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Professor Heinz Neudecker and matrix differential calculus

Author

Listed:
  • Shuangzhe Liu

    (University of Canberra)

  • Götz Trenkler

    (Dortmund University of Technology)

  • Tõnu Kollo

    (University of Tartu)

  • Dietrich Rosen

    (Swedish University of Agricultural Sciences
    Linköping University)

  • Oskar Maria Baksalary

    (Adam Mickiewicz University)

Abstract

The late Professor Heinz Neudecker (1933–2017) made significant contributions to the development of matrix differential calculus and its applications to econometrics, psychometrics, statistics, and other areas. In this paper, we present an insightful overview of matrix-oriented findings and their consequential implications in statistics, drawn from a careful selection of works either authored by Professor Neudecker himself or closely aligned with his scientific pursuits. The topics covered include matrix derivatives, vectorisation operators, special matrices, matrix products, inequalities, generalised inverses, moments and asymptotics, and efficiency comparisons within the realm of multivariate linear modelling. Based on the contributions of Professor Neudecker, several results related to matrix derivatives, statistical moments and the multivariate linear model, which can literally be considered to be his top three areas of research enthusiasm, are particularly included.

Suggested Citation

  • Shuangzhe Liu & Götz Trenkler & Tõnu Kollo & Dietrich Rosen & Oskar Maria Baksalary, 2024. "Professor Heinz Neudecker and matrix differential calculus," Statistical Papers, Springer, vol. 65(4), pages 2605-2639, June.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:4:d:10.1007_s00362-023-01499-w
    DOI: 10.1007/s00362-023-01499-w
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    References listed on IDEAS

