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Volumn 72, Issue 358, 1977, Pages 320-338

Maximum likelihood approaches to variance component estimation and to related problems

Author keywords

Maximum likelihood; Maximum likelihood computations; Mixed linear models; Restricted maximum likelihood; Variance component estimation

Indexed keywords


EID: 84890913931     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.1977.10480998     Document Type: Article
Times cited : (1635)

References (80)
  • 2
    • 0041390997 scopus 로고
    • Statistical Inference for Covariance Matrices with Linear Structure
    • P.R. Krishnaiah, New York: Academic Press
    • Anderson, T.W. (1969), “Statistical Inference for Covariance Matrices with Linear Structure,” in Multivariate Analysis-II, ed. P.R. Krishnaiah, New York: Academic Press, 55–66.
    • (1969) Multivariate Analysis-Ii , pp. 55-66
    • Anderson, T.W.1
  • 3
    • 0042894013 scopus 로고
    • Estimation of Covariance Matrices Which Are Linear Combinations or Whose Inverses Are Linear Combinations of Given Matrices
    • R.C. Bose, I.M. Chakravarti, ‘P.C. Mahalanobis, C.R. Rao, and K.J.C. Smith, Chapel Hill, North Carolina: University of North Carolina Press
    • —— (1970), “Estimation of Covariance Matrices Which Are Linear Combinations or Whose Inverses Are Linear Combinations of Given Matrices,” in Essays in Probability and Statistics, eds. R.C. Bose, I.M. Chakravarti, ‘P.C. Mahalanobis, C.R. Rao, and K.J.C. Smith, Chapel Hill, North Carolina: University of North Carolina Press, 1–24.
    • (1970) Essays in Probability and Statistics , pp. 1-24
    • Anderson, T.W.1
  • 5
    • 0000531364 scopus 로고
    • Asymptotically Efficient Estimation of Covariance Matrices with Linear Structure
    • —— (1973), “Asymptotically Efficient Estimation of Covariance Matrices with Linear Structure,” Annals of Statistics, 1, 135–41.
    • (1973) Annals of Statistics , vol.1 , pp. 135-141
    • Anderson, T.W.1
  • 6
    • 0014746832 scopus 로고
    • Comparison of Gradient Methods for the Solution of Nonlinear Parameter Estimation Problems
    • Bard, Yonathan (1970), “Comparison of Gradient Methods for the Solution of Nonlinear Parameter Estimation Problems,” SIAM Journal on Numerical Analysis, 7, 157–86.
    • (1970) SIAM Journal on Numerical Analysis , vol.7 , pp. 157-186
    • Bard, Y.1
  • 10
    • 0000533741 scopus 로고
    • The Created Response Surface Technique for Optimizing Nonlinear, Restrained Systems
    • Carroll, C.W. (1961), “The Created Response Surface Technique for Optimizing Nonlinear, Restrained Systems,” Operations Research, 9, 169–84.
    • (1961) Operations Research , vol.9 , pp. 169-184
    • Carroll, C.W.1
  • 11
    • 0016919827 scopus 로고
    • Restricted Maximum Likelihood (REML) Estimation of Variance Components in the Mixed Model
    • Corbeil, Robert R., and Searle, Shayle R. (1976a), “Restricted Maximum Likelihood (REML) Estimation of Variance Components in the Mixed Model,” Technometrics, 18, 31–38.
    • (1976) Technometrics , vol.18 , pp. 31-38
    • Corbeil, R.R.1    Searle, S.R.2
  • 12
    • 0017045481 scopus 로고
    • A Comparison of Variance Component Estimators
    • ——, and Searle, Shayle R. (1976b), “A Comparison of Variance Component Estimators,” Biometrics, 32, 779–91.
