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Assume to observe 2 quantiles, x and y, associated with the z% probability. These quantiles are generated by 2 non-independent standardized student-t distributions X and Y.

In case of linear combination of x and y with the Correlation matrix, its distribution function is known?

Can I assume any initial condition on x and y in order to have a known distribution function of the linear combination?

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  • $\begingroup$ Could you please describe any kind of experimental setup or procedure that actually observes specified quantiles? How do you know the correlation? Even when you do, that does not determine the distribution, so what distribution family do you have in mind specifically? $\endgroup$
    – whuber ♦
    May 14 at 21:30
  • $\begingroup$ Thanks whuber. In this procedure, I have a statistical model for quantiles. Furthermore, I am able to assume any distribution family. Sample correlation matrix is given. Honestly, I cannot figure out how to set assumptions in order to have a (generalized) student-t linear combination of standardized student-t quantiles and a given sample correlation matrix. $\endgroup$
    – 3463495
    May 15 at 11:33

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Answers are in McNeil, Frey and Embrechts, "Quantitative Risk Management", Princeton 2005.

Seems that, assuming a student-t multivariate distribution, linear combination of marginals is also standardized student-t.

See also Linear Combination of multivariate t distribution

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  • $\begingroup$ This doesn't answer the question you asked: it assumes unstated information. With the usual multivariate Student t there's really nothing to ask because that family is constructed to have this property. $\endgroup$
    – whuber ♦
    May 15 at 14:31
  • $\begingroup$ IMHO, this do answer the question because I stated was looking for a known distribution function under any reasonable condition (multivariate student-t). Additional details are in my previous comment. $\endgroup$
    – 3463495
    May 15 at 16:50
  • $\begingroup$ Please, let me see any other unstated information. Thanks. $\endgroup$
    – 3463495
    May 15 at 16:56
  • $\begingroup$ "Non-independent" does not mean "multivariate t." There are plenty of multivariate distributions with Student t marginals that are not this particular "multivariate t." $\endgroup$
    – whuber ♦
    May 15 at 17:41

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