Rolf E. Bargmann

A Method for the Evaluation of Cumulative Probabilities of Bivariate Distributions using the Pearson Family

A technique is developed for the approximation of bivariate cumulative distribution function values through the use of the bivariate Pearson family when moments of the distribution to be approximated are known. The method utilizes a factorization of the joint density function into the product of a marginal density function and an associated conditional density, permitting the expression of the double integral in a form amenable to the use of specialized Gaussian-type quadrature techniques for numerical evaluation of cumulative probabilities. Such an approach requires moments of truncated Pearson distributions, for which a recurrence relation is presented, and moments of the conditional distributions for which quartic expressions are used. It is shown that results are of high precision when this technique is employed to evaluate cumulative distribution functions that are of the Pearson class.

Orthogonal Central Composite Designs of the Third Order in the Evaluation of Sensitivity and Plant Growth Simulation Models

Abstract The U.S. Department of Agriculture has used response-surface techniques as applied by Baker and Bargmann (1981) to plant process simulation models as an aid in the identification of interrelationships among yield and various single-valued and functional parameters. Orthogonal cubic surfaces have provided insight into higher order relationships as well as a measure of the relative sensitivity of yield to experimentally determined parameter values. Several examples investigate the effectiveness of those higher order surfaces and illustrate how less precise (and less costly) measurements may be possible in building and using these simulation models.

Efficient algorithms for statistical estimation in compartmental analysis: Modelling 60Co kinetics in an aquatic microcosm

Abstract A mathematical model of radionuclide kinetics in a laboratory microcosm was built by systems identification techniques. Insight into the functioning of the system was obtained from analysis of the model. Methods employed have allowed movements of radioisotopes not directly observable in the experimental system to be distinguished. Results are generalized to whole ecosystems.

Discussion of “The Internal Correlation: Its Applications in Statistics and Psychometrics” by G. W. Joe and J. L. Mendoza

The use of the internal correlation for data analysis (principal components, ridge regression, and other methods to correct multicollinearity) by Joe and Mendoza (1989) is interesting and could be implemented, even on a personal computer, with relatively little effort. One advantage of this original application is that exact distributions of the sample statistic do not seem to be required. The application that Schuenemeyer (1975) studied in his dissertation concerned the splitting of a correlation matrix into two subsets to maximize canonical correlations. At the time, we also studied the logical extension of the Lancaster (1957) scaling techniques to threedimensional and higher order contingency tables, that is, scaling of nominal data involving three and more variables. As in the current paper, the computer programs were prepared easily. The algorithm that produced the three or more sets of canonical weights associated with the internal (maximum eccentricity) statistic (by the Fletcher Powell technique) proceeded rapidly, and distributional questions were not involved. Bargmann (1980) presented details of this procedure and illustrations. On the other hand, application of this technique to factor analysis posed several problems: (a) As the internal correlation statistic applied to the matrix of partial correlations (after extraction of factors) attains its minimum, several largest and smallest roots become nearly equal (and thus convergence of a procedure involving derivatives is painfully slow); and (b) the distribution of the test statistic is needed for deciding when to stop extracting factors. Many attempts were made subsequently (including an abandoned dissertation after more than a years work) to get some grasp of the distribution of characteristic roots of a correlation matrix, without success to date. P. L. Hsu (1939) solved the related problem for covariance matrices, and Krishnaiah and Schuurman (1974) prepared tables. Joe and Mendoza again call for a solution to this problem (one other application requiring percentage points would be the problem of obtaining confidence intervals on the largest possible canonical correlation, under all splits). Because in factor analysis the maximum-likelihood (maximum determinant) solution is readily available, and the asymptotic distribution of the determinant of the partial correlation matrix can be expanded to a high