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.
References
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Anderson, T.W., Ghurye, S.G.: Unique factorization of products of bivariate normal cumulative distribution functions. Ann. Inst. Stat. Math.30 63–69 (1979)
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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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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DOI: https://doi.org/10.1007/BF01197886
Keywords
- Normal Distribution
- Stochastic Process
- Probability Theory
- Random Vector
- Mathematical Biology