Efficient use of exact samples

Propp and Wilson (Random Structures and Algorithms (1996) 9: 223–252, Journal of Algorithms (1998) 27: 170–217) described a protocol called coupling from the past (CFTP) for exact sampling from the steady-state distribution of a Markov chain Monte Carlo (MCMC) process. In it a past time is identified from which the paths of coupled Markov chains starting at every possible state would have coalesced into a single value by the present time; this value is then a sample from the steady-state distribution.Unfortunately, producing an exact sample typically requires a large computational effort. We consider the question of how to make efficient use of the sample values that are generated. In particular, we make use of regeneration events (cf. Mykland et al. Journal of the American Statistical Association (1995) 90: 233–241) to aid in the analysis of MCMC runs. In a regeneration event, the chain is in a fixed reference distribution– this allows the chain to be broken up into a series of tours which are independent, or nearly so (though they do not represent draws from the true stationary distribution).In this paper we consider using the CFTP and related algorithms to create tours. In some cases their elements are exactly in the stationary distribution; their length may be fixed or random. This allows us to combine the precision of exact sampling with the efficiency of using entire tours.Several algorithms and estimators are proposed and analysed.     

On perfect simulation for some mixtures of distributions

This paper studies the implementation of the coupling from the past (CFTP) method of Propp and Wilson (1996) in the set-up of two and three component mixtures with known components and unknown weights. We show that monotonicity structures can be exhibited in both cases, but that CFTP can still be costly for three component mixtures. We conclude with a simulation experiment exhibiting an almost perfect sampling scheme where we only consider a subset of the exhaustive set of starting values.     

Modified chain sampling plans for lot inspection by variables and attributes

The purpose of acceptance sampling is to develop decision rules to accept or reject production lots based on sample data. When testing is destructive or expensive, dependent sampling procedures cumulate results from several preceding lots. This chaining of past lot results reduces the required size of the samples. A large part of these procedures only chain past lot results when defects are found in the current sample. However, such selective use of past lot results only achieves a limited reduction of sample sizes. In this article, a modified approach for chaining past lot results is proposed that is less selective in its use of quality history and, as a result, requires a smaller sample size than the one required for commonly used dependent sampling procedures, such as multiple dependent sampling plans and chain sampling plans of Dodge. The proposed plans are applicable for inspection by attributes and inspection by variables. Several properties of their operating characteristic-curves are derived, and search procedures are given to select such modified chain sampling plans by using the two-point method.     

Perfect sampling algorithm for small <Emphasis Type="Italic">m</Emphasis>×<Emphasis Type="Italic">n</Emphasis> contingency tables

A Markov chain is proposed that uses coupling from the past sampling algorithm for sampling m×n contingency tables. This method is an extension of the one proposed by Kijima and Matsui (Rand. Struct. Alg., 29:243–256, 2006). It is not polynomial, as it is based upon a recursion, and includes a rejection phase but can be used for practical purposes on small contingency tables as illustrated in a classical 4×4 example.     

Perfect simulation of positive Gaussian distributions

We provide an exact simulation algorithm that produces variables from truncated Gaussian distributions on ( ₊) ^p via a perfect sampling scheme, based on stochastic ordering and slice sampling, since accept-reject algorithms like the one of Geweke (1991) and Robert (1995) are difficult to extend to higher dimensions.     

Simulation algorithms for integrals of a class of sampling distributions arising in population genetics

Efficient stochastic algorithms are presented in order to simulate allele configurations distributed according to a family π _A , 0<A<∞, of exchangeable sampling distributions arising in population genetics. Each distribution π _A has two parameters n and k, the sample size and the number of alleles, respectively. For A→0, the distribution π _A is induced from neutral sampling, whereas for A→∞, it is induced from Maxwell–Boltzmann sampling. Three different Monte Carlo methods (independent sampling procedures) are provided, based on conditioning, sequential methods and a generalization of Pitmans ‘Chinese restaurant process’. Moreover, an efficient Markov chain Monte Carlo method is provided. The algorithms are applied to the homozygosity test and to the Ewens–Watterson–Slatkin test in order to test the hypothesis of selective neutrality.     

