Matrix Differential Calculus with Applications in Statistics and Econometrics

Matrix Differential Calculus with Applications in Statistics and Econometrics

Volumn , Issue , 2019, Pages 1-479

  Magnus, Jan R ^a   Neudecker, Heinz ^b  

^a VU UNIVERSITY AMSTERDAM   (Netherlands)

^b UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85144859648     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9781119541219     Document Type: Book

References (241)

1. Abadir, K. M., R. D. H. Heijmans and J. R. Magnus (2019). Statistics, Volume 2 of Econometric Exercises (eds K. M. Abadir, J. R. Magnus and P. C. B. Phillips), Cambridge University Press, New York.
2. Abadir, K. M. and J. R. Magnus (2005). Matrix Algebra, Volume 1 of Econometric Exercises (eds K. M. Abadir, J. R. Magnus and P. C. B. Phillips), Cambridge University Press, New York.
3. Abdullah, J., H. Neudecker and S. Liu (1992). Problem 92.4.6, Econometric Theory, 8, 584. Solution in Econometric Theory, 9, 703.
4. Abrahamse, A. P. J. and J. Koerts (1971). New estimators of disturbances in regression analysis, Journal of the American Statistical Association, 66, 71-74.
5. Adachi, K. (2016). Matrix-based Introduction to Multivariate Data Analysis, Springer Nature, Singapore.
6. Aitken, A. C. (1935). On least squares and linear combinations of observations, Proceedings of the Royal Society of Edinburgh, A, 55, 42-48.
7. Aitken, A. C. (1939). Determinants and Matrices, Oliver and Boyd, Edinburgh and London.
8. Albert, A. (1973). The Gauss-Markov theorem for regression models with possibly singular covariances, SIAM Journal of Applied Mathematics, 24, 182-187.
9. Amano, T. and M. Tanigushi (2011). Control variate method for stationary processes, Journal of Econometrics, 165, 20-29.
10. Amir-Móez, A. R. and A. L. Fass (1962). Elements of Linear Spaces, Pergamon Press, Oxford.
11. Anderson, T.W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd edition, John Wiley, New York.
12. Ando, T. (1979). Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra and Its Applications, 26, 203-241.
13. Ando, T. (1983). On the arithmetic-geometric-harmonic-mean inequalities for positive definite matrices, Linear Algebra and Its Applications, 52, 31-37.
14. Andrew, A. L., K.-W. E. Chu and P. Lancaster (1993). Derivatives of eigenvalues and eigenvectors of matrix functions, SIAM Journal on Matrix Analysis and Applications, 14, 903-926.
15. Apostol, T. M. (1974). Mathematical Analysis, 2nd edition, Addison-Wesley, Reading.
16. Aubry, A., A. De Maio, L. Pallotta and A. Farina (2012). Maximum likelihood estimation of a structured covariance matrix with a condition number constraint, IEEE Transactions on Signal Processing, 60, 3004-3021.
17. Balestra, P. (1973). Best quadratic unbiased estimators of the variancecovariance matrix in normal regression, Journal of Econometrics, 1, 17-28.
18. Balestra, P. (1976). La Derivation Matricielle, Collection de l'Institut de Mathematiques Economiques, No. 12, Sirey, Paris.
19. Bargmann, R. E. and D. G. Nel (1974). On the matrix differentiation of the characteristic roots of matrices, South African Statistical Journal , 8, 135-144.
20. Baron, M. E. (1969). The Origins of the Infinitesimal Calculus, Pergamon Press, Oxford.
21. Barten, A. P. (1969). Maximum likelihood estimation of a complete system of demand equations, European Economic Review, 1, 7-73.
22. Beckenbach, E. F. and R. Bellman (1961). Inequalities, Springer-Verlag, Berlin.
23. Bellman, R. (1970). Introduction to Matrix Analysis, 2nd edition, McGraw- Hill, New York.
24. Ben-Israel, A. and T. N. Greville (1974). Generalized Inverses: Theory and Applications, John Wiley, New York.
25. Bentler, P. M. and S.-Y. Lee (1978). Matrix derivatives with chain rule and rules for simple, Hadamard, and Kronecker products, Journal of Mathematical Psychology, 17, 255-262.
26. Binmore, K. G. (1982). Mathematical Analysis: A Straightforward Approach, 2nd edition, Cambridge University Press, Cambridge.
27. Black, J. and Y.Morimoto (1968). A note on quadratic forms positive definite under linear constraints, Economica, 35, 205-206.
