Let X1,...,Xn be i.i.d. random variables on [0,1] with unknown distribution P. The problem is to test P = P0, the uniform distribution on [0, 1]. Assume a parametric model with densities of the form (14.35) for some fixed positive integer k. Set T0(x) = 1 and assume the functions T1,...,Tk are chosen so that T0,...,Tk is a set of orthonormal functions on L2(P0). Assume that sup x,j

|Tj (x)| < ∞ , so that Ck(θ) is well-defined for all k-vectors θ. Let Λn be a probability distribution over values of θ and let A(φn, Λn) denote the average power of a test φn with respect to Λn; that is, A(φn, Λn) = 

θ

Eθ(φn)dΛn(θ) .

In particular, let Λn be the k-dimensional normal distribution with mean vector 0 and covariance matrix equal to n−1 times the identity matrix. Among tests φn such that E0(φn) → α, find one that maximizes limn A(φn, Λn)

and find a simple expression for this limiting average power.