On the distributions of the estimated process capability indices

The exact distributions of the estimated process capability indices are presented and their means, variances, and mean-squared errors are given. The basic assumption is that the process measurements are taken from a normal distribution. Theresults in this article are useful in evaluating process capability.     

AN EXACT CONDITIONAL TEST FOR COVARIANCE SELECTION MODELS

An exact conditional test is developed for testing the absence of an edge in a graphical covariance selection model and is shown to be equivalent to a test based on the partial correlation coefficient. An example is given.     

Improved minimax estimation of a normal precision matrix

Let S_{p × p} have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ^?1 under the loss functions L₁(σ) tr (σ) - log |σ| and L₂(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.     

On the distribution of quadratic forms in normal random variables

This paper provides necessary and sufficient conditions for a quadratic form in singular normal random variables to be distributed as a given linear combination of independent noncentral chi-square variables. Using this result, an extension of Cochran's theorem to quadratic forms of noncentral chi-square variables is derived.     

Distribution of multivariate quadratic forms under certain covariance structures

Necessary and sufficient conditions are given for the covariance structure of all the observations in a multivariate factorial experiment under which certain multivariate quadratic forms are independent and distributed as a constant times a Wishart. It is also shown that exact multivariate test statistics can be formed for certain covariance structures of the observations when the assumption of equal covariance matrices for each normal population is relaxed. A characterization is given for the dependency structure between random vectors in which the sample mean and sample covariance matrix have certain properties.     

Gammaization and wishartness of dependent quadratic forms

In this paper two equivalent sets of necessary and sufficient conditions are derived for dependent quadratic forms to be distributed as multivariate gamma distribution. The procedure also gives a set of necessary and sufficient conditions for principal minors of generalized quadratic forms to be jointly distributed as the joint distribution of principal minors of a Kishart matrix.     

Maximum likelihood estimation of parameter structures for the Wishart distribution using constraints

Maximum likelihood estimation under constraints for estimation in the Wishart class of distributions, is considered. It provides a unified approach to estimation in a variety of problems concerning covariance matrices. Virtually all covariance structures can be translated to constraints on the covariances. This includes covariance matrices with given structure such as linearly patterned covariance matrices, covariance matrices with zeros, independent covariance matrices and structurally dependent covariance matrices. The methodology followed in this paper provides a useful and simple approach to directly obtain the exact maximum likelihood estimates. These maximum likelihood estimates are obtained via an estimation procedure for the exponential class using constraints.     

Continuous Time Wishart Process for Stochastic Risk

Risks are usually represented and measured by volatility–covolatility matrices. Wishart processes are models for a dynamic analysis of multivariate risk and describe the evolution of stochastic volatility–covolatility matrices, constrained to be symmetric positive definite. The autoregressive Wishart process (WAR) is the multivariate extension of the Cox, Ingersoll, Ross (CIR) process introduced for scalar stochastic volatility. As a CIR process it allows for closed-form solutions for a number of financial problems, such as term structure of T-bonds and corporate bonds, derivative pricing in a multivariate stochastic volatility model, and the structural model for credit risk. Moreover, the Wishart dynamics are very flexible and are serious competitors for less structural multivariate ARCH models.