28. Bloomfield, P. and G. S. Watson (1975). The inefficiency of least squares, Biometrika, 62, 121-128.
29. Bodewig, E. (1959). Matrix Calculus, 2nd edition, North-Holland, Amsterdam.
30. Boik, R. J. (2008). Newton algorithms for analytic rotation: an implicit function approach, Psychometrika, 73, 231-259.
31. Boullion, T. L. and P. L. Odell (1971). Generalized Inverse Matrices, John Wiley, New York.
32. Bozdogan, H. (1990). On the information-based measure of covariance complexity and its application to the evaluation of multivariate linear models, Communications in Statistics-Theory and Methods, 19, 221-278.
33. Bozdogan, H. (1994). Mixture-model cluster analysis using model selection criteria and a new informational measure of complexity, in: Multivariate Statistical Modeling (ed. H. Bozdogan), Volume II, Proceedings of the First US/Japan Conference on the Frontiers of Statistical Modeling: An Informational Approach, Kluwer, Dordrecht, pp. 69-113.
34. Brinkhuis, J. (2015). On the use of coordinate-free matrix calculus, Journal of Multivariate Analysis, 133, 377-381.
35. Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures, South African Statistical Journal , 8, 1-24. Reprinted in: Latent Variables in Socio-economic Models (eds D. J. Aigner and A. S. Goldberger), North-Holland, Amsterdam, 1977, pp. 205-226.
36. Cavicchioli, M. (2017). Asymptotic Fisher information matrix of Markov switching VARMA models, Journal of Multivariate Analysis, 157, 124-135.
37. Chipman, J. S. (1964). On least squares with insufficient observations, Journal of the American Statistical Association, 59, 1078-1111.
38. Chipman, J. S. (1996). "Proofs" and proofs of the Eckart-Young theorem, in: Stochastic Processes and Functional Analysis (eds J. Goldstein, N. Gretsky and J. Uhl), Marcel Dekker, New York, pp. 80-81.
39. Cho, J. S. and H. White (2014). Testing the equality of two positive-definite matrices with application to information matrix testing, in: Essays in Honor of Peter C. B. Phillips (Advances in Econometrics, Volume 33) (eds Y. Chang, T. B. Fomby and J. Y. Park), Emerald Group Publishing Limited, pp. 491-556.
40. Clarke, M. R. B. (1970). A rapidly convergent method for maximum likelihood factor analysis, British Journal of Mathematical and Statistical Psychology, 23, 43-52.
41. Courant, R. and D. Hilbert (1931). Methoden der Mathematischen Physik, reprinted by Interscience, New York.
42. Cramer, J. S. (1986). Econometric Applications of Maximum Likelihood Methods, Cambridge University Press, Cambridge.
43. Debreu, G. (1952). Definite and semidefinite quadratic forms, Econometrica, 20, 295-300.
44. De Luca, G., J.R. Magnus and F. Peracchi (2018). Weighted-average least squares estimation of generalized linear models, Journal of Econometrics, 204, 1-17.
45. Dieudonńe, J. (1969). Foundations of Modern Analysis, 2nd edition, Academic Press, New York.
46. Don, F. J. H. (1986). Linear Methods in Non-Linear Models, unpublished Ph.D. Thesis, University of Amsterdam.
47. Dubbelman, C., A. P. J. Abrahamse and J. Koerts (1972). A new class of disturbance estimators in the general linear model, Statistica Neerlandica, 26, 127-142.
48. Duda, R. O., P. E. Hart and D. G. Stork (2001). Pattern Classification, 2nd edition, Wiley-Interscience, New York.
49. Dunajeva, O. (2004). The second-order derivatives of matrices of eigenvalues and eigenvectors with an application to generalized F-statistic, Linear Algebra and Its Applications, 388, 159-171.
50. Dwyer, P. S. (1967). Some applications of matrix derivatives in multivariate analysis, Journal of the American Statistical Association, 62, 607-625.
51. Dwyer, P. S. and M. S. Macphail (1948). Symbolic matrix derivatives, Annals of Mathematical Statistics, 19, 517-534.
52. Eckart, C. and G. Young (1936). The approximation of one matrix by another of lower rank, Psychometrika, 1, 211-218.
53. Faliva, M. (1983). Identificazione e Stima nel Modello Lineare ad Equazioni Simultanee, Vita e Pensiero, Milan.
54. Fan, K. (1949). On a theorem of Weyl concerning eigenvalues of linear transformations, I, Proceedings of the National Academy of Sciences of the USA, 35, 652-655.
55. Fan, K. (1950). On a theorem of Weyl concerning eigenvalues of linear transformations, II, Proceedings of the National Academy of Sciences of the USA, 36, 31-35.
56. Fan, K. and A. J. Hoffman (1955). Some metric inequalities in the space of matrices, Proceedings of the American Mathematical Society, 6, 111-116.