    • (1976) Biometrics , vol.32 , pp. 779-791
    • Corbeil, R.R.1    Searle, S.R.2
  • 13
    • 0014265812 scopus 로고
    • An Iterative Procedure for Estimating Fixed Effects and Variance Components in Mixed Model Situations
    • Cunningham, E.P., and Henderson, Charles R. (1968), “An Iterative Procedure for Estimating Fixed Effects and Variance Components in Mixed Model Situations,” Biometrics, 24, 13–25.
    • (1968) Biometrics , vol.24 , pp. 13-25
    • Cunningham, E.P.1    Henderson, C.R.2
  • 14
    • 0000984609 scopus 로고
    • Linear Dynamic Recursive Estimation from the Viewpoint of Regression Analysis
    • Duncan, David B., and Horn, Susan D. (1972), “Linear Dynamic Recursive Estimation from the Viewpoint of Regression Analysis,” Journal of the American Statistical Association, 67, 815–21.
    • (1972) Journal of the American Statistical Association , vol.67 , pp. 815-821
    • Duncan, D.B.1    Horn, S.D.2
  • 15
    • 84949161149 scopus 로고
    • Steins Estimation Rule and Its Competitors—An Empirical Bayes Approach
    • Efron, Bradley, and Morris, Carl (1973), “Stein’s Estimation Rule and Its Competitors—An Empirical Bayes Approach,” Journal of the American Statistical Association, 68, 117–30.
    • (1973) Journal of the American Statistical Association , vol.68 , pp. 117-130
    • Efron, B.1    Morris, C.2
  • 16
    • 0002256959 scopus 로고
    • Multivariate Empirical Bayes and Estimation of Covariance Matrices
    • ——, and Morris, Carl (1976), “Multivariate Empirical Bayes and Estimation of Covariance Matrices,” Annals of Statistics, 4, 22–32.
    • (1976) Annals of Statistics , vol.4 , pp. 22-32
    • Efron, B.1    Morris, C.2
  • 17
    • 84950646173 scopus 로고
    • Some Remarks on Herbachs Paper, ‘Optimum Nature of the F-Test for Model II in the Balanced Case’
    • Gautschi, Werner (1959), “Some Remarks on Herbach’s Paper, ‘Optimum Nature of the F-Test for Model II in the Balanced Case,”’ Annals of Mathematical Statistics, 30, 960–3.
    • (1959) Annals of Mathematical Statistics , vol.30 , pp. 960-963
    • Gautschi, W.1
  • 19
    • 0001237594 scopus 로고
    • Maximization by Quadratic Hill Climbing
    • Goldfeld, S.M., Quandt, R.E., and Trotter, H.F. (1966), “Maximization by Quadratic Hill Climbing,” Econometrica, 34, 541–51.
    • (1966) Econometrica , vol.34 , pp. 541-551
    • Goldfeld, S.M.1    Quandt, R.E.2    Trotter, H.F.3
  • 23
    • 0014099014 scopus 로고
    • Maximum-Likelihood Estimation for the Mixed Analysis of Variance Model
    • Hartley, H.O., and Rao, J.N.K. (1967), “Maximum-Likelihood Estimation for the Mixed Analysis of Variance Model,” Biometrika, 54, 93–108.
    • (1967) Biometrika , vol.54 , pp. 93-108
    • Hartley, H.O.1    Rao, J.N.K.2
  • 24
    • 4243108018 scopus 로고
    • A Computer Program for the Mixed Analysis of Variance Model Based on Maximum Likelihood
    • T.A. Bancroft, Ames, Iowa: Iowa State University Press
    • ——, and Vaughn, William K. (1972), “A Computer Program for the Mixed Analysis of Variance Model Based on Maximum Likelihood,” in Statistical Papers in Honor of George W. Snedecor, ed. T.A. Bancroft, Ames, Iowa: Iowa State University Press.
    • (1972) Statistical Papers in Honor of George W. Snedecor
    • Hartley, H.O.1    Vaughn, W.K.2
  • 25
    • 0009034742 scopus 로고
    • Quadratic Unbiased Estimation of Variance Components for the One-Way Classification
    • (Correction, (1970), Biometrika, 57, 226.)
