Asymptotic Power of Certain Test Criteria (Based on First and Second Differences) for Serial Correlation between Successive Observations

Abstract

Given $x_i, i = 1, 2, \cdots, n$, a sequence of $n$ observations, the following statistics for measuring dispersion are defined as usual: \begin{equation*} \begin{align*}s^2 &= \frac{1}{n - 1} \sum^n_{i = 1} (x_i - \bar x)^2,\qquad\big(\bar x = \sum^n_{i = 1} x_i/n\big) \\ \tag{1}\delta^2 &= \frac{1}{n - 1} \sum^{n - 1}_{i = 1} (x_i - x_{i + 1})^2,\qquad d = \frac{1}{n - 1} \sum^{n - 1}_{i = 1} |x_i - x_{i + 1}|, \\ \delta^2_2 &= \frac{1}{n - 2} \sum^{n - 2}_{i = 1} (x_i - 2x_{i + 1} + x_{i + 2}),\quad d_2 = \frac{1}{n - 2} \sum^{n - 2}_{i = 1} |x_i - 2x_{i + 1} + x_{i + 2}|.\end{align*}\end{equation*} Further, we define the following ratio criteria: \begin{equation*}\begin{align*} w^2 &= \delta^2/s^2,\qquad W = d/s,\\ \tag{2} w^2_2 &= \delta^2_2/s^2,\qquad W_2 = d_2/s, \\ u^2 &= \delta^2_2/\delta^2,\qquad U = d_2/d.\end{align*}\end{equation*} Von Neumann [12] proposed the ratio $w^2 = \delta^2/s^2$ for testing the randomness of the sequence against the alternatives of serial correlation or trend. $W = d/s$, which has the advantage of being simpler for computation, was suggested by Kamat [8] for the same purpose. Similarly, we can construct two more ratio criteria, $w^2_2 = \delta^2_2/s^2$ and $W_2 = d_2/s$, by using the second order differences. And, finally, using both first and second successive differences, it is possible to construct a third type of ratio criteria, $w^2_2 = \delta^2_2/s^2$ and $W_2 = d_2/s$, by using the second order differences. And, finally, using both first and second successive differences, it is possible to construct a third type of ratio criteria, $u^2 = \delta^2_2/\delta^2$ and $U = d_2/d$, which may also be used for detecting serial correlation or trend in successive observations of the sequence. (See Tintner [11]). Under the hypothesis of randomness, when the $x_i$ are independent normal $(\mu, \sigma)$, the distributions of the ratio statistics $w^2, w^2_2, W$ and $W_2$ are known, either in exact or approximate form (Von Neumann [12], Kamat [8] and [9].) Under the same hypothesis, Kamat [9] has shown that they are all asymptotically normal. (See also Anderson [1], Hsu [6], and Dixon [3].) The distributions, however, do not appear to have been discussed under any alternative hypotheses of non-randomness. As to comparisons of their discriminating powers, Anderson [1] has shown that, against the alternatives defined by the density function \begin{equation*}\begin{align*}p(x_1, x_2, \cdots, x_n) &= K \exp\big\lbrack -\frac{1}{2\sigma^2}\big\{(1 + \rho^2) \sum^n_{i = 1} (x_i - \mu)^2 \\ -2\rho \sum^{n - 1}_{i = 1} (x_i - \mu)(x_{i + 1} - \mu)\big\}\big\rbrack, \\ \text{or by} \\ p(x_1, x_2, \cdots, x_n) &= K \exp\big\lbrack -\frac{1}{2\sigma^2}\big\{(1 + \rho^2 - \rho)((x_1 - \mu)^2 + (x_n - \mu)^2) \\ &+ (1 + \rho^2) \sum^{n - 1}_{i = 2} (x_i - \mu)^2 - 2\rho \sum^{n - 1}_{i = 1} (x_i - \mu_(x_{i + 1} -\mu) \big\} \big\rbrack, \end{align*}\end{equation*} a statistic linearly dependent on $w^2$ is the best criterion (in the Neymann-Pearson framework) to test the hypothesis of randomness, that is to test $\rho = 0$. As mentioned by Anderson, these density functions are modifications of the density function of the circular autoregressive model, $x_i - \mu = \rho(x_{i - 1} - \mu) + e_i$, where the $e_i$ are independent normal $(0, \sigma),$ and are obtained from it by modifying the end terms in squares and products. For the autoregressive model, however, he has shown that no such best criterion exists. We are not aware of any other work on the power of these criteria. In this paper, we discuss the distributions of these ratio criteria under another alternative of serial dependence between successive observations, specified in (4) below, when the size of the sample sequence is large. It should be noted that this alternative of serial dependence is different from the alternatives considered by Anderson, and that it is not related to the autoregressive model mentioned above. First, in Section 2, we show that all these ratio criteria are asymptotically normally distributed under this alternative hypothesis. In Section 3, we obtain the means and the variances of the limiting normal distributions. In Section 4, we present numerically the power for three samples sizes, $n = 100, 200 \text{and} 400,$ and for some positive values of the serial correlation coefficient. Power curves are also exhibited for $n = 200 \text{and} 400.$ Finally, in Section 5, we compare the relative efficiencies of these ratio criteria by using the Pitman criterion of asymptotic relative efficiency, as extended by Noether [10].