Bartlett Decomposition and Wishart Distribution

Kshirsagar, A. M.

doi:10.1214/aoms/1177706379

Cited by 51 publications

(20 citation statements)

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“…These coefficients were plotted to yield a p(~3) distribution as shown in Figure 1 and averaged to give €p(~). The process of R-generation works by means of the Bartlett decomposition of the Wishart distribution (Bartlett, 1933;Kshirsogar, 1959;Wijsman, 1957), and is outlined in Appendix 1. This approach to sampling from N(_~c, ~) has been discussed and employed by Browne (1968) and Herzberg (1969), both of whom have carried out tests of the simulated data showing that the required assumptions are met.…”

Section: Methodsmentioning

confidence: 99%

The Relative Efficiency of Regression and Simple Unit Predictor Weights in Applied Differential Psychology

Schmidt

1971

Educational and Psychological Measurement

206

124

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A very common problem in the behavioral and social sciences is the prediction of the standing of a person or thing on one variable, usually designated the criterion, from his or its standing on a number of other variables, usually called the predictors. Leastsquared error multiple regression weights are most commonly used in weighting the predictors into a composite. These weights, which minimize, over the cases in the sample, the sum of the squared deviations of the observed from the predicted criterion score, are calculated from the normal equations which express the minimization conditions (Anderson, 1958). The fitting of the regression weights to the idiosyncracies of the initial sample leads to a decrease in effectiveness when these weights are applied to a new sample in which these particular idiosyncracies are not present. This &dquo;shrinkage&dquo; is often substantial in practical situations (e.g., see: Kurtz, 1948;Cureton, 1950;and Kirkpatrick, 1951), especially when the initial sample is small. And small samples, as Lawshe and Schucker (1959) point out, are the rule rather than the exception in many areas of applied psychology.Certain other approaches to the weighting problem produce 1 This study is based on a dissertation submitted in partial fulfillment of the requirements for the PhD degree at Purdue University. The author is indebted to

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Section: Methodsmentioning

confidence: 99%

The Relative Efficiency of Regression and Simple Unit Predictor Weights in Applied Differential Psychology

Schmidt

1971

Educational and Psychological Measurement

206

124

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“…A population covariance matrix that contains nine parameters with four constraints was constructed based on the 2-factor solution (Sorbom & J6reskog, 1976 For each of the sample sizes 30, 60, 90, 120 and 150, sixty sample covariance matrices were generated by the method of Kshirsagar (1959). A population covariance matrix that contains nine parameters with four constraints was constructed based on the 2-factor solution (Sorbom & J6reskog, 1976 For each of the sample sizes 30, 60, 90, 120 and 150, sixty sample covariance matrices were generated by the method of Kshirsagar (1959).…”

Section: Numerical Resultsmentioning

confidence: 99%

On Testing Functional Constraints in Structural Equation Models

Lee¹

1985

Biometrika

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARYIn the context of a robust generalized least squares approach, a new statistic for testing the validity of constraints in structural equation models is proposed. It is shown that the new test requires significantly less computational effort than the traditional one. Extension to models in several populations is also considered. Illustrative applications based on real data are given.Some key words: Chi-squared distribution; Convergence in probability; Covariance structure; Generalized least squares.

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“…Replacing 0 by r in -E(L)/P(L), then 4 4 Now following Kshirsagar's (1969) and that RO, for fixed S22, is distributed as…”

Section: Cn2(ý E)mentioning

confidence: 99%

Some Inference Problems Associated with the Complex Multivariate Normal Distribution

Young¹

1971

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(Security clausificeaton of title, body of abxttart and lndexing annc ratJon rnu.t be entered when the ovetall report is clahssfied) I. OnICINATING "S4. CONTRACT OR GRANT NO. Se. ORIGINATOR'S REPORT NUMBERIS)N00014-68-A-0515b. PROJECT NO. NR 042-260 C.ob. OTHER REPORT NO($) (Any other numbers Mtat may be alssiged this report) d. DISTRIDUTION ITATECENTThis document has been approved for public release and sale; its distribution is unlimited. Reproduction in whole or in part is permitted for any purpose of the United States Government. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITYOffice of Naval Research ASXIACTConsider k = q+l complex multivariate normal populations with the same variancecovariance matrix but with different means.The problem of the number of discriminant functions needed to discriminate among the k-populations is considered and shown to be equivalent to the rank of the mean-space. A test for this dimensionality being r is presented. Also developed is the statistic for testing the goodness of fit of a single hypothetical discriminant function. As in the case of the real normal populations, the test statistic for this single hypothetical function is presented as the product of two independent tactors, whose distributions are given; one measuring the direction aspect of the hypothetical function, and the other measures the collinearity aspect that is the necessity of only one function.Also included is the Bartlett decompositicn of a complex Wishart matrix and some results pertaining to the coherence of complex random variables.

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Bartlett Decomposition and Wishart Distribution

Cited by 51 publications

References 1 publication

The Relative Efficiency of Regression and Simple Unit Predictor Weights in Applied Differential Psychology

The Relative Efficiency of Regression and Simple Unit Predictor Weights in Applied Differential Psychology

On Testing Functional Constraints in Structural Equation Models

Some Inference Problems Associated with the Complex Multivariate Normal Distribution

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