    • Harville, David A. (1969a), “Quadratic Unbiased Estimation of Variance Components for the One-Way Classification,” Biometrika, 56, 313–26. (Correction, (1970), Biometrika, 57, 226.)
    • (1969) Biometrika , vol.56 , pp. 313-326
    • Harville, D.A.1
  • 27
    • 85041975654 scopus 로고
    • Bayesian Inference for Variance Components Using Only Error Contrasts
    • —— (1974), “Bayesian Inference for Variance Components Using Only Error Contrasts,” Biometrika, 61, 383–5.
    • (1974) Biometrika , vol.61 , pp. 383-385
    • Harville, D.A.1
  • 29
    • 0000317021 scopus 로고
    • Extension of the Gauss-Markov Theorem to Include the Estimation of Random Effects
    • —— (1976), “Extension of the Gauss-Markov Theorem to Include the Estimation of Random Effects,” Annals of Statistics, 4, 384–95.
    • (1976) Annals of Statistics , vol.4 , pp. 384-395
    • Harville, D.A.1
  • 30
    • 0001422082 scopus 로고
    • Computing Maximum Likelihood Estimates for the Mixed A.O.V. Model Using the W Transformation
    • Hemmerle, William J., and Hartley, H.O. (1973), “Computing Maximum Likelihood Estimates for the Mixed A.O.V. Model Using the W Transformation,” Technometrics, 15, 819–31.
    • (1973) Technometrics , vol.15 , pp. 819-831
    • Hemmerle, W.J.1    Hartley, H.O.2
  • 31
    • 0001421956 scopus 로고
    • Estimation of Variance and Covariance Components
    • Henderson, Charles R. (1953), “Estimation of Variance and Covariance Components,” Biometrics, 9, 226–52.
    • (1953) Biometrics , vol.9 , pp. 226-252
    • Henderson, C.R.1
  • 32
    • 0000517795 scopus 로고
    • Selection Index and Expected Genetic Advance
    • National Academy of Sciences–National Research Council Publication No. 982
    • —— (1963), “Selection Index and Expected Genetic Advance,” in Statistical Genetics and Plant Breeding, National Academy of Sciences–National Research Council Publication No. 982, 141–63.
    • (1963) Statistical Genetics and Plant Breeding , pp. 141-163
    • Henderson, C.R.1
  • 36
    • 0016704147 scopus 로고
    • Best Linear Unbiased Estimation and Prediction Under a Selection Model
    • —— (1975), “Best Linear Unbiased Estimation and Prediction Under a Selection Model,” Biometrics, 31, 423–47.
    • (1975) Biometrics , vol.31 , pp. 423-447
    • Henderson, C.R.1
  • 37
    • 0000867123 scopus 로고
    • Properties of Model II-Type Analysis of Variance Tests, A: Optimum Nature of the F-Test for Model II in the Balanced Case
    • Herbach, Leon 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,” Annals of Mathematical Statistics, 30, 939–59.
    • (1959) Annals of Mathematical Statistics , vol.30 , pp. 939-959
    • Herbach, L.H.1
  • 38
    • 0016749476 scopus 로고
    • Some Analytical and Numerical Comparisons of Estimators for the Mixed A.O.V. Model
    • Hocking, Ronald R., and Kutner, Michael H. (1975), “Some Analytical and Numerical Comparisons of Estimators for the Mixed A.O.V. Model,” Biometrics, 31, 19–28.
    • (1975) Biometrics , vol.31 , pp. 19-28
    • Hocking, R.R.1    Kutner, M.H.2
  • 39
    • 0016919826 scopus 로고
    • Newton-Raphson and Related Algorithms for Maximum Likelihood Variance Component Estimation
    • Jennrich, R.I., and Sampson, P.F. (1976), “Newton-Raphson and Related Algorithms for Maximum Likelihood Variance Component Estimation,” Technometrics, 18, 11–17.
