A reformulation of the criteria for the independence of quadratic functions in normal variables

The Craig-Sakamoto theorem establishes a sufficient and necessary condition for the independence of two quadratic forms in normal variates, fascinating many statisticians and mathematicians, who continuously seek for simple and better proofs of the theorem and its extensions. In this article, we present a simple proof of a unified theorem on the independence of linear and quadratic functions in general normal variates.     

Control Variates for the Metropolis–Hastings Algorithm

Abstract.  We propose new control variates for variance reduction in estimation of mean values using the Metropolis–Hastings algorithm. Traditionally, states that are rejected in the Metropolis–Hastings algorithm are simply ignored, which intuitively seems to be a waste of information. We present a setting for construction of zero mean control variates for general target and proposal distributions and develop ideas for the standard Metropolis–Hastings and reversible jump algorithms. We give results for three simulation examples. We get best results for variates that are functions of the current state x and the proposal y , but we also consider variates that in addition are functions of the Metropolis–Hastings acceptance/rejection decision. The variance reduction achieved varies depending on the target distribution and proposal mechanisms used. In simulation experiments, we typically achieve relative variance reductions between 15% and 35%.     

Generating random variates from a bicompositional Dirichlet distribution

A composition is a vector of positive components summing to a constant. The sample space of a composition is the simplex, and the sample space of two compositions, a bicomposition, is a Cartesian product of two simplices. We present a way of generating random variates from a bicompositional Dirichlet distribution defined on the Cartesian product of two simplices using the rejection method. We derive a general solution for finding a dominating density function and a rejection constant and also compare this solution to using a uniform dominating density function. Finally, some examples of generated bicompositional random variates, with varying number of components, are presented.     

A Method for Generating High-Dimensional Multivariate Binary Variates

Examples are given of the need for simulating correlated binary variates with different given marginal expectations and pairwise correlations. An algorithm is then presented for generating such variates. The algorithm may be used to generate variates of any dimension.     

Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n ^-1/2 B ^-1, where n is the sample size and B is the bootstrap sample size.We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.     

Improved point and confidence interval estimators of mean response in simulation when control variates are used

In discrete event simulation, the method of control variates is often used to reduce the variance of estimation for the mean of the output response. In the present paper, it is shown that when three or more control variates are used, the usual linear regression estimator of the mean response is one of a large class of unbiased estimators, many of which have smaller variance than the usual estimator. In simulation studies using control variates, a confidence interval for the mean response is typically reported as well. Intervals with shorter width have been proposed using control variates in the literature. The present paper however develops confidence intervals which not only have shorter width but also have higher coverage probability than the usual confidence interval     

The mean of the conditional,truncated multinormal distribution

An expression is derived for the mean of the conditional, truncated multinormal distribution in the general case, and for the situation where the conditioning variates have iden¬tical correlation. This equicorrelated case occurs in some models of duration and in panel data. An example is taken from economics, where a cross section of firms consider the adoption of several technological innovations. The results of the paper are used to estimate the firm-specific.     

On the maximal invariance of manova step down procedure statistics

Subbaiah and Mudhol kar (1978) remark the general mu1tivariate linear hypothesis testing step down procedure statistics do not appear to be maximal invariants under nonsingular lower triangular matrix transformations of the original variates. This paper proves the maximal invariance of these statistics. The invariance results are essential to study the power functions of the step down procedures for MANOVA problems. An example is given to show that such power function studies are very involved.     

Estimation of nonlinear simulation metamodels using control variates

The method of control variates has been intensively used for reducing the variance of estimated (linear) regression metamodels in simulation experiments. In contrast to previous studies, this article presents a procedure for applying multiple control variates when the objective is to estimate and validate a nonlinear regression metamodel for a single response, in terms of selected decision variables. This procedure includes robust statistical regression techniques for estimation and validation. Assuming joint normality of the response and controls, confidence intervals and hypothesis tests for the metamodel parameters are obtained. Finally, results for measuring the efficiency of the use of control variates are discussed.     

Exact power comparison of three criteria to assess the independence between two sets of variates

Powers of the three criteria are evaluated for testing the hypothesis of the independence between a -set and a q-set of variates in a (p + q) -variate normal population. They are: (1) the likelihood ratio type criterion, Wt (2) the largest root criterion, r1, and (3) criterion of the sum of roots, V. For p= 2, Pillai and Jayachandran, and others have studied for the restricted range of the alternative hypothesis. Recently the power of the largest root was investigated in detail by Sugiyama and %%. In this paper, their power functions are compared in a wide range of the alternative hypotheses. The powers of rl and V are locally optimum, but the W shows a large power in a wide range.     

Multilevel multivariate modelling of childhood growth, numbers of growth measurements and adult characteristics

Summary.  A general latent normal model for multilevel data with mixtures of response types is extended in the case of ordered responses to deal with variates having a large number of categories and including count data. An example is analysed by using repeated measures data on child growth and adult measures of body mass index and glucose. Applications are described that are concerned with the flexible prediction of adult measurements from collections of growth measurements and for studying the relationship between the number of measurement occasions and growth trajectories.     

