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

Abstract

Let $\rho(x, y)$ be a positive definite symmetric kernel defined over the unit square such that $\rho(x, y) = K(x, y) - \sum^k_{i=1} \psi_i(x)\psi_i (y), 0 \leqq x, y \leqq 1$, where $K(x, y)$ is a bounded symmetric positive definite kernel defined over the unit square, and $\psi_i(x) \in L_2(0, 1)$. Methods of finding Fredholm determinant $D(\lambda)$ of $\rho(x, y)$ in terms of the eigenvalues and the eigenfunctions of $K(x, y)$ are given. A kernel of the type of $\rho(x, y)$ 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 $X_1, \cdots, X_n$ be $n$ independent observations (random variables) from a population with a continuous distribution function $G(x)$. Suppose for every $\theta = (\theta_1, \cdots, \theta_k) \in \mathbf{I,I}$ being an open interval in the $k$-dimensional Euclidean space $R^k, F(x, \theta)$ is a continuous distribution function. Let $\hat{\theta}_n$ be an estimate of $\theta$ obtained from the sample. A test of the hypothesis $H: G(x) = F(x, \theta)$ for some unspecified $\theta \in \mathbf{I}$ based on the statistic $C_n^2 = n \int^{+\infty}_{-\infty} \lbrack F_n(x) - F(x, \hat{\theta}_n) \rbrack^2 dF(x, \hat{\theta}_n),$ is considered and the characteristic function of the asymptotic distribution of $C_n^2$ is shown to be the Fredholm determinant of a kernel of the type of $\rho(x, y)$ 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 $C_n^2$.