Summary
The distribution of a quadratic form in a normal sample plays a very important role in multivariate statistical analysis. In many cases, statistics are functions of a quadratic form or special types of it.
In the univariate case, the distribution of a quadratic form was treated by many authors and was derived by using the Laguerre polynomials expansion or the Dirichlet series expansion, etc. [6], [7], [8]. In this paper, the distribution of a quadratic form in the multivariate case will be given in terms of zonal polynomials which were developed for multivariate analysis by A. T. James [3], [4] and A. G. Constantine [2]. Recently, the author’s attention was called to C. G. Khatri [9] which deals with the same problem also by using zonal polynomials. However, the present paper treats the problem from another point of view.
The distribution of a quadratic form enables us to derive the distribution of a linear combination of several Wishart matrices. The distributions and probability functions of certain statistics of a quadratic form are given.
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References
T. Hayakawa, “ On the distribution of a quadratic form in a multivariate normal sample,”Research Memorandum, No. 2, The Inst. Statist. Math., March 1966.
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C. G. Khatri, “ On certain distribution problems based on positive definite quadratic functions in normal vectors,”Ann. Math. Statist., 37 (1966), 468–479.
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The first version of the paper, [0], was published in March, 1966.
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Hayakawa, T. On the distribution of a quadratic form in a multivariate normal sample. Ann Inst Stat Math 18, 191–201 (1966). https://doi.org/10.1007/BF02869529
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DOI: https://doi.org/10.1007/BF02869529