Duncan J. Murdoch

Extension of Fill's perfect rejection sampling algorithm to general chains

We provide an extension of the perfect sampling algorithm of Fill (1998) to general chains, and describe how use of bounding processes can ease computational burden. Along the way, we unearth a simple connection between the Coupling From The Past (CFTP) algorithm originated by Propp and Wilson (1996) and our extension of Fills algorithm.

P-Values are Random Variables

P-values are taught in introductory statistics classes in a way that confuses many of the students, leading to common misconceptions about their meaning. In this article, we argue that p-values should be taught through simulation, emphasizing that p-values are random variables. By means of elementary examples we illustrate how to teach students valid interpretations of p-values and give them a deeper understanding of hypothesis testing.

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.

A First Course in Statistical Programming with R: Programming with R

Programming involves writing relatively complex systems of instructions. There are two broad styles of programming: the imperative style (used in R, for example) involves stringing together instructions telling the computer what to do. The declarative style (used in HTML in web pages, for example) involves writing a description of the end result, without giving the details about how to get there. Within each of these broad styles, there are many subdivisions, and a given program may involve aspects of several of them. For example, R programs may be procedural (describing what steps to take to achieve a task), modular (broken up into self-contained packages), object-oriented (organized to describe operations on complex objects), and/or functional (organized as a collection of functions which do specific calculations without having external side-effects), among other possibilities. In this book we will concentrate on the procedural aspects of programming. As described in Chapter 1, R statements mainly consist of expressions to be evaluated. Most programs are very repetitive, but the amount of repetition depends on the input. In this chapter we start by describing several flow control statements that control how many times statements are repeated. The remainder of the chapter gives advice on how to design and debug programs. Flow control The for() loop One of the goals of this book is to introduce stochastic simulation. Simulations are often very repetitive: we want to see patterns of behaviour, not just a single instance. The for() statement allows one to specify that a certain operation should be repeated a fixed number of times. Syntax for (name in vector) { commands } This sets a variable called name equal to each of the elements of vector, in sequence. For each value, whatever commands are listed within the curly braces will be performed. The curly braces serve to group the commands so that they are treated by R as a single command. If there is only one command to execute, the braces are not needed.

Perfect sampling for queues and network models

We review Propp and Wilsons [1996] CFTP algorithm and Wilsons [2000] ROCFTP algorithm. We then use these to construct perfect samplers for several queueing and network models: Poisson arrivals and exponential service times, several types of customers, and a trunk reservation protocol for accepting new customers; a similar protocol on a network switching model; a queue with a general arrival process; and a queue with both general arrivals and service times. Our samplers give effective ways to generate random samples from the steady-state distributions of these queues.

Correspondence analysis with incomplete paired data using Bayesian imputation

In this paper we consider the analysis of incomplete tables using Correspondence Analysis (CA). We focus on a dataset concerning congenital heart disease (Fraser and Hunter 1975), in which the data forms a square table, but only a symmetrized version of the off-diagonal entries was reported. We use Markov chain Monte Carlo (MCMC) on a hierarchical Bayes model to estimate the underlying rates, and use CA to study the relationships in the completed table.

Uniqueness, consistency and optimality in spherical regression experiments

For designed experiments based on the spherical regression model of Chang (Ann. Statist. 14 (1986) 907) we provide results on the minimum number of covariate directions that are necessary and sufficient for uniqueness and consistency of least squares estimates and on minimizing confidence regions.

Perfect and nearly perfect sampling of work-conserving queues

In this paper, we explore algorithms for perfect and nearly perfect sampling from the stationary distribution of the waiting times in various Poisson arrival multi-class and multi-server queues with non-preemptive work-conserving service disciplines. The service duration distributions of these classes may be identical or may vary from class to class. The algorithms follow the idea of dominated coupling from the past (Kendall, Adv Appl Probab 32:844–865 2000) and are variations on an algorithm of Sigman (J Appl Prob 48A:37–43, 2011). A coupled first come first serve queue is constructed for each work-conserving queue. When the service duration distributions do not vary, we achieve perfect simulation by finding times when the system is known to be totally idle. When the distributions differ, the totally idle times may be impossible to determine exactly, but we can achieve simulations with a specified error limit

Efficient p-value estimation in massively parallel testing problems

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