The problem of discriminating hypotheses on the parameters of a~generalized moving summation process

G. V. Proskurin

Diskretnaya Matematika, Tome 5 (1993) no. 3, pp. 44-63.

Résumé

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 $L$.