Some statistical properties of Hadamard products of random matrices

The mean of the Hadamard product of two linear combinations of a random matrix is presented in terms of the mean and variance of the random matrix for any distribution. The variance is given for the normal distribution. Further, the means of four Hadamard products of matrix bilinear forms in a normally distributed random matrix are given. Finally, the mean of a quadruple Hadamard product of linear combinations is derived under normality. Received: January 5, 2000; revised version: April 3, 2000     

A bayesian analysis of the mixed linear model

The mixed model is defined. The exact posterior distribution for the fixed effect vector is obtained. The exact posterior distribution for the error variance is obtained. The exact posterior mean and variance of a Bayesian estimator for the variances of random effects is also derived. All computations are non-iterative and avoid numerical integrations.     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.     

A Theorem on Products of Random Variables, With Application to Regression

It is shown that the product of two independent random variables in the domain of attraction of the normal distribution is also in the domain of attraction of the normal distribution, while if the product is in the domain of attraction of the normal distribution and one of the variables has finite variance, the other is in the domain of attraction of the normal distribution. This result is applied to prove the asymptotic normality of the regression coefficient in a linear regression when the error variance is not necessarily finite.     

Alternative expectation formulas for real-valued random vectors

Abstract

When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The random product can be that of strictly increasing functions of random variables. For continuous cases, the derivation based on iterated integrals is employed. It is shown that Hoeffding’s covariance lemma is algebraically equal to a special case of this result. For discrete cases, the elements of a random vector can be non-integers and/or unequally spaced. A discrete version of Hoeffding’s covariance lemma is derived for real-valued random variables.     

A Note on a Partition of the Likelihood Ratio Test for Autoregressive Covariance Structure

A sequence of nested hypotheses is presented for the examination of the assumption of autoregressive covariance structure in, for example, a repeated measures experiment. These hypotheses arise naturally by specifying the joint density of the underlying vector random variable as a product of conditional densities and the density of a subset of the vector random variable. The tests for all but one of the nested hypotheses are well known procedures, namely analysis of variance F-tests and Bartlett's test of equality of variances. While the procedure is based on tests of hypotheses, it may be viewed as an exploratory tool which can lead to model identification. An example is presented to illustrate the method.     

Unbiased tests for some one-sided testing problems

Unbiased tests are found for various testing problems. In the first model considered we test homogeneity of k + 1 independent one-parameter exponential family populations vs. the tree-top ordering alternative. The tree-top alternative is appropriate for one-sided comparisons for treatments with a control. In the next set of models normality is assumed. In one such model k independent populations have different unknown means but have an unknown common variance. An independent estimate of the variance exists. We test homogeneity of means against the alternative of no homogeneity. We also consider the alternative of an ordering of the means as well as the tree-top ordering. The final model considered is when we take a random sample from a multivariate normal population with unknown mean vector and an unknown covariance matrix of the intraclass type. We test the hypothesis that the mean vector is the zero vector against the one-sided alternative that each mean is nonnegative (with at least one positive).     

The conditional variance for gamma mixtures of normal distributions

This paper is concerned with obtaining an expression for the conditional variance-covariance matrix when the random vector is gamma scaled of a multivariate normal distribution. We show that the conditional variance is not degenerate as in the multivariate normal distribution, but depends upon a positive function for which various asymptotic properties are derived. A discussion section is included commenting on the usefulness of these results     

On the sample characterization criterion for normal distributions

Conventionally, it was shown that the underlying distribution is normal if and only if the sample mean and sample variance from a random sample are independent. This paper focusses on the normal population characterization theorem by showing that, if the joint distribution of a skew normal sample follows certain multivariate skew normal distribution, the sample mean and sample variance are still independent.     