    • (1976) Technometrics , vol.18 , pp. 11-17
    • Jennrich, R.I.1    Sampson, P.F.2
  • 41
    • 0347868838 scopus 로고
    • Technical Report No. 10, Department of Statistics, University of Kentucky, Lexington, Kentucky
    • LaMotte, Lynn R. (1970), “A Class of Estimators of Variance Components,” Technical Report No. 10, Department of Statistics, University of Kentucky, Lexington, Kentucky.
    • (1970) A Class of Estimators of Variance Components
    • LaMotte, L.R.1
  • 42
    • 0141473000 scopus 로고
    • Technical Report No. 22, Department of Statistics, University of Kentucky, Lexington, Kentucky
    • —— (1971), “Locally Best Quadratic Estimators of Variance Components,” Technical Report No. 22, Department of Statistics, University of Kentucky, Lexington, Kentucky.
    • (1971) Locally Best Quadratic Estimators of Variance Components
    • LaMotte, L.R.1
  • 43
    • 0015828315 scopus 로고
    • Quadratic Estimation of Variance Components
    • —— (1973), “Quadratic Estimation of Variance Components,” Biometrics, 29, 311–30.
    • (1973) Biometrics , vol.29 , pp. 311-330
    • LaMotte, L.R.1
  • 44
    • 0000118161 scopus 로고
    • Elimination of Linear Parameters in Nonlinear Regression
    • Lawton, William H., and Sylvestre, Edward A. (1971), “Elimination of Linear Parameters in Nonlinear Regression,” Technometrics, 13, 461–7.
    • (1971) Technometrics , vol.13 , pp. 461-467
    • Lawton, W.H.1    Sylvestre, E.A.2
  • 45
    • 0000873069 scopus 로고
    • A Method for the Solution of Certain NonLinear Problems in Least Squares
    • Levenberg, K. (1944), “A Method for the Solution of Certain NonLinear Problems in Least Squares,” Quarterly of Applied Mathematics, 2, 164–8.
    • (1944) Quarterly of Applied Mathematics , vol.2 , pp. 164-168
    • Levenberg, K.1
  • 47
    • 0000169232 scopus 로고
    • An Algorithm for Least Squares Estimation of Nonlinear Parameters
    • Marquardt, Donald W. (1963), “An Algorithm for Least Squares Estimation of Nonlinear Parameters,” SIAM Journal, 11, 431–41.
    • (1963) SIAM Journal , vol.11 , pp. 431-441
    • Marquardt, D.W.1
  • 50
    • 0000423624 scopus 로고
    • The Combination of Information in Generally Balanced Designs
    • Ser. B
    • Nelder, J.A. (1968), “The Combination of Information in Generally Balanced Designs,” Journal of the Royal Statistical Society, Ser. B, 30, 303–11.
    • (1968) Journal of the Royal Statistical Society , vol.30 , pp. 303-311
    • Nelder, J.A.1
  • 52
    • 0001516410 scopus 로고
    • Invariant Quadratic Unbiased Estimation for Two Variance Components
    • Olsen, Anthony, Seely, Justus, and Birkes, David (1976), “Invariant Quadratic Unbiased Estimation for Two Variance Components,” Annals of Statistics, 4, 878–90.
    • (1976) Annals of Statistics , vol.4 , pp. 878-890
    • Olsen, A.1    Seely, J.2    Birkes, D.3
  • 53
    • 77956889208 scopus 로고
    • Recovery of InterBlock Information when Block Sizes Are Unequal
    • Patterson, H.D., and Thompson, Robin (1971), “Recovery of InterBlock Information when Block Sizes Are Unequal,” Biometrika, 58, 545–54.
    • (1971) Biometrika , vol.58 , pp. 545-554
    • Patterson, H.D.1    Thompson, R.2
  • 55
    • 0014712205 scopus 로고
    • A Survey of Numerical Methods for Unconstrained Optimization
    • Powell, M.J.D. (1970), “A Survey of Numerical Methods for Unconstrained Optimization,” SIAM Review, 12, 79–97.
