General Formulae for the Central Moments of Certain Serial Correlation Coefficient Approximations

Abstract

The distributions considered in this paper have the probability density functions ($\operatorname{pdf's})$ \begin{equation*}\tag{1.1}\frac{(1 - x^2)^{\frac{1}{2}(n - 1)}(1 + r^2 - 2rx)^{-\frac{1}{2}n}}{B\lbrack\frac{1}{2}(n + 1), \frac{1}{2}\rbrack}, \quad -1 \leqq x \leqq 1,\end{equation*} and \begin{equation*}\tag{1.2}\frac{(n + 1)(1 - x^2)^{\frac{1}{2}n - 1}(1 + r^2 - 2rx)^{\frac{1}{2}(1 - n)}(1 - x)}{(n - nr + 1 + r)B(\frac{1}{2}n, \frac{1}{2})}, \quad -1 \leqq x \leqq 1,\end{equation*} and are respectively the Madow-Leipnik approximation distribution for the serial correlation coefficient, circularly defined with known non-null mean, and Daniels' (1956) modified approximation distribution for the coefficient, circularly defined with fitted mean. (Note correction of misprints in (1.2).) General expressions for the uncorrected moments (u.m's) of (1.1) were given by Kendall (1957) and by White (1957); these may also be obtained from Leipnik's (1958) Neumann-type series for the characteristic function (ch.f.). They have the form of polynomials in $r$, with coefficients involving the Hermite polynomial coefficients. The first four central moments (c.m's) of (1.1) have been derived from the u.m's by Jenkins (1956), Kendall (1957) and White (1957). General expressions for the u.m's and c.m's of (1.2) do not seem to appear in the literature. In this paper we consider firstly the u.m's of (1.1) about $x = 1$; these are shown to be proportional to Gaussian hypergeometric functions, and to lead to representations of the c.m's and of the ch.f. by hypergeometric functions in two variables. The u.m's of (1.2) about $x = 1$ are closely related to those of (1.1), yielding corresponding formulas for its c.m's and ch.f.