Daniel Dvorkin (danielmedic) wrote in stat_geeks,
Daniel Dvorkin

Hougaard-Weibull question

Okay, so I have a question about the Hougaard multivariate Weibull distribution that I'm hoping someone can help me answer. The distribution is as given in [1], and is most easily defined by the survival function. Let T = (T1, ..., Tn) be a vector of r.v.s with marginal Weibull distributions, and let t = (t1, ..., tn) be a vector of observations. Then the multivariate survival function is given by:

S(t) = P(T1 > t1, ..., Tn > tn) = exp{-(Σi=1,...,nεitiγ)α}

for constants α, γ, ε1, ..., εn > 0.

Hougaard claims that this is only a legitimate survival function with the additional constraint α ≤ 1. What I'm trying to understand is why. It seems to me that for any positive α, the expression obeys all the rules for a proper survival function: S(0, ..., 0) = 1, the limit as any ti goes to infinity is 0, and S is strictly decreasing in the ti's.

Now, I've read the derivation (partly in [1], partly in [2]) and I understand that S was derived via a positive stable frailty distribution, and that this derivation imposes the constraint. I also understand that in general, Archimedean copulas, of which this is an example, require concave generator functions, and although I haven't gone through the math I can guess that α > 1 might violate this requirement for some values of t. But again, looking at the specific S given above, I still don't see how any positive value of α can make it not be a legitimate survival function. Honestly, how it was derived seems kind of irrelevant to its legitimacy; once you've got the function, if it meets the requirements, why not use it?

Any insight that anyone can offer on this will be greatly appreciated.

[1] A Class of Multivariate Failure Time Distributions, P. Hougaard (1986), Biometrika 73(3):671-678

[2] Survival Models for Heterogeneous Populations Derived from Stable Distributions, P. Hougaard (1986), Biometrika 73(2):387-396

x-posted to statisticians
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