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    1. Liu, Shuangzhe & Ma, Tiefeng & Polasek, Wolfgang, 2014. "Spatial system estimators for panel models: A sensitivity and simulation study," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 101(C), pages 78-102.
    2. Kollo, Tõnu, 2008. "Multivariate skewness and kurtosis measures with an application in ICA," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2328-2338, November.
    3. H. Neudecker, 1967. "On matrix procedures for optimizing differentiable scalar functions of matrices," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 21(1), pages 101-107, March.
    4. Zhang, Zhihua, 2007. "Pseudo-inverse multivariate/matrix-variate distributions," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1684-1692, September.
    5. Heinz Neudecker & Shuangzhe Liu, 2001. "Some statistical properties of Hadamard products of random matrices," Statistical Papers, Springer, vol. 42(4), pages 475-487, October.
    6. Neudecker, Heinz & Polasek, Wolfgang & Liu, Shuangzhe, 1995. "The heteroskedastic linear regression model and the Hadamard product a note," Journal of Econometrics, Elsevier, vol. 68(2), pages 361-366, August.
    7. Kollo, T. & Neudecker, H., 1993. "Asymptotics of Eigenvalues and Unit-Length Eigenvectors of Sample Variance and Correlation Matrices," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 283-300, November.
    8. Heinz Neudecker & Shuangzhe Liu, 2001. "Statistical properties of the Hadamard product of random vectors," Statistical Papers, Springer, vol. 42(4), pages 529-533, October.
    9. Magnus, J.R. & Neudecker, H., 1980. "The elimination matrix : Some lemmas and applications," Other publications TiSEM 0e3315d3-846c-4bc5-928e-f, Tilburg University, School of Economics and Management.
    10. Turkington,Darrell A., 2013. "Generalized Vectorization, Cross-Products, and Matrix Calculus," Cambridge Books, Cambridge University Press, number 9781107032002, June.
    11. Khatri, C. G. & Rao, C. Radhakrishna, 1981. "Some extensions of the Kantorovich inequality and statistical applications," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 498-505, December.
    12. Satorra, Albert & Neudecker, Heinz, 1994. "On the Asymptotic Optimality of Alternative Minimum-Distance Estimators in Linear Latent-Variable Models," Econometric Theory, Cambridge University Press, vol. 10(5), pages 867-883, December.
    13. Neudecker, Heinz & Satorra, Albert & van de Velden, Michel, 1997. "A Fundamental Matrix Result on Scaling in Multivariate Analysis," Econometric Theory, Cambridge University Press, vol. 13(06), pages 890-890, December.
    14. Shuangzhe Liu & Chris Heyde & Wing-Keung Wong, 2011. "Moment matrices in conditional heteroskedastic models under elliptical distributions with applications in AR-ARCH models," Statistical Papers, Springer, vol. 52(3), pages 621-632, August.
    15. Yonghui Liu & Chaoxuan Mao & Víctor Leiva & Shuangzhe Liu & Waldemiro A. Silva Neto, 2022. "Asymmetric autoregressive models: statistical aspects and a financial application under COVID-19 pandemic," Journal of Applied Statistics, Taylor & Francis Journals, vol. 49(5), pages 1323-1347, April.
    16. Heinz Neudecker & Götz Trenkler, 2006. "Problem 4/SP06: Estimation of the Hadamard and cross products of two mean vectors in multivariate analysis," Statistical Papers, Springer, vol. 47(3), pages 481-485, June.
    17. Wang, Song-Gui & Shao, Jun, 1992. "Constrained Kantorovich inequalities and relative efficiency of least squares," Journal of Multivariate Analysis, Elsevier, vol. 42(2), pages 284-298, August.
    18. H. Neudecker & F.A.G. Windmeijer, 1991. "R2 in Seemingly Unrelated Regression Equations," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 45(4), pages 405-411, December.
    19. Magnus, J.R. & Neudecker, H., 1985. "Matrix differential calculus with applications to simple, Hadamard, and Kronecker products," Other publications TiSEM 1b2f1740-bfd1-4ea5-986c-9, Tilburg University, School of Economics and Management.
    20. Neudecker, Heinz & Satorra, Albert, 1996. "The algebraic equality of two asymptotic tests for the hypothesis that a normal distribution has a specified correlation matrix," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 99-103, October.
    21. von Rosen, Dietrich, 1989. "Maximum likelihood estimators in multivariate linear normal models," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 187-200, November.
    22. Albert Satorra & Heinz Neudecker, 2015. "A Theorem on the Rank of a Product of Matrices with Illustration of Its Use in Goodness of Fit Testing," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 938-948, December.
    23. Przystalski, Marcin, 2014. "Estimation of the covariance matrix in multivariate partially linear models," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 380-385.
    24. S. Liu & H. Neudecker, 1997. "Kantorovich inequalities and efficiency comparisons for several classes of estimators in linear models," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 51(3), pages 345-355, November.
    25. Derrick S. Tracy & Rana P. Singh, 1972. "A new matrix product and its applications in partitioned matrix differentiation," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 26(4), pages 143-157, December.
    26. Oskar Maria Baksalary & Götz Trenkler, 2022. "An alternative look at the linear regression model," Statistical Papers, Springer, vol. 63(5), pages 1499-1509, October.
    27. Neudecker, H., 1985. "On The Dispersion Matrix Of A Matrix Quadratic Form Connected With The Noncentral Wishart Distribution," University of Amsterdam, Actuarial Science and Econometrics Archive 293021, University of Amsterdam, Faculty of Economics and Business.
    28. D. Jochems & H. Neudecker, 1959. "Micro-economic business test data compared with traditional statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 2(1), pages 46-61, December.
    29. Neudecker, Heinz & Satorra, Albert & Trenkler, Götz & Liu, Shuangzhe, 1995. "A Kronecker Matrix Inequality with a Statistical Application," Econometric Theory, Cambridge University Press, vol. 11(03), pages 654-655, June.
    30. Liu, Shuangzhe & Leiva, Víctor & Zhuang, Dan & Ma, Tiefeng & Figueroa-Zúñiga, Jorge I., 2022. "Matrix differential calculus with applications in the multivariate linear model and its diagnostics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    31. Shuangzhe Liu & Chris Heyde, 2008. "On estimation in conditional heteroskedastic time series models under non-normal distributions," Statistical Papers, Springer, vol. 49(3), pages 455-469, July.
    32. Liu, Shuangzhe & Neudecker, Heinz, 2009. "On pseudo maximum likelihood estimation for multivariate time series models with conditional heteroskedasticity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(8), pages 2556-2565.
    33. Magnus, Jan R., 2010. "On the concept of matrix derivative," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2200-2206, October.
    34. repec:wop:ubisop:0081 is not listed on IDEAS
    35. H. Neudecker, 1981. "On the matrix formulation of Kaiser's varimax criterion," Psychometrika, Springer;The Psychometric Society, vol. 46(3), pages 343-345, September.
    36. Satorra, Albert & Neudecker, Heinz, 1992. "A Matrix Equality Applicable in the Analysis of Mean-and-Covariance Structures," Econometric Theory, Cambridge University Press, vol. 8(04), pages 581-582, December.
    Full references (including those not matched with items on IDEAS)

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