Abstract
Approximations of density functions are considered in the multivariate case. The results are presented with the help of matrix derivatives, powers of Kronecker products and Taylor expansions of functions with matrix argument. In particular, an approximation by the Wishart distribution is discussed. It is shown that in many situations the distributions should be centred. The results are applied to the approximation of the distribution of the sample covariance matrix and to the distribution of the non-central Wishart distribution.
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References
Barndorff-Nielsen, O. E. and Cox, D. R. (1989).Asymptotic Techniques for Use in Statistics, Chapman and Hall, London.
Cornish, E. A. and Fisher, R. A. (1937). Moments and cumulants in the specification of distributions,Internat. Statist. Rev.,5, 307–322.
Finney, D. J. (1963). Some properties of a distribution specified by its cumulants,Technometrics,5, 63–69.
Hall, P. (1983). Chi-squared approximations to the distribution of a sum of independent random variables,Ann. Probab. 11, 1028–1036.
Kollo, T. (1991).Matrix Derivative in Multivariate Statistics, Tartu University Press, Tartu (in Russian).
Kollo, T. and Neudecker, H. (1993). Asymptotics of eigenvalues and unit-length eigenvalues of sample variance and correlation matrices,J. Multivariate Anal.,47, 283–300.
Magnus, J. R. and Neudecker, H. (1988).Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley, Chichester.
McCullagh, P. (1987).Tensor Methods in Statistics, Chapman and Hall, London.
Muirhead, R. J. (1982).Aspects of Multivariate Statistical Theory, Wiley, New York.
Skovgaard, I. (1986). On the multivariate Edgeworth expansions,Internat. Statist. Rev.,54, 169–186.
Srivastava, M. S. and Khatri, C. G. (1979).An Introduction to Multivariate Statistics, North Holland, New York.
Steyn, H. S. and Roux, J. J. J. (1972). Approximations for the noncentral Wishart distribution,South African Journal of Statistics,6, 165–173.
Tan, W. Y. (1979). On the approximation of noncentral Wishart distribution by Wishart distribution,Metron,37, 50–58.
Tan, W. Y. (1980). On approximating multivariate distributions,Multivariate Statistical Analysis, Proceedings of the Research Seminar at Daihousie University, 237–249, North Holland, Amsterdam.
Traat, I. (1984). Moments of the sample covariance matrix.Proceedings of the Computer Centre of Tartu State University,51, 108–126 (in Russian).
Traat, I. (1986). Matrix calculus for multivariate distributions,Acta et Commentationes Universitatis Tartuensis,733, 64–84.
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Kollo, T., von Rosen, D. Approximating by the Wishart distribution. Ann Inst Stat Math 47, 767–783 (1995). https://doi.org/10.1007/BF01856546
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DOI: https://doi.org/10.1007/BF01856546