57. Farebrother, R.W. (1977). Necessary and sufficient conditions for a quadratic form to be positive whenever a set of homogeneous linear constraints is satisfied, Linear Algebra and Its Applications, 16, 39-42.
58. Fischer, E. (1905). Ueber quadratische Formen mit reellen Koeffizienten, Monatschrift für Mathematik und Physik, 16, 234-249.
59. Fisher, F. M. (1966). The Identification Problem in Econometrics, McGraw- Hill, New York.
60. Fleming,W. H. (1977). Functions of Several Variables, 2nd edition, Springer- Verlag, New York.
61. Foster, D. W. (2001). Author Unknown: Tales of a Literary Detective, Henry Holt & Company, New York.
62. Gantmacher, F. R. (1959). The Theory of Matrices, Volumes I & II, Chelsea, New York.
63. Gao, F., T. Cui and A. Nallanathan (2008). Optimal training design for channel estimation in decode-and-forward relay networks with individual and total power constraints, IEEE Transactions on Signal Processing, 56, 5937-5949.
64. Gauss, K. F. (1809). Werke, 4, 1-93, Göttingen.
65. Godfrey, L. G. and M. R. Wickens (1982). A simple derivation of the limited information maximum likelihood estimator, Economics Letters, 10, 277-283.
66. Golub, G. H. and V. Pereyra (1973). Differentiation of pseudoinverses, separable nonlinear least square problems and other tales, in: Generalized Inverses and Applications (ed. M. Z. Nashed), Academic Press, New York.
67. Graham, A. (1981). Kronecker Products andMatrix Calculus, Ellis Horwood, Chichester.
68. Greenacre, M. J. (1984). Theory and Applications of Correspondence Analysis, Academic Press, London.
69. Greene, W. H. (1993). Econometric Analysis, 2nd edition, Macmillan, New York.
70. Greub, W. andW. Rheinboldt (1959). On a generalization of an inequality of L. V. Kantorovich, Proceedings of the American Mathematical Society, 10, 407-415.
71. Gustafson, K. (2002). Operator trigonometry of statistics and econometrics, Linear Algebra and Its Applications, 354, 141-158.
72. Hadley, G. (1961). Linear Algebra, Addison-Wesley, Reading.
73. Hardy, G. H., J. E. Littlewood and G. Polya (1952). Inequalities, 2nd edition, Cambridge University Press, Cambridge.
74. Hayashi, K. and Y. Yung (1999). Standard errors for the class of orthomaxrotated factor loadings: some matrix results, Psychometrika, 64, 451-460.
75. Hearon, J. Z. and J. W. Evans (1968). Differentiable generalized inverses, Journal of Research of the National Bureau of Standards, B, 72B, 109-113.
76. Heijmans, R. D. H. and J. R. Magnus (1986). Asymptotic normality of maximum likelihood estimators obtained from normally distributed but dependent observations, Econometric Theory, 2, 374-412.
77. Henderson, H. V. and S. R. Searle (1979). Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics, Canadian Journal of Statistics, 7, 65-81.
78. Hjørungnes, A. and D. Gesbert (2007). Complex-valued matrix differentiation: techniques and key results, IEEE Transactions on Signal processing, 55, 2740-2746.
79. Hogg, R. V. and A. T. Craig (1970). Introduction to Mathematical Statistics, 3rd edition, Collier-Macmillan, London.
80. Holly, A. (1985). Problem 85.1.2, Econometric Theory, 1, 143-144. Solution in Econometric Theory, 2, 297-300.
81. Holly, A. and J. R. Magnus (1988). A note on instrumental variables and maximum likelihood estimation procedures, Annales d'Economie et de Statistique, 10, 121-138.
82. Horn, R. A. and C. R. Johnson (1991). Topics in Matrix Analysis, Cambridge University Press, Cambridge/ New York.
83. Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components, Journal of Educational Psychology, 24, 417-441 and 498-520.
84. Hsiao, C. (1983). Identification, in: Handbook of Econometrics, Volume I (eds Z. Griliches and M. D. Intriligator), North-Holland, Amsterdam, pp. 223-283.
85. Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis, Psychometrika, 23, 187-200.
86. Kaiser, H. F. (1959). Computer program for varimax rotation in factor analysis, Journal of Educational and Psychological Measurement , 19, 413-420.
87. Kalaba, R., K. Spingarn and L. Tesfatsion (1980). A new differential equation method for finding the Perron root of a positive matrix, Applied Mathematics and Computation, 7, 187-193.
