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  • 1.
    Bodnar, Taras
    et al.
    Department of Mathematics, Humboldt University of Berlin, Berlin, Germany.
    Mazur, Stepan
    Department of Statistics, European University Viadrina, Frankfurt an der Oder, Germany.
    Okhrin, Yarema
    Department of Statistics, University of Augsburg, Augsburg, Germany.
    On the exact and approximate distributions of the product of a Wishart matrix with a normal vector2013In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 122, p. 70-81Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the distribution of the product of a Wishart random matrix and a Gaussian random vector. We derive a stochastic representation for the elements of the product. Using this result, the exact joint density for an arbitrary linear combination of the elements of the product is obtained. Furthermore, the derived stochastic representation allows us to simulate samples of arbitrary size by generating independently distributed chi-squared random variables and standard multivariate normal random vectors for each element of the sample. Additionally to the Monte Carlo approach, we suggest another approximation of the density function, which is based on the Gaussian integral and the third order Taylor expansion. We investigate, with a numerical study, the properties of the suggested approximations. A good performance is documented for both methods. 

  • 2.
    Bodnar, Taras
    et al.
    Department of Mathematics, Stockholm University, Stockholm, Sweden.
    Mazur, Stepan
    Department of Statistics, Lund University, Lund, Sweden.
    Podgórski, Krzysztof
    Department of Statistics, Lund University, Lund, Sweden.
    Singular inverse Wishart distribution and its application to portfolio theory2016In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 143, p. 314-326Article in journal (Refereed)
    Abstract [en]

    The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented. 

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