An adaptive estimatcr frequently used in material fatigue testing

Small sample sizes in material fatigue tests give rise to an adaptive estimator of the 100(1-P)% normal percentile, which is the 100P% survival load. The mean squared relative efficiency of the best invariant estimator of normal percentiles to the adaptive estimator is dependent upon the unknown parameters only through the coefficient of variation. The adaptive estimator is shown to be more locally efficient than tl-:ebest invariant estimator over a subset of the parameter space. However, in the extreme values of P the coverage probabilities of the adaptive estimator provide little more protection than a traditional point estimator over the range of preference based on mean squared relative efficiency.     

The Decline and Fall of Type II Error Rates

For general linear models with normally distributed random errors, the probability of a Type II error decreases exponentially as a function of sample size. This potentially rapid decline reemphasizes the importance of performing power calculations.     

Moments of the trace of a noncentral wishart matrix

Several articles have appeared on the moments of the trace of a noncentral Wishart matrix. Partial results are given in these papers but the representations of these partial results are often too cumbersome for any practical use. In this paper general results are given in compact form which are readily usable in practical problems.     

On the Dirichlet distributions on symmetric matrices

In this paper, we introduce a generalization of the Dirichlet distribution on symmetric matrices which represents the multivariate version of the Connor and Mosimann generalized real Dirichlet distribution. We establish some properties concerning this generalized distribution. We also extend to the matrix Dirichlet distribution a remarkable characterization established in the real case by Darroch and Ratcliff.     

Polynomial estimation of eigenvalues

In estimating the eigenvalues of the covariance matrix of a multivariate normal population, the usual estimates are the eigenvalues of the sample covariance matrix. It is well known that these estimates are biased. This paper investigates obtaining improved eigenvalue estimates through improved estimates of the characteristic polynomial, which is a function of the sample eigenvalues. A numerical study investigates the improvements evaluated under both a square error and an entropy loss function.     

Decision-theoretic estimation of generalized variance and generalized precision

Consider a random data matrix X=(X₁,...,X_k):pXk with independent columns [sathik] and an independent p X p Wishart matrix [sathik]. Estimators dominating the best affine equivariant estimators of [sathik] are obtained under four types of loss functions. Improved estimators (Testimators) of generalized variance and generalized precision are also considered under convex entropy loss (CEL).     

The Density Functions of the Singular Quaternion Normal Matrix and the Singular Quaternion Wishart Matrix

In this article, we give the density functions of the singular quaternion normal matrix and the singular quaternion Wishart matrix. Furthermore, we also give the density functions of certain singular quaternion β-matrix and the singular quaternion F-matrix in terms of the density function of the singular quaternion Wishart matrix and hypergeometric functions of quaternion matrix argument.     

Zonal Polynomials and Hypergeometric Functions of Quaternion Matrix Argument

We define zonal polynomials of quaternion matrix argument and deduce some impor-tant formulae of zonal polynomials and hypergeometric functions of quaternion matrix argument. As an application, we give the distributions of the largest and smallest eigenvalues of a quaternion central Wishart matrix W ～ ?W(n, Σ), respectively.     

Cross-entropy method for estimation of posterior expectation in Bayesian VAR models

In this article, an importance sampling (IS) method for the posterior expectation of a non linear function in a Bayesian vector autoregressive (VAR) model is developed. Most Bayesian inference problems involve the evaluation of the expectation of a function of interest, usually a non linear function of the model parameters, under the posterior distribution. Non linear functions in Bayesian VAR setting are difficult to estimate and usually require numerical methods for their evaluation. A weighted IS estimator is used for the evaluation of the posterior expectation. With the cross-entropy (CE) approach, the IS density is chosen from a specified family of densities such that the CE distance or the Kullback–Leibler divergence between the optimal IS density and the importance density is minimal. The performance of the proposed algorithm is assessed in an iterated multistep forecasting of US macroeconomic time series.