A History of the Development of Craig's Theorem

Craig's theorem on the independence of quadratic forms in normal variates is traced from its first form, for iid standard normal variates, to the form for variates following an arbitrary nonsingular joint normal distribution. This article gives the main thrust of the development and makes recommendations on coverage of the theorem in courses and textbooks. The history of Craig's theorem is not a happy one. The authors of the earlier articles in the literature tended to make errors of a linear-algebraic nature. Authors of more recently published textbooks have given incorrect or misleadingly incomplete coverage of Craig's theorem and its proof.     

Simulating Controlled Variate and Rank Correlations Based on the Power Method Transformation

The power method transformation is a popular algorithm used for simulating correlated non normal continuous variates because of its simplicity and ease of execution. Statistical models may consist of continuous and (or) ranked variates. In view of this, the methodology is derived for simulating controlled correlation structures between non normal (a) variates, (b) ranks, and (c) variates with ranks in the context of the power method. The correlation structure between variate-values and their associated rank-order is also derived for the power method. As such, a measure of the potential loss of information is provided when ranks are used in place of variate-values. The results of a Monte Carlo simulation are provided to confirm and demonstrate the methodology.     

On the distribution of products of independent beta variates

This paper concerns an inquiry into the problem of generating closed form expressions for the cumulative distribution and probability density functions of products of independent beta variates. Recursive analytical procedures for constructing the equational forms of these functions-from their Mellin inversion integral representations, via the Cauchy residue theorem-are described. A numerical example illustrating details of the construction of a computable form of the distribution function of the product of three independent beta variates is also included.     

On the hyper-dirichlet type 1 and hyper-liouville distributions

This paper concerns the characterization of a new family of multivariate beta distribution functions - the hyper-Dirichlet type 1 distribution. This family describes the joint density function of the terminal variates of an arbitrary tree constructed from finite sequences of probability vectors having independent Dirichlet type 1 distributions. Expressions for the general properties of the hyper-Dirichlet type 1 distribution are presented. In addition, the hyper-Liouville distribution is described and its properties are discussed as well as a generalization of the Liouville integral identity.     

THE MONTE CARLO EVALUATION OF ORTHANT PROBABILITIES FOR MULTIVARIATE NORMAL DISTRIBUTIONS

The calculation of multivariate normal orthant probabilities is practically impossible when the number of variates is greater than five or six, except in very special cases. A transformation of the integral is obtained which enables quite accurate Monte Carlo estimates to be obtained for a fairly high number of dimensions, particularly if control variates are used.     

Time-dependent coefficients in a multi-event model for survival analysis

In an earlier work, a matrix-valued counting process model was proposed to deal with the occurrence of multiple events for each experimental unit, appealing to a Cox-type model to accommodate plausible concomitant variates. This model is generalized here to a more flexible one that incorporates time-dependent concomitant variates that might have time-dependent coefficients.     

Variable selection and interpretation in canonical correlation analysis

The canonical variates in canonical correlation analysis are often interpreted by looking at the weights or loadings of the variables in each canonical variate and effectively ignoring those variables whose weights or loadings are small. It is shown that such a procedure can be misleading. The related problem of selecting a subset of the original variables which preserves the information in the most important canonical variates is also examined. Because of different possible definitions of ‘the information in canonical variates’, any such subset selection needs very careful consideration.     

A simple approach for generating correlated binary variates ∗

Correlated binary data arise frequently in medical as well as other scientific disciplines; and statistical methods, such as generalized estimating equation (GEE), have been widely used for their analysis. The need for simulating correlated binary variates arises for evaluating small sample properties of the GEE estimators when modeling such data. Also, one might generate such data to simulate and study biological phenomena such as tooth decay or periodontal disease. This article introduces a simple method for generating pairs of correlated binary data. A simple algorithm is also provided for generating an arbitrary dimensional random vector of non-negatively correlated binary variates. The method relies on the idea that correlations among the random variables arise as a result of their sharing some common components that induce such correlations. It then uses some properties of the binary variates to represent each variate in terms of these common components in addition to its own elements. Unlike most previous approaches that require solving nonlinear equations or use some distributional properties of other random variables, this method uses only some properties of the binary variate. As no intermediate random variables are required for generating the binary variates, the proposed method is shown to be faster than the other methods. To verify this claim, we compare the computational efficiency of the proposed method with those of other procedures.     

Variance reduction through negative correlation,a simulation study

A new method for reducing the variance in simalation experiments is presented and evaluated. The method can be regarded as an extension of the antithetic variates principle. We create negative correlation within all pairs of random number while the original mcihod creates correlation only within some pairs. Our method is compared to ihe antithetic variates and crude Monte-Carlo techniques by repeated simulations. The results obtained are ail in favour of our method.