Abstract
This research paper deals with an extension of the non-central Wishart introduced in 1944 by Anderson and Girshick, that is the non-central Riesz distribution when the scale parameter is derived from a discrete vector. It is related to the matrix of normal samples with monotonous missing data. We characterize this distribution by means of its Laplace transform and we give an algorithm for generating it. Then we investigate, based on the method of the moment, the estimation of the parameters of the proposed model. The performance of the proposed estimators is evaluated by a numerical study.
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References
Alam, K., Mitra, A.: On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhya, Ser. B., 52, 133–143 (1990)
Anderson, T. W., Girshick, M. A.: Some extensions of the Wishart distribution. Ann. Statist., 15, 345–357 (1944)
Anderson, T. W.: The non-central Wishart distribution and certain problems of multivariate statistic. Ann. Statist., 17 (4), 409–431 (1946)
Andersson, S. A., Klein, T.: On Riesz and Wishart distributions associated with decomposable undirected graphs. J. Multivariate Anal., 101 (4), 789–810 (2010)
Bodnara, T., Mazur, S., Podgórski, K.: Singular inverse Wishart distribution and its application to portfolio theory. J. Multivariate Anal., 143, 314–326 (2016)
Dawid, A. P.: Some matrix-variate distribution theory: Notational considerations and a Bayesian application. Biometrika, 68 (1), 265–274 (1981)
Faraut, J., Koranyi, A.: Analysis on symmetric cones, Oxford University Press, Oxford, 1994
Ferreira, J. T., Bekker, A.: A unified complex noncentral Wishart type distribution inspired by massive MIMO systems. J. Stat. Distrib. Appl., 6 (1), 1–19 (2019)
Ghorbel, E., Louati, M.: The multiparameter tdistribution. Filomat, 33 (13), 4137–4150 (2019)
Ghorbel, E., Kammoun, K., Louati, M.: Bayesian estimation of the precision matrix with monotone missing data. Lithuanian Math. J., 60 (4), 470–481 (2020)
Gindikin, S. G.: Analysis in homogeneous domains. Russian Math. Surveys, 19 (4), 1–89 (1964)
Graczyk, P., Letac, G., Massam, H.: The moments of the complex Wishart ditribution and the symmetric group. Ann. Statist., 41, 287–309 (2003)
Haff, L. R.: Further identities for the Wishart distribution with applications in regression. Scand. J. Statist., 9 (2), 215–224 (1981)
Johnson, N. L., Kotz, S.: Distribution in Statistics: Continuous Multivariate Distributions, New York, John Wiley, 1972
Kabe, D. G.: On the exact distributions of the GCL estimators in a leading three-equation case. J. Amer. Statist. Assoc., 59, 881–894 (1964)
Kammoun, K., Louati, M., Masmoudi, A.: Maximum likelihood estimator of the scale parameter for the Riesz distribution. Statist. Probab. Lett., 126, 127–131 (2017)
Letac, G., Massam, H.: The noncentral Wishart as an exponential family and its moments. J. Multivariate Anal., 99, 1393–1417 (2008)
Leung, P. L.: An identity for the noncentral Wishart distribution with application. J. Multivariate Anal., 48, 107–114 (1994)
Louati, M.: Mixture of the Riesz distribution with respect to the generalized multivariate gamma distribution. J. Korean Statist. Soc., 42, 83–93 (2013)
Louati, M., Masmoudi, A.: Moment for the inverse Riesz distributions. Statist. Probab. Lett., 102, 30–37 (2015)
Tourneret, J.-Y., Ferrari, A., Letac, G.: The noncentral Wishart distribution: Properties and application to speckle imaging, IEEE/SP: 13th Workshop on Statistical Signal Processing, Bordeaux, France, doi: https://doi.org/10.1109/SSP.2005.1628726 (2005)
Veleva, E.: Testing a normal covariance matrix for small samples with monotone missing data. Appl. Math. Sci., 3(54), 2695–2702 (2009)
Von Rosen, D.: Moments for the inverted Wishart distribution. Scand. J. Statist., 15, 97–109 (1988)
Wishart, J.: The generalized product moment distribution in samples from a normal multivariate population. Biometrics, 20, 32–52 (1928)
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Kammoun, K. An Extension of the Non-central Wishart Distribution with Integer Shape Vector. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-2549-8
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DOI: https://doi.org/10.1007/s10114-024-2549-8
Keywords
- Cholesky decomposition
- Laplace transform
- method of moments
- non-central Wishart distribution
- Riesz distribution