank‐Ordered Set Samples: Age Groups of Fish from Length‐Frequency Data

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank‐ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank‐ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length‐frequency data.     

A generalization of the adaptive rejection sampling algorithm

Rejection sampling is a well-known method to generate random samples from arbitrary target probability distributions. It demands the design of a suitable proposal probability density function (pdf) from which candidate samples can be drawn. These samples are either accepted or rejected depending on a test involving the ratio of the target and proposal densities. The adaptive rejection sampling method is an efficient algorithm to sample from a log-concave target density, that attains high acceptance rates by improving the proposal density whenever a sample is rejected. In this paper we introduce a generalized adaptive rejection sampling procedure that can be applied with a broad class of target probability distributions, possibly non-log-concave and exhibiting multiple modes. The proposed technique yields a sequence of proposal densities that converge toward the target pdf, thus achieving very high acceptance rates. We provide a simple numerical example to illustrate the basic use of the proposed technique, together with a more elaborate positioning application using real data.     

Adaptive rejection sampling with fixed number of nodes

The adaptive rejection sampling (ARS) algorithm is a universal random generator for drawing samples efficiently from a univariate log-concave target probability density function (pdf). ARS generates independent samples from the target via rejection sampling with high acceptance rates. Indeed, ARS yields a sequence of proposal functions that converge toward the target pdf, so that the probability of accepting a sample approaches one. However, sampling from the proposal pdf becomes more computational demanding each time it is updated. In this work, we propose a novel ARS scheme, called Cheap Adaptive Rejection Sampling (CARS), where the computational effort for drawing from the proposal remains constant, decided in advance by the user. For generating a large number of desired samples, CARS is faster than ARS.     

Simulation of Constrained Variables in Engineering Risk Analyses

The problem of sampling random variables with overlapping pdfs subject to inequality constraints is addressed. Often, the values of physical variables in an engineering model are interrelated. This mutual dependence imposes inequality constraints on the random variables representing these parameters. Ignoring the interdependencies and sampling the variables independently can lead to inconsistency/bias. We propose an algorithm to generate samples of constrained random variables that are characterized by typical continuous probability distributions and are subject to different kinds of inequality constraints. The sampling procedure is illustrated for various representative cases and one realistic application to simulation of structural natural frequencies.     

Parallelizing MCMC for Bayesian spatiotemporal geostatistical models

When MCMC methods for Bayesian spatiotemporal modeling are applied to large geostatistical problems, challenges arise as a consequence of memory requirements, computing costs, and convergence monitoring. This article describes the parallelization of a reparametrized and marginalized posterior sampling (RAMPS) algorithm, which is carefully designed to generate posterior samples efficiently. The algorithm is implemented using the Parallel Linear Algebra Package (PLAPACK). The scalability of the algorithm is investigated via simulation experiments that are implemented using a cluster with 25 processors. The usefulness of the method is illustrated with an application to sulfur dioxide concentration data from the Air Quality System database of the U.S. Environmental Protection Agency.     

Maximum Likelihood Inference for Multivariate Frailty Models Using an Automated Monte Carlo EM Algorithm

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.     

Ml estimation in the poisson binomial distribution with grouped data via the em algorithm

The maximum likelihood estimation of parameters of the Poisson binomial distribution, based on a sample with exact and grouped observations, is considered by applying the EM algorithm (Dempster et al, 1977). The results of Louis (1982) are used in obtaining the observed information matrix and accelerating the convergence of the EM algorithm substantially. The maximum likelihood estimation from samples consisting entirely of complete (Sprott, 1958) or grouped observations are treated as special cases of the estimation problem mentioned above. A brief account is given for the implementation of the EM algorithm when the sampling distribution is the Neyman Type A since the latter is a limiting form of the Poisson binomial. Numerical examples based on real data are included.     

Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm

Two new implementations of the EM algorithm are proposed for maximum likelihood fitting of generalized linear mixed models. Both methods use random (independent and identically distributed) sampling to construct Monte Carlo approximations at the E-step. One approach involves generating random samples from the exact conditional distribution of the random effects (given the data) by rejection sampling, using the marginal distribution as a candidate. The second method uses a multivariate t importance sampling approximation. In many applications the two methods are complementary. Rejection sampling is more efficient when sample sizes are small, whereas importance sampling is better with larger sample sizes. Monte Carlo approximation using random samples allows the Monte Carlo error at each iteration to be assessed by using standard central limit theory combined with Taylor series methods. Specifically, we construct a sandwich variance estimate for the maximizer at each approximate E-step. This suggests a rule for automatically increasing the Monte Carlo sample size after iterations in which the true EM step is swamped by Monte Carlo error. In contrast, techniques for assessing Monte Carlo error have not been developed for use with alternative implementations of Monte Carlo EM algorithms utilizing Markov chain Monte Carlo E-step approximations. Three different data sets, including the infamous salamander data of McCullagh and Nelder, are used to illustrate the techniques and to compare them with the alternatives. The results show that the methods proposed can be considerably more efficient than those based on Markov chain Monte Carlo algorithms. However, the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension.     

Maximum Likelihood Estimation of Logistic Regression Parameters under Two-phase, Outcome-dependent Sampling

Outcome-dependent sampling increases the efficiency of studies of rare outcomes, examples being case—control studies in epidemiology and choice–based sampling in econometrics. Two-phase or double sampling is a standard technique for drawing efficient stratified samples. We develop maximum likelihood estimation of logistic regression coefficients for a hybrid two-phase, outcome–dependent sampling design. An algorithm is given for determining the estimates by repeated fitting of ordinary logistic regression models. Simulation results demonstrate the efficiency loss associated with alternative pseudolikelihood and weighted likelihood methods for certain data configurations. These results provide an efficient solution to the measurement error problem with validation sampling based on a discrete surrogate.     

Stochastic approximation Monte Carlo importance sampling for approximating exact conditional probabilities

Importance sampling and Markov chain Monte Carlo methods have been used in exact inference for contingency tables for a long time, however, their performances are not always very satisfactory. In this paper, we propose a stochastic approximation Monte Carlo importance sampling (SAMCIS) method for tackling this problem. SAMCIS is a combination of adaptive Markov chain Monte Carlo and importance sampling, which employs the stochastic approximation Monte Carlo algorithm (Liang et al., J. Am. Stat. Assoc., 102(477):305–320, 2007) to draw samples from an enlarged reference set with a known Markov basis. Compared to the existing importance sampling and Markov chain Monte Carlo methods, SAMCIS has a few advantages, such as fast convergence, ergodicity, and the ability to achieve a desired proportion of valid tables. The numerical results indicate that SAMCIS can outperform the existing importance sampling and Markov chain Monte Carlo methods: It can produce much more accurate estimates in much shorter CPU time than the existing methods, especially for the tables with high degrees of freedom.     

Bayesian temporal density estimation with autoregressive species sampling models

We propose a novel Bayesian nonparametric (BNP) model, which is built on a class of species sampling models, for estimating density functions of temporal data. In particular, we introduce species sampling mixture models with temporal dependence. To accommodate temporal dependence, we define dependent species sampling models by modeling random support points and weights through an autoregressive model, and then we construct the mixture models based on the collection of these dependent species sampling models. We propose an algorithm to generate posterior samples and present simulation studies to compare the performance of the proposed models with competitors that are based on Dirichlet process mixture models. We apply our method to the estimation of densities for the price of apartment in Seoul, the closing price in Korea Composite Stock Price Index (KOSPI), and climate variables (daily maximum temperature and precipitation) of around the Korean peninsula.     