    • (1970) SIAM Review , vol.12 , pp. 79-97
    • Powell, M.J.D.1
  • 57
    • 0001099716 scopus 로고
    • Estimation of Heteroscedastic Variances in Linear Models
    • —— (1970), “Estimation of Heteroscedastic Variances in Linear Models,” Journal of the American Statistical Association, 65, 161–72.
    • (1970) Journal of the American Statistical Association , vol.65 , pp. 161-172
    • Rao, C.R.1
  • 58
    • 36949007083 scopus 로고
    • Estimation of Variance and Covariance Components—MINQUE Theory
    • —— (1971a), “Estimation of Variance and Covariance Components—MINQUE Theory,” Journal of Multivariate Analysis, 1, 257–75.
    • (1971) Journal of Multivariate Analysis , vol.1 , pp. 257-275
    • Rao, C.R.1
  • 59
    • 0001290965 scopus 로고
    • Minimum Variance Quadratic Unbiased Estimation of Variance Components
    • —— (1971b), “Minimum Variance Quadratic Unbiased Estimation of Variance Components,” Journal of Multivariate Analysis, 1, 445–56.
    • (1971) Journal of Multivariate Analysis , vol.1 , pp. 445-456
    • Rao, C.R.1
  • 60
    • 0000101514 scopus 로고
    • Estimation of Variance and Covariance Components in Linear Models
    • —— (1972), “Estimation of Variance and Covariance Components in Linear Models,” Journal of the American Statistical Association, 67, 112–15.
    • (1972) Journal of the American Statistical Association , vol.67 , pp. 112-115
    • Rao, C.R.1
  • 61
    • 84867131744 scopus 로고
    • On Expectations, Variances, and Covariances of ANOVA Mean Squares by ‘Synthesis’
    • Rao, J.N.K. (1968), “On Expectations, Variances, and Covariances of ANOVA Mean Squares by ‘Synthesis’,” Biometrics, 24, 963–78.
    • (1968) Biometrics , vol.24 , pp. 963-978
    • Rao, J.N.K.1
  • 62
    • 0006691216 scopus 로고
    • One-Way Variances in a Two-Way Classification
    • Russell, Thomas S., and Bradley, Ralph A. (1958), “One-Way Variances in a Two-Way Classification,” Biometrika, 45, 111–29.
    • (1958) Biometrika , vol.45 , pp. 111-129
    • Russell, T.S.1    Bradley, R.A.2
  • 63
    • 0012421639 scopus 로고
    • Another Look at Hendersons Methods of Estimating Variance Components
    • (with discussion)
    • Searle, Shayle R., (1968), “Another Look at Henderson’s Methods of Estimating Variance Components” (with discussion), Biometrics, 24, 749–87.
    • (1968) Biometrics , vol.24 , pp. 749-787
    • Searle, S.R.1
  • 64
    • 0010818254 scopus 로고
    • Large Sample Variances of Maximum Likelihood Estimators of Variance Components Using Unbalanced Data
    • —— (1970), “Large Sample Variances of Maximum Likelihood Estimators of Variance Components Using Unbalanced Data,” Biometrics, 26, 505–24.
    • (1970) Biometrics , vol.26 , pp. 505-524
    • Searle, S.R.1
  • 65
    • 0004242197 scopus 로고
    • New York: John Wiley & Sons
    • —— (1971a), Linear Models, New York: John Wiley & Sons.
    • (1971) Linear Models
    • Searle, S.R.1
  • 66
    • 0002603402 scopus 로고
    • Topics in Variance Component Estimation
    • —— (1971b), “Topics in Variance Component Estimation,” Biometrics, 27, 1–76.
