Least Squares Estimation of Covariance Matrices in Balanced Multivariate Variance Components Models

References

• Amemiya , Y. 1985 . What Should be Done When an Estimated Between-Group Covariance Matrix is Not Nonnegative Definite? . The American Statistician , 39 : 112 – 117 .
   Web of Science ®Google Scholar
• Anderson , B. M. , Anderson , T. W. and Olkin , I. 1986 . Maximum Likelihood Estimators and Likelihood Ratio Criteria in Multivariate Components of Variance . The Annals of Statistics , 14 : 405 – 417 .
   Web of Science ®Google Scholar
• Bhargava , A. K. and Disch , D. 1982 . Exact Probabilities of Obtaining Estimated Non-Positive Definite Between-Group Covariance Matrices . Journal of Statistical Computation and Simulation , 15 : 27 – 32 .
   Web of Science ®Google Scholar
• Bollen , K. 1989 . Structural Equation Modelling , New York : John Wiley .
   Google Scholar
• Boyle , J. P. and Dykstra , R. L. 1986 . “ A Method for Finding Projections Onto the Intersection of Convex Sets in Hilbert Spaces ” . In Advances in Order Restricted Statistical Inference , Edited by: Dykstra , R. L. , Robertson , T. and Wright , F. T. 28 – 47 . New York : Springer-Verlag .
   Google Scholar
• Calvin , J. A. and Dykstra , R. L. 1991 . Maximum Likelihood Estimation of a Set of Covariance Matrices Under Löwner Order Restrictions With Applications to Balanced Multivariate Variance Components Models . The Annals of Statistics , 19 : 850 – 869 .
   Web of Science ®Google Scholar
• Calvin , J. A. and Sedransk , J. 1991 . Bayesian and Frequentist Predictive Inference For The Patterns of Care Studies . Journal of the American Statistical Association , 86 : 36 – 48 .
   Web of Science ®Google Scholar
• Chaubey , Y. P. 1983 . A Non-Negative Estimator of Variance Component Closest to MINQUE . Sankhyā , 45 : 201 – 211 . Ser. A
   Google Scholar
• Dykstra , R. L. 1983 . An Algorithm for Restricted Least Squares Estimation . Journal of the American Statistical Association , 78 : 837 – 842 .
   Web of Science ®Google Scholar
• Harville , D. 1977 . Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems . Journal of the American Statistical Association , 72 : 320 – 340 .
   Web of Science ®Google Scholar
• Hayduk , L. 1987 . Structural Equation Modeling With LISREL: Essentials and Advances , Baltimore : Johns Hopkins University Press .
   Google Scholar
• Henderson , C. R. 1953 . Estimation of Variance and Covariance Components . Biometrics , 9 : 226 – 252 .
   Web of Science ®Google Scholar
• Herbach , L. H. 1959 . Properties of Model II-Type Analysis of Variance Tests, A: Optimum Nature of the F-Test for Model II in the Balanced Case . The Annals of Mathematical Statistics , 30 : 939 – 959 .
   Google Scholar
• Hill , W. G. and Thompson , R. 1978 . Probabilities of Non-Positive Definite Between-Group or Genetic Covariance Matrices . Biometrics , 34 : 429 – 439 .
   Web of Science ®Google Scholar
• Jöreskog , K. G. 1981 . Analysis of Covariance Structures . Scandinavian Journal of Statistics , 8 : 65 – 83 .
   Web of Science ®Google Scholar
• Jöreskog , K. G. and Sörbom . 1986 . LISREL VI: Analysis of Linear Structural Relationships by Maximum Likelihood, Instrumental Variables, and Least Squares , Mooresville , IN : Scientific Software .
   Google Scholar
• Klotz , J. H. and Putter , J. 1969 . Maximum Likelihood Estimation of Multivariate Covariance Components for the Balanced One-Way Layout . The Annals of Mathematical Statistics , 40 : 1100 – 1105 .
   Google Scholar
• Lee , S. K. 1980 . Estimation of Covariance Structure Models With Parameters Subject to Functional Restraints . Psychometrika , 45 : 309 – 324 .
   Web of Science ®Google Scholar
• Löwner , K. 1934 . Uber Monotone Matrix Funktionen . Mathematische Zeitschrift , 38 : 177 – 216 .
   Google Scholar
• Muirhead , R. J. 1982 . Aspects of Multivariate Statistical Theory , New York : John Wiley .
   Google Scholar
• Rao , C. R. 1971a . Estimation of Variance Covariance Components–MINQUE Theory . Journal of Multivariate Analysis , 1 : 257 – 275 .
   Google Scholar
• Rao , C. R. 1971b . Minimum Variance Quadratic Unbiased Estimation of Variance Components . Journal of Multivariate Analysis , 1 : 445 – 456 .
   Google Scholar
• Rao , C. R. and Kleffe , J. 1988 . Estimation of Variance Components and Applications , Amsterdam : North-Holland .
   Google Scholar
• Rindskopf , D. 1983 . Parameterizing Inequality Constraints on Unique Variances in Linear Structural Models . Psychometrika , 48 : 73 – 83 .
   Web of Science ®Google Scholar
• Rindskopf , D. 1984 . Using Phantom and Imaginary Latent Variables to Parameterize Constraints in Linear Structural Models . Psychometrika , 49 : 37 – 47 .
   Web of Science ®Google Scholar
• Robertson , T. , Wright , F. T. and Dykstra , R. L. 1988 . Order Restricted Statistical Inference , London : John Wiley .
   Google Scholar
• Rockafellar , R. T. 1970 . Convex Analysis , Princeton , NJ : Princeton University Press .
   Google Scholar
• Schwertman , N. C. and Allen , D. M. 1979 . Smoothing an Indefinite Variance–Covariance Matrix . Journal of Statistical Computation and Simulation , 9 : 183 – 194 .
   Google Scholar
• Searle , S. R. 1971 . Linear Models , New York : John Wiley .
   Google Scholar
• Swamy , P. A. V.B. 1971 . Statistical Inference in Random Coefficient Regression Models , New York : Springer-Verlag .
   Google Scholar
• Thompson , W. A. 1962 . The Problem of Negative Estimates of Variance Components . The Annals of Mathematical Statistics , 33 : 273 – 289 .
   Google Scholar