The problem of discriminating hypotheses on the parameters of a generalized moving summation process
G. V. Proskurin
Abstract:
We consider a random process
$$\chi_t=L(x^1_t, x^1_{t+1},\dots, x^1_{t+n_1-1},\dots, x^r_t,\dots,x^r_{t+n_r-1}),\quad t=1, \dots,T,$$
where
$x^i_\tau$,
$i=1,\dots,r$,
$\tau=1,2,\dots$, are independent, identically distributed random variables,
$x^i_\tau\in\{0,1\}$,
$P\{x^i_\tau=0\}=(1+\theta)/2$,
$L$ is a linear Boolean function. It is proved that the lognormal distribution is the limit distribution of the likelihood ratio statistic for testing a simple hypothesis
$\theta=\delta>0$ on the basis of the sample
$\chi_t$,
$t=1,\dots,T$, against a simple hypothesis
$\theta=0$ as
$\delta\to0$.
Algorithms for calculating the parameters of the function
$L$, which determine the value of
$T$ sufficient to distinguish the hypotheses with errors tending to zero, are presented. It is shown that if
$r\geqslant 2$,
$\sum_{i=1}^r n_i\to\infty$, then the sufficient value of
$T$ is no less than
$\delta^{2k(L)}$ in order, where
$k(L)=O(n/\log n)$ depends on
$L$.
UDC:
519.2 Received: 27.10.1992