Approximate Moments for the Serial Correlation Coefficient

The first order Gaussian auto-regressive process (xₜ) may be defined by the stochastic difference equation {equation*}{1}x_t = ρ xₜ₋₁ + u_t,{equation*} where the u's are NID(0, 1) and ρ is an unknown parameter. The choice of a statistic as an estimator for ρ depends on the initial conditions imposed on the difference equation (1). The so-called "circular" model is obtained by considering a sample of size N and then assuming that xN + 1 = x₁. An appropriate estimator for ρ in this case is the circular serial correlation coefficient {equation*}{2} r = {∑^Nt = 1 x_txt + 1}{∑^Nt = 1 x^2_t} (xN + 1 = x_1).{equation*} Leipnik [1] has derived an approximate density function {equation*}{3} f(t) = {Γ(N + 2/2)}{Γ(N + 1/2 Γ(1/2)} (1 - 2tρ + ρ^2)-N/2(1 - t^2)(N - 1)/2{equation*} for the estimator r. Leipnik also evaluated the first two moments of this distribution. In this paper a formula is obtained which gives E(rᵏ) as a polynomial of degree k in ρ.