Testing Departures from Gamma, Rayleigh and Truncated Normal Distributions

This paper provides necessary and sufficient conditions for a solution to likelihood equations for an exponential family of distributions, which includes Gamma, Rayleigh and singly truncated normal distributions. Furthermore, the maximum likelihood estimator is obtained as a limit case when the equations have no solution. These results provide a way to test departures from Rayleigh and singly truncated normal distributions using the likelihood ratio test. A new easy way to test departures from a Gamma distribution is also introduced.     

On the Convergence of Successive Substitution Sampling

Abstract

The problem of finding marginal distributions of multidimensional random quantities has many applications in probability and statistics. Many of the solutions currently in use are very computationally intensive. For example, in a Bayesian inference problem with a hierarchical prior distribution, one is often driven to multidimensional numerical integration to obtain marginal posterior distributions of the model parameters of interest. Recently, however, a group of Monte Carlo integration techniques that fall under the general banner of successive substitution sampling (SSS) have proven to be powerful tools for obtaining approximate answers in a very wide variety of Bayesian modeling situations. Answers may also be obtained at low cost, both in terms of computer power and user sophistication. Important special cases of SSS include the “Gibbs sampler” described by Gelfand and Smith and the “IP algorithm” described by Tanner and Wong. The major problem plaguing users of SSS is the difficulty in ascertaining when “convergence” of the algorithm has been obtained. This problem is compounded by the fact that what is produced by the sampler is not the functional form of the desired marginal posterior distribution, but a random sample from this distribution. This article gives a general proof of the convergence of SSS and the sufficient conditions for both strong and weak convergence, as well as a convergence rate. We explore the connection between higher-order eigenfunctions of the transition operator and accelerated convergence via good initial distributions. We also provide asymptotic results for the sampling component of the error in estimating the distributions of interest. Finally, we give two detailed examples from familiar exponential family settings to illustrate the theory.     

Multivariate Normal Slice Sampling

By introducing auxiliary variables, the traditional Markov chain Monte Carlo method can be improved in certain cases by implementing a “slice sampler.” In the current literature, this sampling technique is used to sample from multivariate distributions with both single and multiple auxiliary variables. When the latter is employed, it generally updates one component at a time.

In this article, we propose two variations of a new multivariate normal slice sampling method that uses multiple auxiliary variables to perform multivariate updating. These methods are flexible enough to allow for truncation to a rectangular region and/or exclusion of any n-dimensional hyper-quadrant. We present results of our methods and existing state-of-the-art slice samplers by comparing efficiency and accuracy. We find that we can generate approximately iid samples at a rate that is more efficient than other methods that update all dimensions at once. Supplemental materials are available online.     

Antithetic Methods for Gibbs Samplers

In this article we propose a modification to the output from Metropolis-within-Gibbs samplers that can lead to substantial reductions in the variance over standard estimates. The idea is simple: at each time step of the algorithm, introduce an extra sample into the estimate that is negatively correlated with the current sample, the rationale being that this provides a two-sample numerical approximation to a Rao–Blackwellized estimate. As the conditional sampling distribution at each step has already been constructed, the generation of the antithetic sample often requires negligible computational effort. Our method is implementable whenever one subvector of the state can be sampled from its full conditional and the corresponding distribution function may be inverted, or the full conditional has a symmetric density. We demonstrate our approach in the context of logistic regression and hierarchical Poisson models. The data and computer code used in this article are available online.     

Convergence rates of the blocked Gibbs sampler with random scan in the Wasserstein metric

ABSTRACT

This paper establishes explicit estimates of convergence rates for the blocked Gibbs sampler with random scan under the Dobrushin conditions. The estimates of convergence in the Wasserstein metric are obtained by taking purely analytic approaches.     

Image Restoration Using Modifications of Simulated Annealing

Abstract

This article uses a modified version of the simulated annealing algorithm to restore degraded spatial patterns. Standard simulated annealing is used to find an image that is a posterior mode when the number of images under consideration precludes sequential search for a maximum. I incorporate jumping probabilities of the annealing algorithm without randomization. The convergence of our algorithm is proven under a practical annealing schedule. The same idea is also implemented to improve the performance of other modifications of simulated annealing. These include forcing proportions of labels in an image, using posterior marginals, and incorporating an edge process. This article also studies nonlinear presmoothing of the observations.     

三参数正态双卵模型Gibbs抽样算法研究

项目反应理论作为一种现代的教育和心理测量方法,凭借其强大的优势和先进性,在实际测量中应用越来越广泛.能否有效地估计模型中的参数是项目反应模型得以应用的前提.本文基于数据扩充技术给出了一种适用于三参数正态双卵模型的Gibbs抽样算法,有效的实现三参数正态双卵模型的贝叶斯分析.最后,通过计算机模拟研究和实例分析对该算法的有效性进行了验证.     

Bivariate gamma distributions, sums and ratios

Exact distributions of R = X +Y and W = X/(X +Y ) and the corresponding moment properties are derived when X and Y follow five flexible bivariate gamma distributions. The expressions turn out to involve several special functions.     

An Improvement of Normal Approximation of Randomized Orthogonal Array Sampling Designs

This article considers a class of nonlocal stochastic functional differential equations with infinite delay whose coefficients are dependent the pth moment and establishes the existence-and-uniqueness theorem under the conditions that are similar to the classical linear growth condition and the Lipschitz condition. Compared with the existing results, the conditions of this article are easier to test.     

