Elliptically contoured model and factorization of wllks1 A

In a series of papers, Kshirsagar (1964, 1971) and McHenry and Kshirsagar (1977), factorize Wilks' A into a number of factors and find the independent null multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of independent or dependent variables. Essentially the factors of Wilks' A are the factors of certain multivariate beta distributed matrix or its determinant. The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distribution of these factors. The present paper generalizes Kshirsagar's (1971) normal theory to the elliptically contoured model, and shows that his results are null robust for the elliptically contoured model.     

ON THE NONCENTRAL DISTRIBUTION OF A RANDOM MATRIX USEFUL IN CLASSIFICATION THEORY

To prove the optimality properties of the maximum likelihood (and also minimum distance) discriminant rule Rogers (1980, p. 98) embeds the maximum likelihood discriminant function in a Cauchy-Schwartz inequality. This embedding procedure of Rogers (1980) may be used to derive a new distribution for Anderson's (1958) classification statistic.     

The exact noncentral distribution of the generalized multiple correlation matrix

Given p×n X N(βY, ∑?I), β, ∑ unknown, the noncentral multivariate beta density of the matrix L = [(YY′)^-1/2Y X′ (XX′)^-1XY′ (YY′)^-1/2] is desired. Khatri (1964) finds this density when β is of rank unity. The present paper derives the noncentral density of L and the density of the roots matrix of L for full rank β. The dual case density of L is also obtained. The derivations are based on generalized Sverdrup's lemma, Kabe (1965), and the relationship between primal and dual density of L is explicitly established.     

Which countries protect intellectual property? The case of software piracy

Using data on software piracy, we examine how protection of intellectual property varies across countries. Consistent with other studies, we find that intellectual property receives greater protection in developed economies; high-income countries have lower piracy rates. We also find that protection depends on cultural factors. Countries with an individualist culture have lower piracy rates than do countries with a collectivist culture. Piracy rates are also lower in countries that have strong institutions that enforce contracts and protect property from expropriation. These results suggest that national policies toward intellectual property reflect not only economic concerns but also national culture and institutions.     

On a superiority problem in misspecified restricted linear models

Razzaghi (1987) conjectures that a wrong choice of covariance matrix in a restricted linear model results in loss of efficiency, This conjecture is proved to be correct.     

A quadratic programming approach to the construction of simultaneous confidence intervals

Richmond (1982) uses a linear programming approach to the construction of simultaneous confidence intervals for a set of linear estimable parametric functions of the normal mean vector. We present a quadratic programming approach which constructs narrower confidence intervals than the linear programming approach given by Richmond (1982).     

Goodness-of-fit tests of hypothesized canonical variables

In the literature goodness-of-fit tests for canonical variables are available either in the x space or in the y space, see e.g., Bartlett (1951), Kshirsagar (1971, 1972), Radcliffe (1968), and Williams (1952). Here we present goodness-of-fit tests for canonical variables in both the x and y spaces. The results appear as extensions of the results of the above authors.     

On a criterion for extrapolation in multivariate normal regression

In a full (or less than full) rank normal multivariate regression model, the existence of an unbiased estimate of the distribution associated to any future observation vector is studied. A necessary and sufficient condition in terms of the point at which the future observation vector will be taken introduced to consider the validity of extrapolation. The results given here generalize some results of Silvey [3] and the related results of O’Reilly [1].