Distribution of a Serial Correlation Coefficient Near the Ends of the Range

Abstract

If $y_1, \cdots, y_N$ are observations on a stationary time series at equal intervals of time and it is known that $Ey_t = 0$ for $t = 1, \cdots, N$, the most natural definition of a serial correlation coefficient with lag unity would be $$r^{\ast} = \Bigg(\sum^{N - 1}_{i = 1} y_iy_{i + 1}\Bigg) \Bigg\lbrack\Bigg(\sum^{N - 1}_{i = 1} y^2_i\Bigg)\Bigg(\sum^{N - 1}_{i = 1} y^2_{i + 1}\Bigg)\Bigg\rbrack^{-1/2}$$ if the denominator $\neq 0$. This is the ordinary correlation coefficient between $(y_1, \cdots, y_{N - 1}$ and $(y_2, \cdots, y_N)$, except that instead of taking deviations from the sample mean, we have taken deviations from the population means. Due to the seemingly unsurmountable mathematical difficulties involved in obtaining the distribution of $r^{\ast}$ even on the hypothesis of independence and normality of the observations, several alternative definitions have been proposed as approximations to $r^{\ast}$. However, it is desirable to consider some relevant properties of the distribution of $r^{\ast}$. In this paper the distribution of $r^{\ast}$ near the extremities of its range will be considered. The observations will be assumed to be distributed as independent $N(0, 1)$ variates. There is no loss of generality in assuming the variance to be unity as $r^{\ast}$ is independent of the scale parameter. A geometrical approach suggested by Hotelling seemed to be particularly suitable in obtaining the order of contact of the distribution curve at $r^{\ast} = \pm 1$. Hotelling [1] shows how to determine the order of contact of frequency curves of some statistics with the variate axis at the ends of the range even though the actual distributions are unknown. It will be shown here that if for a number $r_0$ in $\lbrack 0, 1\rbrack$ and close to $1, P(r^{\ast} \geqq r_0)$ is expanded in a series of powers of $(1 - r_0)$, the first non-zero coefficient is that of the power $(N - 2)/2$. Upper and lower bounds for the coefficient of this power will be calculated. The lower bound is positive and the upper bound gives an approximation for an upper bound on $P(r^{\ast} \geqq r_0)$.