Random sample size. Let N be a random variable with a power series distribution P(N = n) = a(n)λn C(λ) , n = 0, 1,... (λ > 0, unknown).

When N = n, a sample X1,..., Xn from the exponential family (3.19) is observed.

On the basis of (N, X1,..., X N ) there exists a UMP unbiased test of H : Q(θ) ≤ c.