88. Kalaba, R., K. Spingarn and L. Tesfatsion (1981a). Individual tracking of an eigenvalue and eigenvector of a parameterized matrix, Nonlinear Analysis, Theory, Methods and Applications, 5, 337-340.
89. Kalaba, R., K. Spingarn and L. Tesfatsion (1981b). Variational equations for the eigenvalues and eigenvectors of nonsymmetric matrices, Journal of Optimization Theory and Applications, 33, 1-8.
90. Kano, Y. and M. Ihara (1994). Identification of inconsistent variates in factor analysis, Psychometrika, 59, 5-20.
91. Kantorovich, L. V. (1948). Functional analysis and applied mathematics, Uspekhi Matematicheskikh Nauk, 3, 89-185. Translated by C. D. Benster, National Bureau of Standards, Report 1509, 7 March 1952.
92. Kapteyn, A., H. Neudecker and T.Wansbeek (1986). An approach to n-mode components analysis, Psychometrika, 51, 269-275.
93. Kato, T. (1976). Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, Berlin/ Heidelberg/ New York.
94. Kleibergen, F. and E. Zivot (2003). Bayesian and classical approaches to instrumental variable regression, Journal of Econometrics, 114, 29-72.
95. Koerts, J. and A. P. J. Abrahamse (1969). On the Theory and Applications of the General Linear Model , Rotterdam University Press, Rotterdam.
96. Kollo, T. (1991). The Matrix Derivative in Multivariate Statistics, Tartu University Press (in Russian).
97. Kollo, T. and H. Neudecker (1993). Asymptotics of eigenvalues and unitlength eigenvectors of sample variance and correlation matrices, Journal of Multivariate Analysis, 47, 283-300. (Corrigendum in Journal of Multivariate Analysis, 51, 210.)
98. Kollo, T. and H. Neudecker (1997a). Asymptotics of Pearson-Hotelling principal component vectors of sample variance and correlation matrices, Behaviormetrika, 24, 51-69.
99. Kollo, T. and H. Neudecker (1997b). The derivative of an orthogonal matrix of eigenvectors of a symmetric matrix, Linear Algebra and Its Applications, 264, 489-493.
100. Koopmans, T. C., H. Rubin and R. B. Leipnik (1950). Measuring the equation systems of dynamic economics, in: Statistical Inference in Dynamic Economic Models (ed. T. C. Koopmans), Cowles Foundation for Research in Economics, Monograph 10, John Wiley, New York, Chapter 2.
101. Kreijger, R. G. and H. Neudecker (1977). Exact linear restrictions on parameters in the general linear model with a singular covariance matrix, Journal of the American Statistical Association, 72, 430-432.
102. Lancaster, P. (1964). On eigenvalues of matrices dependent on a parameter, Numerische Mathematik, 6, 377-387.
103. Lancaster, T. (1984). The covariance matrix of the information matrix test, Econometrica, 52, 1051-1052.
104. Lastovicka, J. L. (1981). The extension of component analysis to four-mode matrices, Psychometrika, 46, 47-57.
105. Lawley, D. N. and A. E. Maxwell (1971). Factor Analysis as a Statistical Method, 2nd edition, Butterworths, London.
106. Leamer, E. E. (1978). Specification Searches, John Wiley, New York.
107. Lemonte, A. J. (2011). Improved maximum-likelihood estimation in a regression model with general parametrization, Journal of Statistical Computation and Simulation, 81, 1027-1037.
108. Liu, S. (1995). Contributions to Matrix Calculus and Applications in Econometrics, Ph.D. Thesis, University of Amsterdam.
109. Liu, S. and H. Neudecker (1995). Matrix-trace Cauchy-Schwarz inequalities and applications in canonical correlation analysis, Statistical Papers, 36, 287-298.
110. Liu, S. and H. Neudecker (1999). A survey of Cauchy-Schwarz and Kantorovich type matrix inequalities, Statistical Papers, 40, 55-73.
111. Liu, S. and W. Polasek (1997). Two matrix inequalities involving the Moore- Penrose inverse, Problem 97.2.3, Econometric Theory, 13, 463-464. Solution by H. Neudecker and M. van de Velden in Econometric Theory, 14, 290-291.
112. Lu, Y. Y. (1998). Computing the logarithm of a symmetric positive definite matrix, Applied Numerical Mathematics, 26, 483-496.
113. Luenberger, D. G. (1969). Optimization by Vector Space Methods, John Wiley, New York.
114. Lyhagen, J. (2012). A note on the representation of E[x ? xx'] and E[xx' ? xx'] for the random vector x, Statistical Papers, 53, 697-701.
115. MacDuffee, C. C. (1933). The Theory of Matrices, reprinted by Chelsea, New York.
116. MacRae, E. C. (1974). Matrix derivatives with an application to an adaptive linear decision problem, The Annals of Statistics, 2, 337-346.
