John F. Monahan

Monte Carlo Comparison of ANOVA, MIVQUE, REML, and ML Estimators of Variance Components

For the one-way classification random model with unbalanced data, we compare five estimators of σ2 a and σ2 e , the among- and within-treatments variance components: analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and two minimum variance quadratic unbiased (MIVQUE) estimators. MIVQUE(0) is MIVQUE with a priori values = 0 and = 1; MIVQUE(A) is MIVQUE with the ANOVA estimates used as a prioris, We enforce nonnegativity for all estimators, setting any negative estimate to zero in accord with usual practice. The estimators are compared through their biases and MSEs, estimated by Monte Carlo simulation. Our results indicate that the ANOVA estimators perform well, except with seriously unbalanced data when σ2 a /σ2 e > 1; ML is excellent when σ2 a /σ2 e < 0.5, and MIVQUE(A) is adequate; further iteration to the REML estimates is unnecessary. When σ2 a /σ2 e ≥ 1, MIVQUE(0) (the default for SASS PROCEDURE VARCOMP) is poor for estimating σ2 a and very poor for σ2 e ,...

Computer Generation of Random Variables Using the Ratio of Uniform Deviates

The ratio-of-uniforms method for generating random variables having continuous nonuniform distributions is presented. In thin method a point is generated uniformly over a particular region of the plane. The ratio of the coordinate values of thin point yields a deviate with the desired distribution. Algorithms which utilize this techmque are generally short and often as fast as longer algorithms.

Fully Bayesian analysis of ARMA time series models

Abstract Statistical analysis of autoregressive-moving average (ARMA) models is an important non-standard problem. No classical approach is widely accepted; legitimacy for most classical approaches is based solely on asymptotic grounds, while small sample sizes are common. The only obstacle to the Bayesian approach are designing a structure through which prior information can be incorporated and designing a practical computational method. The objective of this work is to overcome these two obstacles. In addition to the standard results, the Bayesian approach gives a different method of determining the order of the ARMA model, that is (p, q).

Spherical-Radial Integration Rules for Bayesian Computation

Abstract The common numerical problem in Bayesian analysis is the numerical integration of the posterior. In high dimensions, this problem becomes too formidable for fixed quadrature methods, and Monte Carlo integration is the usual approach. Through the use of modal standardization and a spherical-radial transformation, we reparameterize in terms of a radius r and point z on the surface of the sphere in d dimensions. We propose two types of methods for spherical-radial integration. A completely randomized method uses randomly placed abscissas for the radial integration and for the sphere surface. A mixed method uses fixed quadrature (i.e., Simpsons rule) on the radius and randomized spherical integration. The mixed methods show superior accuracy in comparisons, require little or no assumptions, and provide diagnostics to detect difficult problems. Moreover, if the posterior is close to the multivariate normal, then the mixed methods can give remarkable accuracy.

A stochastic algorithm for high-dimensional integrals over unbounded regions with Gaussian weight

Abstract Details are given for a Fortran implementation of an algorithm that uses stochastic spherical–radial rules for the numerical computation of multiple integrals over unbounded regions with Gaussian weight. The implemented rules are suitable for high-dimensional problems. A high-dimensional example from a computational finance application is used to illustrate the use of the rules.

New methods for generating student's t and gamma variables

Algorithms based on ratio-of-uniforms method are developed for generating random variables from the Studentst and gamma families.ZusammenfassungAlgorithmen, die auf der „Ratio of Uniforms”-Methode basieren, werden entwickelt, um Zufallsveränderliche der Studentst und Gamma-Familien zu erzeugen.

Computer methods for sampling from student's t distribution

Several new algorithms for generating deviates from the t family for the degrees of freedom parameter > or =1 are presented. Both acceptance-rejection and probability mixing procedures are developed. The new algorithms outperform traditional methods for generating deviates from the t family. Recommendations are made concerning choosing an algorithm suited to its application.

Predation of Gypsy Moth (Lepidoptera: Lymantriidae) Pupae in Three Ecosystems Along the Southern Edge of Infestation

Abstract The predation potential of small mammals, in particular mice, Peromyscus spp., and invertebrates, was evaluated from 1992 to 1995 near the leading edge of gypsy moth, Lymantria dispar (L.), spread into the southeastern United States. Two study sites were established in each of three geographic areas: the coastal plain, Piedmont, and mountains. All sites were mixed hardwood stands with varying amounts of oak, Quercus spp., and all were classified for gypsy moth susceptibility. Small mammal density was estimated using Sherman live-traps and pitfall traps within these 4.68-ha sites in early and late summer. Each site contained 75 trapping stations located on a 25-m grid. Predation was measured by offering freeze-dried gypsy moth pupae near trapping stations at four heights (0, 0.25, 1.0, and 2.0 m) on different tree boles. Pupal predation was monitored for three consecutive nights. Vertebrate predation was positively correlated with good mast production in the previous autumn. Predation data showed that when mice were at high densities they were the major source of pupal predation. However, within these southern sites, when densities of Peromyscus spp. were low, predation by invertebrates was occasionally greater than predation by vertebrates. These data suggest that in some years invertebrates may retard gypsy moth buildup when small mammals are scarce due to mast crop failures.

An algorithm for generating chi random variables

An algorithm is presented for generating random variables from the chi family of distributions withdegrees of freedom parameter LY 2 1. It is based on the ratio of uniforms method and can be usedeffectively for the gamma family.

Accuracy in random number generation

The generation of continuous random variables on a digital computer encounters a problem of accuracy caused by approximations and discretization error. These in turn impose a bias on the simulation results. An ideal discrete approximation of a continuous distribution and a measure of error are proposed. Heuristic analysis of common methods for transforming uniform deviates to other continuous random variables is discussed. Comments and recom- mendations are made for the design of algorithms to reduce the bias and avoid overflow problems.