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Identification of parameters by the distribution of the maximum random variable: The general multivariate normal case
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  • Published: September 1990

Identification of parameters by the distribution of the maximum random variable: The general multivariate normal case

  • Arunava Mukherjea1 &
  • Richard Stephens2 

Probability Theory and Related Fields volume 84, pages 289–296 (1990)Cite this article

Summary

LetX 1,X 2, ...,X r ber independentn-dimensional random vectors each with a non-singular normal distribution with zero means and positive partial correlations. Suppose thatX i =(X i1 , ...,X in ) and the random vectorY=(Y 1, ...,Y n ), their maximum, is defined byY j =max{X ij :1≦i≦r}. LetW be another randomn-vector which is the maximum of another such family of independentn-vectorsZ 1,Z 2, ...,Z s . It is then shown in this paper that the distributions of theZ i 's are simply a rearrangement of those of theZ j 's (and of course,r=s), whenever their maximaY andW have the same distribution. This problem was initially studied by Anderson and Ghurye [2] in the univariate and bivariate cases and motivated by a supply-demand problem in econometrics.

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References

  1. Anderson, T.W.: An introduction to multivariate statistical analysis. New York: Wiley 1958, (See also its 1984 second edition)

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  2. Anderson, T.W., Ghurye, S.G.: Identification of parameters by the distribution of a maximum random variable. J. R. Stat. Soc., Ser. B39, 337–342 (1977)

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  3. Anderson, T.W., Ghurye, S.G.: Unique factorization of products of bivariate normal cumulative distribution functions. Ann. Inst. Stat. Math.30 63–69 (1979)

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  4. Mukherjea, A., Nakassis, A., Miyashita, J.: Identification of parameters by the distribution of the maximum random variable: The Anderson-Ghurye theorem. J. Multivariate Anal.18, 178–186 (1986)

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  5. Savage, I.R.: Mill's ratio for multivariate normal distributions. J. Res. Natl. Bur. Stand.66B, 93–96 (1986)

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Authors and Affiliations

  1. Department of Mathematics, University of Southern Florida, 33620-5700, Tampa, FL, USA

    Arunava Mukherjea

  2. Department of Mathematics, West Carolina University, 28723, Cullowhee, NC, USA

    Richard Stephens

Authors
  1. Arunava Mukherjea
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  2. Richard Stephens
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Cite this article

Mukherjea, A., Stephens, R. Identification of parameters by the distribution of the maximum random variable: The general multivariate normal case. Probab. Th. Rel. Fields 84, 289–296 (1990). https://doi.org/10.1007/BF01197886

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  • Received: 08 August 1988

  • Revised: 30 May 1989

  • Issue Date: September 1990

  • DOI: https://doi.org/10.1007/BF01197886

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Keywords

  • Normal Distribution
  • Stochastic Process
  • Probability Theory
  • Random Vector
  • Mathematical Biology
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