Bayesian spatial prediction of skew and censored data via a hybrid algorithm

A correct detection of areas with excess of pollution relies first on accurate predictions of pollutant concentrations, a task that is usually complicated by skewed histograms and the presence of censored data. The unified skew-Gaussian (SUG) random field proposed by Zareifard and Jafari Khaledi [19] offers a more flexible class of sampling spatial models to account for skewness. In this paper, we adopt a Bayesian framework to perform prediction for the SUG model in the presence of censored data. Owing to the presence of many latent variables with strongly dependent components in the model, we encounter convergence issues when using Monte Carlo Markov Chain algorithms. To overcome this obstacle, we use a computationally efficient inverse Bayes formulas sampling procedure to obtain approximately independent samples from the posterior distribution of latent variables. Then they are applied to update parameters in a Gibbs sampler scheme. This hybrid algorithm provides effective samples, resulting in some computational advantages and precise predictions. The proposed approach is illustrated with a simulation study and applied to a spatial data set which contains right censored data.     

A MCMC algorithm to fit a general exchangeable model

Consider the exchangeable Bayesian hierarchical model where observations y_i are independently distributed from sampling densities with unknown means, the means µi, are a random sample from a distribution g, and the parameters of g are assigned a known distribution h. A simple algorithm is presented for summarizing the posterior distribution based on Gibbs sampling and the Metropolis algorithm. The software program Matlab is used to implement the algorithm and provide a graphical output analysis. An binomial example is used to illustrate the flexibility of modeling possible using this algorithm. Methods of model checking and extensions to hierarchical regression modeling are discussed.     

Bayesian analysis of time series Poisson data

This paper provides a practical simulation-based Bayesian analysis of parameter-driven models for time series Poisson data with the AR(1) latent process. The posterior distribution is simulated by a Gibbs sampling algorithm. Full conditional posterior distributions of unknown variables in the model are given in convenient forms for the Gibbs sampling algorithm. The case with missing observations is also discussed. The methods are applied to real polio data from 1970 to 1983.     

规模以下工业抽样调查中代表性样本的一种探索设计:平衡抽样设计

规下工业抽样调查是社会经济统计调查的重要组成部分,为国民经济核算提供基础数据,而样本代表性直接决定统计推断结果。对企业目录库抽取平衡样本,能够使得样本结构与总体结构相似。平衡样本是指满足如下条件的样本:辅助变量的汉森赫维茨估计等于总体总量真值。平衡抽样设计需要包含丰富辅助信息的完善抽样框,政府统计数据能够为此提供足够的支撑。基于2009年工业企业数据库的实证分析表明,平衡抽样设计对总体总量的估计相对误差很小,特别是估计的均值与总体真值非常接近,近似无偏;与简单随机抽样比较,平衡抽样设计更加有效。     

Generalized inverted exponential distribution under hybrid censoring

The hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. Based on hybrid censored samples, we first derive the maximum likelihood estimators of the unknown parameters and the expected Fisher’s information matrix of the generalized inverted exponential distribution (GIED). Monte Carlo simulations are performed to study the performance of the maximum likelihood estimators. Next we consider Bayes estimation under the squared error loss function. These Bayes estimates are evaluated by applying Lindley’s approximation method, the importance sampling procedure and Metropolis–Hastings algorithm. The importance sampling technique is used to compute the highest posterior density credible intervals. Two data sets are analyzed for illustrative purposes. Finally, we discuss a method of obtaining the optimum hybrid censoring scheme.     

A modified ziggurat algorithm for generating exponentially and normally distributed pseudorandom numbers

The ziggurat algorithm is a very fast rejection sampling method for generating pseudorandom numbers (PRNs) from statistical distributions. In the algorithm, rectangular sampling domains are layered on top of each other (resembling a ziggurat) to encapsulate the desired probability density function. Random values within these layers are sampled and then returned if they lie beneath the graph of the probability density function. Here, we present an implementation where ziggurat layers reside completely beneath the probability density function, thereby eliminating the need for any rejection test within the ziggurat layers. In the new algorithm, small overhanging segments of probability density remain to the right of each ziggurat layer, which can be efficiently sampled with triangularly shaped sampling domains. Median runtimes of the new algorithm for exponential and normal variates is reduced to 58% and 53%, respectively (collective range: 41–93%). An accessible C library, along with extensions into Python and MATLAB/Octave are provided.