117. Madansky, A. (1976). Foundations of Econometrics, North-Holland, Amsterdam.
118. Magnus, J. R. (1985). On differentiating eigenvalues and eigenvectors, Econometric Theory, 1, 179-191.
119. Magnus, J. R. (1987). A representation theorem for (trAp)1/p, Linear Algebra and Its Applications, 95, 127-134.
120. Magnus, J. R. (1988). Linear Structures, Griffin's Statistical Monographs, No. 42, Edward Arnold, London and Oxford University Press, New York.
121. Magnus, J. R. (1990). On the fundamental bordered matrix of linear estimation, in: Advanced Lectures in Quantitative Economics (ed. F. van der Ploeg), Academic Press, London, pp. 583-604.
122. Magnus, J. R. (2010). On the concept of matrix derivative, Journal of Multivariate Analysis, 101, 2200-2206.
123. Magnus, J. R. (2017). Introduction to the Theory of Econometrics, VU University Press, Amsterdam.
124. Magnus, J. R. and H. Neudecker (1979). The commutation matrix: some properties and applications, The Annals of Statistics, 7, 381-394.
125. Magnus, J. R. and H. Neudecker (1980). The elimination matrix: some lemmas and applications, SIAM Journal on Algebraic and Discrete Methods, 1, 422-449.
126. Magnus, J. R. and H. Neudecker (1985). Matrix differential calculus with applications to simple, Hadamard, and Kronecker products, Journal of Mathematical Psychology, 29, 474-492.
127. Magnus, J. R. and H. Neudecker (1986). Symmetry, 0-1 matrices and Jacobians: a review, Econometric Theory, 2, 157-190.
128. Malinvaud, E. (1966). Statistical Methods of Econometrics, North-Holland, Amsterdam.
129. Marcus, M. and H. Minc (1964). A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston.
130. Mardia, K. V., J. T. Kent and J. M. Bibby (1992). Multivariate Analysis, Academic Press, London.
131. Markov, A. A. (1900). Wahrscheinlichkeitsrechnung, Teubner, Leipzig.
132. Marshall, A. and I. Olkin (1979). Inequalities: Theory of Majorization and Its Applications, Academic Press, New York.
133. Mathai, A. M. (1997). Jacobians of Matrix Transformations and Functions of Matrix Arguments, World Scientific, Singapore.
134. McCulloch, C. E. (1982). Symmetric matrix derivatives with applications, Journal of the American Statistical Association, 77, 679-682.
135. McDonald, R. P. and H. Swaminathan (1973). A simple matrix calculus with applications to multivariate analysis, General Systems, 18, 37-54.
136. Milliken, G. A. and F. Akdeniz (1977). A theorem on the difference of the generalized inverses of two nonnegative matrices, Communications in Statistics-Theory and Methods, A6, 73-79.
137. Mirsky, L. (1961). An Introduction to Linear Algebra, Oxford University Press, Oxford.
138. Mohebi, H. and A. Salemi (2007). Analysis of symmetric matrix valued functions, Numerical Functional Analysis and Optimization, 28, 691-715.
139. Mood, A. M., F. A. Graybill and D. C. Boes (1974). Introduction to the Theory of Statistics, 3rd edition, McGraw-Hill, New York.
140. Moore, E. H. (1920). On the reciprocal of the general algebraic matrix (Abstract), Bulletin of the American Mathematical Society, 26, 394-395.
141. Moore, E. H. (1935). General Analysis, Memoirs of the American Philosophical Society, Volume I, American Philosophical Society, Philadelphia.
142. Moore, M. H. (1973). A convex matrix function, American Mathematical Monthly, 80, 408-409.
143. Morrison, D. (1976). Multivariate Statistical Methods, 2nd edition, McGraw- Hill, New York.
144. Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, John Wiley, New York.
145. Needham, T. (1997). Visual Complex Analysis, Oxford University Press, Oxford, UK.
146. Nel, D. G. (1980). On matrix differentiation in statistics, South African Statistical Journal , 14, 137-193.
147. Neudecker, H. (1967). On matrix procedures for optimizing differentiable scalar functions of matrices, Statistica Neerlandica, 21, 101-107.
148. Neudecker, H. (1969). Some theorems on matrix differentiation with special reference to Kronecker matrix products, Journal of the American Statistical Association, 64, 953-963.
