General Autocorrelation Function.

An example of a class of statistics of the form (8.1) where $ \{{\bf p}_m\}$ and $ \{q_m\}$ are all zero are ACF estimates

$\displaystyle r_t = \frac{1}{N}\; \sum_{i=t+1}^N \; x_i \; x_{i-t}, \;\;\; 0\leq t \leq N-1.$ (8.7)

Suppose we are interested only in a selected set of ACF samples at delays $ t_1, t_2 \ldots t_M$. The problem is to obtain the joint PDF of the feature vector

$\displaystyle {\bf z}= [r_{t_1}\; r_{t_2} \ldots r_{t_M}]^\prime,
$

denoted by

$\displaystyle p(r_{t_1}, r_{t_2} \ldots r_{t_M} ; H_0).
$

The elements of $ {\bf z}$ can be written as quadratic forms

$\displaystyle z_m = {\bf x}^\prime {\bf P}_{t_m} {\bf x}, \;\;\; 1\leq m \leq M,
$

where

$\displaystyle \begin{array}{r}
{\bf P}_0 = \frac{1}{N}\;\left[ \begin{array}{c...
...\cdots \\
\vdots & \vdots & \vdots & \vdots
\end{array}\right],
\end{array}$

and so forth. The pattern is such that $ {\bf P}_k$ is nonzero only on the super- and sub-diagonals spaced $ k$ away from the main diagonal.

If the sample mean is subtracted from $ {\bf x}$ prior to calculation of the ACF estimates, the quadratic forms (8.1) still hold, but the elements of $ \{{\bf P}_k\}$ are changed. For example, the $ j,k$-th element of $ {\bf P}_0$ is now $ \delta_{jk} - 1/N$ instead of $ \delta_{jk}$, where $ \delta_{jk}$ is the Kronecker delta; the remaining matrices $ \{{\bf P}_k\}$ are more complicated, but each element in the matrices can be evaluated by means of a single sum.

Baggenstoss 2017-05-19