Perfect simulation of conditionally specified models

We discuss how the ideas of producing perfect simulations based on coupling from the past for finite state space models naturally extend to multivariate distributions with infinite or uncountable state spaces such as auto-gamma, auto-Poisson and autonegative binomial models, using Gibbs sampling in combination with sandwiching methods originally introduced for perfect simulation of point processes.     

Designing of a mixed-chain sampling plan based on the process capability index with chain sampling as the attributes plan

In this article, we propose a new mixed chain sampling plan based on the process capability index C_pk, where the quality characteristic of interest having double specification limits and follows the normal distribution with unknown mean and variance. In the proposed mixed plan, the chain sampling inspection plan is used for the inspection of attribute quality characteristics. The advantages of this proposed mixed sampling plan are also discussed. Tables are constructed to determine the optimal parameters for practical applications by formulating the problem as a non linear programming in which the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operating characteristic curve. The practical application of the proposed mixed sampling plan is explained with an illustrative example. Comparison of the proposed sampling plan is also made with other existing sampling plans.     

Estimation of Finite Population Mean in Stratified Random Sampling with Two Auxiliary Variables under Double Sampling Design

In this article, a chain ratio-product type exponential estimator is proposed for estimating finite population mean in stratified random sampling with two auxiliary variables under double sampling design. Theoretical and empirical results show that the proposed estimator is more efficient than the existing estimators, i.e., usual stratified random sample mean estimator, Chand (1975) chain ratio estimator, Choudhary and Singh (2012) estimator, chain ratio-product-type estimator, Sahoo et al. (1993) difference type estimator, and Kiregyera (1984) regression-type estimator. Two data sets are used to illustrate the performances of different estimators.     

Annealed importance sampling

Simulated annealing—moving from a tractable distribution to a distribution of interest via a sequence of intermediate distributions—has traditionally been used as an inexact method of handling isolated modes in Markov chain samplers. Here, it is shown how one can use the Markov chain transitions for such an annealing sequence to define an importance sampler. The Markov chain aspect allows this method to perform acceptably even for high-dimensional problems, where finding good importance sampling distributions would otherwise be very difficult, while the use of importance weights ensures that the estimates found converge to the correct values as the number of annealing runs increases. This annealed importance sampling procedure resembles the second half of the previously-studied tempered transitions, and can be seen as a generalization of a recently-proposed variant of sequential importance sampling. It is also related to thermodynamic integration methods for estimating ratios of normalizing constants. Annealed importance sampling is most attractive when isolated modes are present, or when estimates of normalizing constants are required, but it may also be more generally useful, since its independent sampling allows one to bypass some of the problems of assessing convergence and autocorrelation in Markov chain samplers.     

On an adaptive version of the Metropolis–Hastings algorithm with independent proposal distribution

In this paper, we present a general formulation of an algorithm, the adaptive independent chain (AIC), that was introduced in a special context in Gåsemyr et al . [ Methodol. Comput. Appl. Probab. 3 (2001)]. The algorithm aims at producing samples from a specific target distribution Π, and is an adaptive, non-Markovian version of the Metropolis–Hastings independent chain. A certain parametric class of possible proposal distributions is fixed, and the parameters of the proposal distribution are updated periodically on the basis of the recent history of the chain, thereby obtaining proposals that get ever closer to Π. We show that under certain conditions, the algorithm produces an exact sample from Π in a finite number of iterations, and hence that it converges to Π. We also present another adaptive algorithm, the componentwise adaptive independent chain (CAIC), which may be an alternative in particular in high dimensions. The CAIC may be regarded as an adaptive approximation to the Gibbs sampler updating parametric approximations to the conditionals of Π.     

Consultant's corner: A consulting problem involving bivariate acceptance sampling by variables

Acceptance sampling techniques are used to monitor the accuracy of gas meters. Random samples of meters are taken from homogeneous lots, and two accuracy measurements are recorded for each meter. In the past, the two measurements were averaged, and an acceptance sampling test applied to the sample of averages. In 1987, the plan was modified so that virtually the same test is applied to the two measurements individually. This new procedure is more stringent than the old procedure. In a study of data sampled over three years, the new plan rejects more lots than does the old plan, leading to greatly increased costs to the gas industry and therefore to the consumer. Theoretical reasons are given for why this occurs, and an alternative plan is proposed.     

