Ronald W. Butler

Saddlepoint approximations with applications

Modern statistical methods use models that require the computation of probabilities from complicated distributions, which can lead to intractable computations. Saddlepoint approximations can be the answer. Written from the user’s point of view, this book explains in clear, simple language how such approximate probability computations are made, taking readers from the very beginnings to current applications. The book aims to make the subject accessible to the widest possible audience by using graduated levels of difficulty in which the core material is presented in chapters 1–6 at an elementary mathematical level. Readers are guided in applying the methods in various computations that will build their skills and deepen their understanding when later complemented with discussion of theoretical aspects. Chapters 7–9 address the p∗ and r∗ formulas of higher order asymptotic inference, developed through the Danish approach to the subject by Barndorff-Nielsen and others. These provide a readable summary of the literature and an overview of the subject beginning with the original work of Fisher. Later chapters address special topics where saddlepoint methods have had substantial impact through particular applications. These include applications in multivariate testing, applications to stochastic systems and applied probability, bootstrap implementation in the transform domain, and Bayesian computation and inference. No previous background in the area is required as the book introduces the subject from the very beginning. Many data examples from real applications show the methods at work and demonstrate their practical value. Ideal for graduate students and researchers in statistics, biostatistics, electrical engineering, econometrics, applied mathematics, and other fields where statistical and probabilistic modeling are used, this is both an entry-level text and a valuable reference.

Bootstrap Methods for Finite Populations

Abstract We show that the familiar bootstrap plug-in rule of Efron has a natural analog in finite population settings. In our method a characteristic of the population is estimated by the average value of the characteristic over a class of empirical populations constructed from the sample. Our method extends that of Gross to situations in which the stratum sizes are not integer multiples of their respective sample sizes. Moreover, we show that our method can be used to generate second-order correct confidence intervals for smooth functions of population means, a property that has not been established for other resampling methods suggested in the literature. A second resampling method is proposed that also leads to second-order correct confidence intervals and is less computationally intensive than our bootstrap. But a simulation study reveals that the second method can be quite unstable in some situations, whereas our bootstrap performs very well.

Saddlepoint Approximations to the CDF of Some Statistics with Nonnormal Limit Distributions

Abstract In standard saddlepoint approximations to the cumulative distribution function of a random variable, the normal distribution has appeared to play a special role. In this article we consider what happens when the normal “base” distribution is replaced by an arbitrary base distribution. Generalized versions of several standard formulas, are presented. The choice of a chi-squared base or an inverse Gaussian base is then considered in detail. The generalized approximations are compared in two examples: a linear combination of chi-squared variables and the first passage time distribution for a random walk. The former example considers approximations using the chi-squared base that are slightly more accurate than their normal-based counterparts. In the latter example, approximations based on the inverse Gaussian are considerably more accurate than their normal-based counterparts.

Stochastic Network Models for Survival Analysis

Abstract We present methodology giving highly accurate approximations for Bayesian predictive densities and distribution functions of first passage times between states of a semi-Markov process with a finite number of states. When the states describe a degenerative disorder with an absorbing end state, such predictive distributions are the survival distributions of a patient. We illustrate these methods with a variety of examples, including data from the San Francisco AIDS study. We achieve our approximations using a three-step sequence. First, we introduce advanced concepts of flowgraph theory, which allow us to compute the moment generating function of the first passage time given the model parameters. Next, we use saddlepoint approximations to convert this into a density or distribution function conditional on the model parameter. Finally, we use Monte Carlo methods to remove dependence on the model parameter. These methods apply quite generally to all finite-state semi-Markov models in discrete or con...

Laplace approximation for Bessel functions of matrix argument

We derive Laplace approximations to three functions of matrix argument which arise in statistics and elsewhere: matrix Bessel Av; matrix Bessel Bv; and the type II confluent hypergeometric function of matrix argument, Ψ. We examine the theoretical and numerical properties of the approximations. On the theoretical side, it is shown that the Laplace approximations to Av, Bv and Ψ given here, together with the Laplace approximations to the matrix argument functions 1F1 and 2F1 presented in Butler and Wood (Laplace approximations to hyper-geometric functions with matrix argument, Ann. Statist. (2002)), satisfy all the important confluence relations and symmetry relations enjoyed by the original functions.

Saddlepoint approximation for moment generating functions of truncated random variables

We consider the problem of approximating the moment generating function (MGF) of a truncated random variable in terms of the MGF of the underlying (i.e., untruncated) random variable. The purpose of approximating the MGF is to enable the application of saddlepoint approximations to certain distributions determined by truncated random variables. Two important statistical applications are the following: the approximation of certain multivariate cumulative distribution functions; and the approximation of passage time distributions in ion channel models which incorporate time interval omission. We derive two types of representation for the MGF of a truncated random variable. One of these representations is obtained by exponential tilting. The second type of representation, which has two versions, is referred to as an exponential convolution representation. Each representation motivates a different approximation. It turns out that each of the three approximations is extremely accurate in those cases to which it is suited. Moreover, there is a simple rule of thumb for deciding which approximation to use in a given case, and if this rule is followed, then our numerical and theoretical results indicate that the resulting approximation will be extremely accurate.

Generalized Inverse Gaussian Distributions and their Wishart Connections

The matrix generalized inverse Gaussian distribution (MGIG) is shown to arise as a conditional distribution of components of a Wishart distributio n. In the special scalar case, the characterization refers to members of the class of generalized inverse Gaussian distributions (GIGs) and includes the inverse Gaussian distribution among others

Calculating the density and distribution function for the singly and doubly noncentral F

Simple, closed form saddlepoint approximations for the distribution and density of the singly and doubly noncentral F distributions are presented. Their overwhelming accuracy is demonstrated numerically using a variety of parameter values. The approximations are shown to be uniform in the right tail and the associated limitating relative error is derived. Difficulties associated with some algorithms used for “exact” computation of the singly noncentral F are noted.

Uniform saddlepoint approximations for ratios of quadratic forms

Ratios of quadratic forms in correlated normal variables which introduce noncentrality into the quadratic forms are considered. The denominator is assumed to be positive (with probability 1). Various serial correlation estimates such as least-squares, Yule–Walker and Burg, as well as Durbin–Watson statistics, provide important examples of such ratios. The cumulative distribution function (c.d.f.) and density for such ratios admit saddlepoint approximations. These approximations are shown to preserve uniformity of relative error over the entire range of support. Furthermore, explicit values for the limiting relative errors at the extreme edges of support are derived.

Saddlepoint Approximation for Multivariate Cumulative Distribution Functions and Probability Computations in Sampling Theory and Outlier Testing

Abstract Four multivariate distributions commonly arise in sampling theory: the multinomial, multivariate hypergeometric, Dirichlet, and multivariate Polya distributions. Second-order saddlepoint approximations are given for approximating these multivariate cumulative distribution functions (cdfs) in their most general settings. Probabilities of rectangular regions associated with these cdfs are also approximated directly using second-order saddlepoint methods. All the approximations follow from characterizations of the multivariate distributions as conditional distributions. Applications to outlier discordancy tests and slippage tests are discussed.