*Proof of Theorem 8.*

We consider the case of continuous *σ*; the proof of the discrete case is similar. Taking negative logs on both sides of and using the power series representation − log(1 − *x*) = *x* + *x*^{2}/2 + *x*^{3}/3 + …, we obtain

Thus, we wish to show that, for suitable *a*_{n} and *b*_{n}, for any *x*, while as *n* → *∞*:

(A.31)
By the von Mises condition, as *n* → *∞*, so we want

Now specializing to *F*_{i}(*x*) = exp[−(*σ*_{i}/*x*^{ρ})] = exp(−*τ*_{i}/*x*^{ρ}), with we have *φ*_{i}(*x*) =*τ*_{i}*x*^{− ρ}, *φ*′_{i} (*x*) = − *ρτ*_{i} *x*^{− (ρ + 1)}, and *ψ*_{i}(*x*) = *x*^{ρ + 1}/*ρτ*_{i}, so that

(A.33)
Letting *u* = *i*/*n* (*i* = 1, …, *n*) and assuming there exists a continuous ...