149. Neudecker, H. (1973). De BLUF-schatter: een rechtstreekse afleiding, Statistica Neerlandica, 27, 127-130.
150. Neudecker, H. (1974). A representation theorem for |A|1/n , METU Journal of Pure and Applied Sciences, 7, 1-2.
151. Neudecker, H. (1975). A derivation of the Hessian of the (concentrated) likelihood function of the factor model employing the Schur product, British Journal of Mathematical and Statistical Psychology, 28, 152-156.
152. Neudecker, H. (1977a). Abrahamse and Koerts' 'new estimator' of disturbances in regression analysis, Journal of Econometrics, 5, 129-133.
153. Neudecker, H. (1977b). Bounds for the bias of the least squares estimator of s2 in the case of a first-order autoregressive process (positive autocorrelation), Econometrica, 45, 1257-1262.
154. Neudecker, H. (1978). Bounds for the bias of the least squares estimator of s2 in the case of a first-order (positive) autoregressive process when the regression contains a constant term, Econometrica, 46, 1223-1226.
155. Neudecker, H. (1980a). A comment on 'Minimization of functions of a positive semidefinite matrix A subject to AX = 0', Journal of Multivariate Analysis, 10, 135-139.
156. Neudecker, H. (1980b). Best quadratic unbiased estimation of the variance matrix in normal regression, Statistische Hefte, 21, 239-242.
157. Neudecker, H. (1981). On the matrix formulation of Kaiser's varimax criterion, Psychometrika, 46, 343-345.
158. Neudecker, H. (1982). On two germane matrix derivatives, The Matrix and Tensor Quarterly, 33, 3-12.
159. Neudecker, H. (1985a). Recent advances in statistical applications of commutation matrices, in: Proceedings of the Fourth Pannonian Symposium on Mathematical Statistics (eds W. Grossman, G. Pflug, I. Vincze and W. Wertz), Volume B, Reidel, Dordrecht, pp. 239-250.
160. Neudecker, H. (1985b). On the dispersion matrix of a matrix quadratic form connected with the noncentral normal distribution, Linear Algebra and Its Applications, 70, 257-262.
161. Neudecker, H. (1989a). A matrix derivation of a representation theorem for (trAp)1/p , Qüestiío, 13, 75-79.
162. Neudecker, H. (1989b). A new proof of the Milliken-Akdeniz theorem, Qüestiío, 13, 81-82.
163. Neudecker, H. (1992). A matrix trace inequality, Journal of Mathematical Analysis and Applications, 166, 302-303.
164. Neudecker, H. (1995). Mathematical properties of the variance of the multinomial distribution, Journal of Mathematical Analysis and Applications, 189, 757-762.
165. Neudecker, H. (1997). An inequality concerning the Hadamard matrix product, Problem 97.2.4, Econometric Theory, 13, 464. Solution in Econometric Theory, 14, 291-292.
166. Neudecker, H. and S. Liu (1993). Best quadratic and positive semidefinite unbiased estimation of the variance of the multivariate normal distribution, Communications in Statistics-Theory and Methods, 22, 2723-2732.
167. Neudecker, H. and S. Liu (1995). Note on a matrix-concave function, Journal of Mathematical Analysis and Applications, 196, 1139-1141.
168. Neudecker, H. and S. Liu (2001a). Some statistical properties of Hadamard products of random matrices, Statistical Papers, 42, 475-487.
169. Neudecker, H. and S. Liu (2001b). Statistical properties of the Hadamard product of random vectors, Statistical Papers, 42, 529-533.
170. Neudecker, H., S. Liu and W. Polasek (1995). The Hadamard product and some of its applications in statistics, Statistics, 26, 365-373.
171. Neudecker, H., W. Polasek and S. Liu (1995). The heteroskedastic linear regression model and the Hadamard product: a note, Journal of Econometrics, 68, 361-366.
172. Neudecker, H. and A. Satorra (1993). Problem 93.3.9, Econometric Theory, 9, 524. Solutions in Econometric Theory, 11, 654-655.
173. Neudecker, H., A. Satorra and M. van de Velden (1997). A fundamental matrix result on scaling in multivariate analysis, Problem 97.5.3, Econometric Theory, 13, 890. Solution in Econometric Theory, 15, 637.
174. Neudecker, H. and M. van de Velden (1998). A determinantal identity, Problem 98.4.1, Econometric Theory, 14, 525. Solution by S. Puntanen, G. P. H. Styan and H. J. Werner in Econometric Theory, 15, 632-633.
175. Neudecker, H. and M. van de Velden (2000). A relationship satisfied by two representations of a positive semidefinite matrix, Problem 00.1.3, Econometric Theory, 16, 143. Solution in Econometric Theory, 17, 281-282.
176. Neudecker, H. and T. Wansbeek (1983). Some results on commutation matrices, with statistical applications. Canadian Journal of Statistics, 11, 221-231.
