**T**= (T

_{1}, ..., T

_{n}) be a vector of r.v.s with marginal Weibull distributions, and let

**t**= (t

_{1}, ..., t

_{n}) be a vector of observations. Then the multivariate survival function is given by:

S(

**t**) = P(T

_{1}> t

_{1}, ..., T

_{n}> t

_{n}) = exp{-(Σ

_{i=1,...,n}ε

_{i}t

_{i}

^{γ})

^{α}}

for constants α, γ, ε

_{1}, ..., ε

_{n}> 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 t

_{i}goes to infinity is 0, and S is strictly decreasing in the t

_{i}'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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