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Identification of the Parameters of a Multivariate Normal Vector by the Distribution of the Maximum

Author

Listed:
  • Ming Dai

    (University of South Florida)

  • Arunava Mukherjea

    (University of South Florida)

Abstract

This paper continues the work started by Basu and Ghosh (J. Mult. Anal. (1978), 8, 413–429), by Gilliland and Hannan (J. Amer. Stat. Assoc. (1980), 75, No. 371, 651–654), and then continued on by Mukherjea and Stephens (Prob. Theory and Rel. Fields (1990), 84, 289–296), and Elnaggar and Mukherjea (J. Stat. Planning and Inference (1990), 78, 23–37). Let (X1, X2,..., Xn) be a multivariate normal vector with zero means, a common correlation ρ and variances σ2 1, σ2 2,..., σ2 n such that the parameters ρ, σ2 1, σ2 2,..., s2 n are unknown, but the distribution of the max{Xi: 1≤i≤n} (or equivalently, the distribution of the min{Xi: 1≤i≤n}) is known. The problem is whether the parameters are identifiable and then how to determine the (unknown) parameters in terms of the distribution of the maximum (or its density). Here, we solve this problem for general n. Earlier, this problem was considered only for n≤3. Identifiability problems in related contexts were considered earlier by numerous authors including: T. W. Anderson and S. G. Ghurye, A. A. Tsiatis, H. A. David, S. M. Berman, A. Nadas, and many others. We also consider here the case where the Xi's have a common covariance instead of a common correlation.

Suggested Citation

  • Ming Dai & Arunava Mukherjea, 2001. "Identification of the Parameters of a Multivariate Normal Vector by the Distribution of the Maximum," Journal of Theoretical Probability, Springer, vol. 14(1), pages 267-298, January.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:1:d:10.1023_a:1007889519309
    DOI: 10.1023/A:1007889519309
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    References listed on IDEAS

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    1. Basu, A. P. & Ghosh, J. K., 1978. "Identifiability of the multinormal and other distributions under competing risks model," Journal of Multivariate Analysis, Elsevier, vol. 8(3), pages 413-429, September.
    2. Mukherjea, Arunava & Stephens, Richard, 1990. "The problem of identification of parameters by the distribution of the maximum random variable: Solution for the trivariate normal case," Journal of Multivariate Analysis, Elsevier, vol. 34(1), pages 95-115, July.
    3. Amemiya, Takeshi, 1974. "A Note on a Fair and Jaffee Model," Econometrica, Econometric Society, vol. 42(4), pages 759-762, July.
    4. Fair, Ray C & Jaffee, Dwight M, 1972. "Methods of Estimation for Markets in Disequilibrium," Econometrica, Econometric Society, vol. 40(3), pages 497-514, May.
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    Cited by:

    1. Das, Bikramjit & Fasen-Hartmann, Vicky, 2024. "On heavy-tailed risks under Gaussian copula: The effects of marginal transformation," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    2. Arup Bose & Rajat Subhra Hazra & Koushik Saha, 2011. "Spectral Norm of Circulant-Type Matrices," Journal of Theoretical Probability, Springer, vol. 24(2), pages 479-516, June.
    3. Bikramjit Das & Vicky Fasen-Hartmann, 2023. "On heavy-tailed risks under Gaussian copula: the effects of marginal transformation," Papers 2304.05004, arXiv.org.

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