Fredholm Determinant of a Positive Definite Kernel of a Special Type and Its Application

Let ρ(x, y) be a positive definite symmetric kernel defined over the unit square such that ρ(x, y) = K(x, y) - ∑ᵏᵢ₌₁ ψᵢ(x)ψᵢ (y), 0 x, y 1, where $K(x, y)$ is a bounded symmetric positive definite kernel defined over the unit square, and ψᵢ(x) ∈ L₂(0, 1). Methods of finding Fredholm determinant D(λ) of ρ(x, y) in terms of the eigenvalues and the eigenfunctions of $K(x, y)$ are given. A kernel of the type of ρ(x, y) arises as the covariance function of a Gaussian process in the limiting distribution of the modified Cramer-Smirnov test statistic in the k-parameter case which may be described as follows: Let X₁, ⋯, Xₙ be n independent observations (random variables) from a population with a continuous distribution function $G(x)$. Suppose for every θ = (θ₁, ⋯, θₖ) ∈ I,I being an open interval in the k-dimensional Euclidean space Rᵏ, F(x, θ) is a continuous distribution function. Let θ̂ₙ be an estimate of θ obtained from the sample. A test of the hypothesis H: G(x) = F(x, θ) for some unspecified θ ∈ I based on the statistic Cₙ² = n ∫+∞-∞ Fₙ(x) - F(x, θ̂ₙ) ² dF(x, θ̂ₙ), is considered and the characteristic function of the asymptotic distribution of Cₙ² is shown to be the Fredholm determinant of a kernel of the type of ρ(x, y) with K(x, y) = min (x, y) - xy whose eigenvalues and eigenfunctions are known. Results are also used to obtain the limiting distribution of l-sample analogue of Cₙ².