On independence of statistics in truncated exponential models

Let X₍₁₎ ≤ … ≤ X_(n) be the order statistics of a sample of size n from the left truncated k-parameter exponential family of distributions with unknown truncation parameter. The densities of this family are of the form f(x:θ,γ₁) = a(x) exp{θ «u(x) }/b (θ,γ₁), c < γ₁ < x < d, θ = (θ₁,…,θ_k), u(x) = (u₁(x),…, U_k(x)). Let Z = (Z₁,…, Z_k) where and g_n be a vector valued function. The existence of a statistic of the form Z* = Z - g_n(X₍₁₎ with distribution independent of X₍₁₎ and depending on θ is characterized. The case k=l is shown to be of particular interest and an application in this case to uniformly most powerful unbiased tests is presented. Right and two-sided truncation models are also discussed.