Resampling estimation when observations are m–dependent

For m–dependent, identically distributed random observation, the bootstrap method provides inconsistent estimators of the distribution and variance of the sample mean. This paper proposes an alternative resampling procedure. For estimating the distribution and variance of a function of the sample mean, the proposed resampling estimators are shown to be strongly consistent.     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

ESTIMATION, PREDICTION AND INFERENCE FOR THE LASSO RANDOM EFFECTS MODEL

The least absolute shrinkage and selection operator (LASSO) can be formulated as a random effects model with an associated variance parameter that can be estimated with other components of variance. In this paper, estimation of the variance parameters is performed by means of an approximation to the marginal likelihood of the observed outcomes. The approximation is based on an alternative but equivalent formulation of the LASSO random effects model. Predictions can be made using point summaries of the predictive distribution of the random effects given the data with the parameters set to their estimated values. The standard LASSO method uses the mode of this distribution as the predictor. It is not the only choice, and a number of other possibilities are defined and empirically assessed in this article. The predictive mode is competitive with the predictive mean (best predictor), but no single predictor performs best across in all situations. Inference for the LASSO random effects is performed using predictive probability statements, which are more appropriate under the random effects formulation than tests of hypothesis.     

Dimension reduction for the conditional mean and variance functions in time series

This paper deals with the nonparametric estimation of the mean and variance functions of univariate time series data. We propose a nonparametric dimension reduction technique for both mean and variance functions of time series. This method does not require any model specification and instead we seek directions in both the mean and variance functions such that the conditional distribution of the current observation given the vector of past observations is the same as that of the current observation given a few linear combinations of the past observations without loss of inferential information. The directions of the mean and variance functions are estimated by maximizing the Kullback–Leibler distance function. The consistency of the proposed estimators is established. A computational procedure is introduced to detect lags of the conditional mean and variance functions in practice. Numerical examples and simulation studies are performed to illustrate and evaluate the performance of the proposed estimators.     

Sum of a Random Number of Correlated Random Variables that Depend on the Number of Summands

The mean and variance of a sum of a random number of random variables are well known when the number of summands is independent of each summand and when the summands are independent and identically distributed (iid), or when all summands are identical. In scientific and financial applications, the preceding conditions are often too restrictive. Here, we calculate the mean and variance of a sum of a random number of random summands when the mean and variance of each summand depend on the number of summands and when every pair of summands has the same correlation. This article shows that the variance increases with the correlation between summands and equals the variance in the iid or identical cases when the correlation is zero or one.     

Nonisomorphic complete sets of orthogonal f-squares and hadamard matrices

Five nonisomorphic classes of Hadamard matrices of order 16 were given by Hall (1961). Three of these Hadamard classes have a 4 ×4 row and column structure; they generate three nonisomorphic complete sets of nine orthogonal F(4;2,2)-squares, one of which shows a previously unreported pattern. The remaining two Hadamard classes do not produce complete sets of F-squares, Each of the five Hadamard classes corresponds to a distinct set of single-degree-of-freedom contrasts in an analysis of variance.     

Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean

Linear vector autoregressive (VAR) models where the innovations could be unconditionally heteroscedastic are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose ordinary least squares (OLS), generalized least squares (GLS) and adaptive least squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residual vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a nonstationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations.     

Peak Points in a Symmetrically Correlated Random Walk

This note reports explicit formulae for the probability distribution and mean and variance of the number of peak points in a symmetrically correlated random walk.     

The exponentiated exponential distribution: a survey

The exponentiated exponential distribution, a most attractive generalization of the exponential distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Rényi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic distribution of the extreme order statistics, reliability, distribution of the sum of exponentiated exponential random variables, distribution of the product of exponentiated exponential random variables and the distribution of the ratio of exponentiated exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated exponential distribution and encourage further research.     

Estimation of a normal variance—A critical review

Consider the problem of estimating the variance based on a random sample from a normal distribution with unknown mean. In this article we review the rich literature developed over the last three decades on the problem of variance estimation in a decision theoretic setup. While examining the developments we point out a few errors that exist in the literature.     

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.