177. Nishisato, S. (1994). Elements of Dual Scaling: An Introduction to Practical Data Analysis, Lawrence Erlbaum, Hillsdale, New Jersey, USA.
178. Norden, R. H. (1972). A survey of maximum likelihood estimation, International Statistical Review, 40, 329-354.
179. Norden, R. H. (1973). A survey of maximum likelihood estimation, Part 2, International Statistical Review, 41, 39-58.
180. Olkin, I. (1983). An inequality for a sum of forms, Linear Algebra and Its Applications, 52/53, 529-532.
181. Patriota, A. G. and A. J. Lemonte (2009). Bias correction in a multivariate normal regression model with general parameterization, Statistics & Probability Letters, 79, 1655-1662.
182. Penrose, R. (1955). A generalized inverse for matrices, Proceedings of the Cambridge Philosophical Society, 51, 406-413.
183. Penrose, R. (1956). On best approximate solutions of linear matrix equations, Proceedings of the Cambridge Philosophical Society, 52, 17-19.
184. Poincaŕe, H. (1890). Sur les equations aux derivees partielles de la physique mathematique, American Journal of Mathematics, 12, 211-294.
185. Polasek, W. (1986). Local sensitivity analysis and Bayesian regression diagnostics, in: Bayesian Inference and Decision Techniques (eds P. K. Goel and A. Zellner), North-Holland, Amsterdam, pp. 375-387.
186. Pollock, D. S. G. (1979). The Algebra of Econometrics, John Wiley, New York.
187. Pollock, D. S. G. (1985). Tensor products and matrix differential calculus, Linear Algebra and Its Applications, 67, 169-193.
188. Prieto, R. E. (2003). A general solution to the maximization of the multidimensional generalized Rayleigh quotient used in linear discriminant analysis for signal classification, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 6, 157-160.
189. Pringle, R. M. and A. A. Rayner (1971). Generalized Inverse Matrices with Applications to Statistics, Griffin's Statistical Monographs and Courses, No. 28, Charles Griffin, London.
190. Przystalski, M. (2014). Estimation of the covariance matrix in multivariate partially linear models, Journal of Multivariate Analysis, 123, 380-385.
191. Rao, A. R. and P. Bhimasankaram (1992). Linear Algebra, Tata McGraw- Hill, New Delhi.
192. Rao, C. R. (1945). Markoff's theorem with linear restrictions on parameters, Sankhya, 7, 16-19.
193. Rao, C. R. (1955). Estimation and tests of significance in factor analysis, Psychometrika, 20, 93-111.
194. Rao, C. R. (1971a). Unified theory of linear estimation, Sankhya, A, 33, 371-477. (Corrigenda in Sankhya, A, 34, 477.)
195. Rao, C. R. (1971b). Estimation of variance and covariance components- MINQUE theory, Journal of Multivariate Analysis, 1, 257-275.
196. Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd edition, John Wiley, New York.
197. Rao, C. R. and S. K. Mitra (1971). Generalized Inverse of Matrices and Its Applications, John Wiley, New York.
198. Rao, C. R. and M. B. Rao (1998). Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, Singapore.
199. Rellich, F. (1953). Perturbation Theory of Eigenvalue Problems, Gordon & Breach, New York.
200. Rogers, G. S. (1980). Matrix Derivatives, Marcel Dekker, New York.
201. Rolle, J.-D. (1994). Best nonnegative invariant partially orthogonal quadratic estimation in normal regression, Journal of the American Statistical Association, 89, 1378-1385.
202. Rolle, J.-D. (1996). Optimization of functions of matrices with an application in statistics, Linear Algebra and Its Applications, 234, 261-275.
203. Roth, W. E. (1934). On direct product matrices, Bulletin of the American Mathematical Society, 40, 461-468.
204. Rothenberg, T. J. (1971). Identification in parametric models, Econometrica, 39, 577-591.
205. Rothenberg, T. J. and C. T Leenders (1964). Efficient estimation of simultaneous equation systems, Econometrica, 32, 57-76.
206. Rudin, W. (1964). Principles of Mathematical Analysis, 2nd edition, McGraw-Hill, New York.
207. Satorra, A. and H. Neudecker (2015). A theorem on the rank of a product of matrices with illustration of its use in goodness of fit testing, Psychometrika, 80, 938-948.
208. Schönemann, P. H. (1985). On the formal differentiation of traces and determinants, Multivariate Behavioral Research, 20, 113-139.
209. Schönfeld, P. (1971). Best linear minimum bias estimation in linear regression, Econometrica, 39, 531-544.
210. Sherin, R. J. (1966). A matrix formulation of Kaiser's varimax criterion, Psychometrika, 31, 535-538.
211. Smith, R. J. (1985). Wald tests for the independence of stochastic variables and disturbance of a single linear stochastic simultaneous equation, Economics Letters, 17, 87-90.