随机变量的和、积、商分布的一种新表示

应用Feller提出的点-集函数并结合二元copula,对二元连续型正值随机变量的和、积、商的分布进行了研究,得到了和、积、商分布的一种新的计算方法.最后给出一个应用实例.     

Sequential Confidence Bands for Quantile Densities Under Truncated and Censored Data

In this paper an asymptotic distribution is obtained for the maximal deviation between the kernel quantile density estimator and the quantile density when the data are subject to random left truncation and right censorship. Based on this result we propose a fully sequential procedure for construct ing a fixed-width confidence band for the quantile density on a finite interval and show that the procedure has the desired coverage probability asymptotically as the width of the band approaches zero.     

两个非独立正态变量的联合分布不服从二维正态分布的示例

围绕着两个非独立的正态变量的联合分布一定服从二维正态分布的问题,从不同角度给出了反例,并加以论证.     

Multidimensional Limit Theorems Allowing Large Deviations for Densities of Regular Variation

The sums of i.i.d. random vectors are considered. It is assumed that the underlying distribution is absolutely continuous and its density possesses the property which can be referred to as regular variation. The asymptotic expressions for the probability of large deviations are established in the case of a normal limiting law. Furthermore, the role of the maximal summand is emphasized.     

Diameter, connectivity, and phase transition of the uniform random intersection graph

We study properties of the uniform random intersection graph model G(n,m,d). We find asymptotic estimates on the diameter of the largest connected component of the graph near the phase transition and connectivity thresholds. Moreover we manage to prove an asymptotically tight bound for the connectivity and phase transition thresholds for all possible ranges of d, which has not been obtained before. The main motivation of our research is the usage of the random intersection graph model in the studies of wireless sensor networks.     

Prediction and Inference for Truncated Spatial Data

Abstract

Spatial data in mining, hydrology, and pollution monitoring commonly have a substantial proportion of zeros. One way to model such data is to suppose that some pointwise transformation of the observations follows the law of a truncated Gaussian random field. This article considers Monte Carlo methods for prediction and inference problems based on this model. In particular, a method for computing the conditional distribution of the random field at an unobserved location, given the data, is described. These results are compared to those obtained by simple kriging and indicator cokriging. Simple kriging is shown to give highly misleading results about conditional distributions; indicator cokriging does quite a bit better but still can give answers that are substantially different from the conditional distributions. A slight modification of this basic technique is developed for calculating the likelihood function for such models, which provides a method for computing maximum likelihood estimates of unknown parameters and Bayesian predictive distributions for values of the process at unobserved locations.     

On Random Sampling Without Replacement from a Finite Population

We consider the three progressively more general sampling schemes without replacement from a finite population: simple random sampling without replacement, Midzuno sampling and successive sampling. We (i) obtain a lower bound on the expected sample coverage of a successive sample, (ii) show that the vector of first order inclusion probabilities divided by the sample size is majorized by the vector of selection probabilities of a successive sample, and (iii) partially order the vectors of first order inclusion probabilities for the three sampling schemes by majorization. We also show that the probability of an ordered successive sample enjoys the arrangement increasing property and for sample size two the expected sample coverage of a successive sample is Schur convex in its selection probabilities. We also study the spacings of a simple random sample from a linearly ordered finite population and characterize in several ways a simple random sample.     

On Moments of Folded and Truncated Multivariate Normal Distributions

Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low-order moments of folded and truncated multivariate normal distributions. Supplementary material for this article is available online.     

Bayesian image restoration,with two applications in spatial statistics

There has been much recent interest in Bayesian image analysis, including such topics as removal of blur and noise, detection of object boundaries, classification of textures, and reconstruction of two- or three-dimensional scenes from noisy lower-dimensional views. Perhaps the most straightforward task is that of image restoration, though it is often suggested that this is an area of relatively minor practical importance. The present paper argues the contrary, since many problems in the analysis of spatial data can be interpreted as problems of image restoration. Furthermore, the amounts of data involved allow routine use of computer intensive methods, such as the Gibbs sampler, that are not yet practicable for conventional images. Two examples are given, one in archeology, the other in epidemiology. These are preceded by a partial review of pixel-based Bayesian image analysis.An earlier version of this article was presented at the symposium on the Analysis of Statistical Information held in the Institute of Statistical Mathematics, Tokyo during December 5–8, 1989.This research was carried out partly at the University of Durham, U.K., with the support of an award by the Complex Stochastic Systems Initiative of the Science and Engineering Research Council.     

On the convergence of random polynomials and multilinear forms

We consider different kinds of convergence of homogeneous polynomials and multilinear forms in random variables. We show that for a variety of complex random variables, the almost sure convergence of the polynomial is equivalent to that of the multilinear form, and to the square summability of the coefficients. Also, we present polynomial Khintchine inequalities for complex gaussian and Steinhaus variables. All these results have no analogues in the real case. Moreover, we study the Lp-convergence of random polynomials and derive certain decoupling inequalities without the usual tetrahedral hypothesis. We also consider convergence on “full subspaces” in the sense of Sjögren, both for real and complex random variables, and relate it to domination properties of the polynomial or the multilinear form, establishing a link with the theory of homogeneous polynomials on Banach spaces.