A new mixed chain sampling plan based on the process capability index for product acceptance

This article proposes a new mixed chain sampling plan based on the process capability index C_pk, where the quality characteristic of interest follows the normal distribution with unknown mean and variance. The advantages of this proposed mixed sampling plan are also discussed. Tables are constructed to determine the optimal parameters for practical applications. In order to construct the tables, the problem is formulated as a nonlinear programming where the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operating characteristic curve. The practical application of the proposed mixed sampling plan is explained with an illustrative example. Comparison of the proposed sampling plan is also made with other existing sampling plans.     

Layer Sampling

Layer sampling is an algorithm for generating variates from a non-normalized multidimensional distribution p( · ). It empirically constructs a majorizing function for p( · ) from a sequence of layers. The method first selects a layer based on the previous variate. Next, a sample is drawn from the selected layer, using a method such as Rejection Sampling. Layer sampling is regenerative. At regeneration times, the layers may be adapted to increase mixing of the Markov chain. Layer sampling may also be used to estimate arbitrary integrals, including normalizing constants.     

A Class of Normalized Random Measures with an Exact Predictive Sampling Scheme

Abstract. In this article, we define and investigate a novel class of non‐parametric prior distributions, termed the class . Such class of priors is dense with respect to the homogeneous normalized random measures with independent increments and it is characterized by a richer predictive structure than those arising from other widely used priors. Our interest in the class is mainly motivated by Bayesian non‐parametric analysis of some species sampling problems concerning the evaluation of the species relative abundances in a population. We study both the probability distribution of the number of species present in a sample and the probability of discovering a new species conditionally on an observed sample. Finally, by using the coupling from the past method, we provide an exact sampling scheme for the system of predictive distributions characterizing the class .     

Some remarks on the traditional way of circular and diagonal circular systematic sampling schemes

ABSTRACT

In this paper, some deficiencies in traditional selection procedure of circular version of systematic sampling schemes are investigated and alternative methods are proposed. We also suggest some rules of thumb for coincidence of units in the sample. The end corrections proposed by Bellhouse and Rao (1975 Bellhouse, D.R., Rao, J.N.K. (1975). Systematic sampling in the presence of a trend. Biometrika. 62:694–697.[Crossref], [Web of Science ®] , [Google Scholar]) and Sampath and Varalakshmi (2008) for circular systematic sampling and diagonal circular systematic sampling, respectively, are also modified.     

On Exact Simulation of Markov Random Fields Using Coupling from the Past

A general framework for exact simulation of Markov random fields using the Propp–Wilson coupling from the past approach is proposed. Our emphasis is on situations lacking the monotonicity properties that have been exploited in previous studies. A critical aspect is the convergence time of the algorithm; this we study both theoretically and experimentically. Our main theoretical result in this direction says, roughly, that if interactions are sufficiently weak, then the expected running time of a carefully designed implementation is O ( N log N ), where N is the number of interacting components of the system. Computer experiments are carried out for random q -colourings and for the Widom–Rowlinson lattice gas model.     

Unbiased estimators for restricted adaptive cluster sampling

In adaptive cluster sampling the size of the final sample is random, thus creating design problems. To get round this, Brown (1994) and Brown & Manly (1998) proposed a modification of the method, placing a restriction on the size of the sample, and using standard but biased estimators for estimating the population mean. But in this paper a new unbiased estimator and an unbiased variance estimator are proposed, based on estimators proposed by Murthy (1957) and extended to sequential and adaptive sampling designs by Salehi & Seber (2001). The paper also considers a restricted version of the adaptive scheme of Salehi & Seber (1997a) in which the networks are selected without replacement, and obtains unbiased estimators. The method is demonstrated by a simple example. Using simulation from this example, the new estimators are shown to compare very favourably with the standard biased estimators.     

Ratio estimator for the population mean using ranked set sampling

We adapt the ratio estimation using ranked set sampling, suggested by Samawi and Muttlak (Biometr J 38:753–764, 1996), to the ratio estimator for the population mean, based on Prasad (Commun Stat Theory Methods 18:379–392, 1989), in simple random sampling. Theoretically, we show that the proposed ratio estimator for the population mean is more efficient than the ratio estimator, in Prasad (1989), in all conditions. In addition, we support this theoretical result with the aid of a numerical example.