212. Stewart, G. W. (1969). On the continuity of the generalized inverse, SIAM Journal of Applied Mathematics, 17, 33-45.
213. Styan, G. P. H. (1973). Hadamard products and multivariate statistical analysis, Linear Algebra and Its Applications, 6, 217-240.
214. Sugiura, N. (1973). Derivatives of the characteristic root of a symmetric or a Hermitian matrix with two applications in multivariate analysis, Communications in Statistics, 1, 393-417.
215. Sydsæter, K. (1974). Letter to the editor on some frequently occurring errors in the economic literature concerning problems of maxima and minima, Journal of Economic Theory, 9, 464-466.
216. Sydsæter, K. (1981). Topics in Mathematical Analysis for Economists, Academic Press, London.
217. Tanabe, K. and M. Sagae (1992). An exact Cholesky decomposition and the generalized inverse of the variance-covariance matrix of the multinomial distribution, with applications, Journal of the Royal Statistical Society, B, 54, 211-219.
218. Ten Berge, J. M. F. (1993). Least Squares Optimization in Multivariate Analysis, DSWO Press, Leiden.
219. Theil, H. (1965). The analysis of disturbances in regression analysis, Journal of the American Statistical Association, 60, 1067-1079.
220. Theil, H. (1971). Principles of Econometrics, John Wiley, New York.
221. Theil, H. and A. L. M. Schweitzer (1961). The best quadratic estimator of the residual variance in regression analysis, Statistica Neerlandica, 15, 19-23.
222. Tracy, D. S. and P. S. Dwyer (1969). Multivariate maxima and minima with matrix derivatives, Journal of the American Statistical Association, 64, 1576-1594.
223. Tracy, D. S. and R. P. Singh (1972). Some modifications of matrix differentiation for evaluating Jacobians of symmetric matrix transformations, in: Symmetric Functions in Statistics (ed. D. S. Tracy), University of Windsor, Canada.
224. Tsai, H. and R. S. Tsay (2010). Constrained factor models, Journal of the American Statistical Association, 105, 1593-1605.
225. Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis, Psychometrika, 31, 279-311.
226. van de Velden, M., I. Iodice d'Enza and F. Palumbo (2017). Cluster correspondence analysis, Psychometrika, 82, 158-185.
227. Von Rosen, D. (1985). Multivariate Linear Normal Models with Special References to the Growth Curve Model , Ph.D. Thesis, University of Stockholm.
228. Wang, S. G. and S. C. Chow (1994). Advanced Linear Models, Marcel Dekker, New York.
229. Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem, Clarendon Press, Oxford.
230. Wilks, S. S. (1962). Mathematical Statistics, 2nd edition, John Wiley, New York.
231. Wolkowicz, H. and G. P. H. Styan (1980). Bounds for eigenvalues using traces, Linear Algebra and Its Applications, 29, 471-506.
232. Wong, C. S. (1980). Matrix derivatives and its applications in statistics, Journal of Mathematical Psychology, 22, 70-81.
233. Wong, C. S. (1985). On the use of differentials in statistics, Linear Algebra and Its Applications, 70, 285-299.
234. Wong, C. S. and K. S. Wong (1979). A first derivative test for the ML estimates, Bulletin of the Institute of Mathematics, Academia Sinica, 7, 313-321.
235. Wong, C. S. and K. S. Wong (1980). Minima and maxima in multivariate analysis, Canadian Journal of Statistics, 8, 103-113.
236. Yang, Y. (1988). A matrix trace inequality, Journal of Mathematical Analysis and Applications, 133, 573-574.
237. Young, W. H. (1910). The Fundamental Theorems of the Differential Calculus, Cambridge Tracts in Mathematics and Mathematical Physics, No. 11, Cambridge University Press, Cambridge.
238. Yuan, K.-H., L. L. Marshall and P. M. Bentler (2002). A unified approach to exploratory factor analysis with missing data, nonnormal data, and in the presence of outliers, Psychometrika, 67, 95-122.
239. Yuen, T. P., H. Wong and K. F. C. Yiu (2018). On constrained estimation of graphical time series models, Computational Statistics and Data Analysis, 124, 27-52.
240. Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias, Journal of the American Statistical Association, 57, 348-368.
241. Zyskind, G. and F. B. Martin (1969). On best linear estimation and a general Gauss-Markov theorem in linear models with arbitrary nonnegative covariance structure, SIAM Journal of Applied Mathematics, 17, 1190-1202.