    • (1971) Biometrics , vol.27 , pp. 1-76
    • Searle, S.R.1
  • 67
    • 0013268257 scopus 로고
    • Prediction, Mixed Models, and Variance Components
    • F. Proschan and R.J. Serfling, Philadelphia, Pennsylvania: Society of Industrial and Applied Mathematics
    • —— (1974), “Prediction, Mixed Models, and Variance Components,” in Reliability and Biometry, eds. F. Proschan and R.J. Serfling, Philadelphia, Pennsylvania: Society of Industrial and Applied Mathematics, 229–66.
    • (1974) Reliability and Biometry , pp. 229-266
    • Searle, S.R.1
  • 69
    • 0016763992 scopus 로고
    • An Example of an Inquadmissible Analysis of Variance Estimator for a Variance Component
    • Seely, Justus, (1975), “An Example of an Inquadmissible Analysis of Variance Estimator for a Variance Component, Biometrika, 62, 689–90.
    • (1975) Biometrika , vol.62 , pp. 689-690
    • Seely, J.1
  • 70
    • 85042251801 scopus 로고
    • Marginal and Conditional Sufficiency
    • Sprott, D.A. (1975), “Marginal and Conditional Sufficiency,” Biometrika, 62, 599–605.
    • (1975) Biometrika , vol.62 , pp. 599-605
    • Sprott, D.A.1
  • 71
    • 0001224549 scopus 로고
    • Iterative Estimation of Variance Components for Nonorthogonal Data
    • Thompson, Robin (1969), “Iterative Estimation of Variance Components for Nonorthogonal Data,” Biometrics, 25, 767–73.
    • (1969) Biometrics , vol.25 , pp. 767-773
    • Thompson, R.1
  • 72
    • 0015895279 scopus 로고
    • The Estimation of Variance and Covariance Components with an Application when Records Are Subject to Culling
    • —— (1973), “The Estimation of Variance and Covariance Components with an Application when Records Are Subject to Culling,” Biometrics, 29, 527–50.
    • (1973) Biometrics , vol.29 , pp. 527-550
    • Thompson, R.1
  • 73
    • 84892221743 scopus 로고
    • A Note on the W Transformation
    • —— (1975), “A Note on the W Transformation,” Technometrics, 17, 511–2.
    • (1975) Technometrics , vol.17 , pp. 511-512
    • Thompson, R.1
  • 74
    • 0000139505 scopus 로고
    • The Problem of Negative Estimates of Variance Components
    • Thompson, W.A., Jr. (1962), “The Problem of Negative Estimates of Variance Components,” Annals of Mathematical Statistics, 33, 273–89.
    • (1962) Annals of Mathematical Statistics , vol.33 , pp. 273-289
    • Thompson, W.A.1
  • 76
    • 0004409676 scopus 로고
    • Best Quadratic Unbiased Estimation of Variance Components from Unbalanced Data in the 1-Way Classification
    • ——, and Searle, Shayle, R. (1971), “Best Quadratic Unbiased Estimation of Variance Components from Unbalanced Data in the 1-Way Classification,” Biometrics, 27, 643–57.
    • (1971) Biometrics , vol.27 , pp. 643-657
    • Townsend, E.C.1    Searle, S.R.2
  • 77
    • 0001255679 scopus 로고
    • Asymptotic Properties of Maximum Likelihood Estimators in Some Nonstandard Cases
    • Weiss, Lionel (1971), “Asymptotic Properties of Maximum Likelihood Estimators in Some Nonstandard Cases,” Journal of the American Statistical Association, 66, 345–50.
    • (1971) Journal of the American Statistical Association , vol.66 , pp. 345-350
    • Weiss, L.1
  • 78
    • 0000395351 scopus 로고
    • Asymptotic Properties of Maximum Likelihood Estimators in Some Nonstandard Cases, II
    • —— (1973), “Asymptotic Properties of Maximum Likelihood Estimators in Some Nonstandard Cases, II,” Journal of the American Statistical Association, 68, 428–30.
    • (1973) Journal of the American Statistical Association , vol.68 , pp. 428-430
    • Weiss, L.1


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