We prove the asymptotic normality of the number of absent s-chains of identical outcomes in the equiprobable polynomial scheme under the condition that the number of trials n and the number of outcomes N tend to infinity in such a way that .
We prove the asymptotic normality of the number of absent noncontinuous chains (chains with gaps) of outcomes of independent trials. The asymptotic normality of the number of absent chains of identical outcomes, which has been proved for equiprobable outcomes, is proved in the nonequiprobable case.
We obtain the expression of the statistic which is a linear function of the logarithm of the likelihood function and investigate properties of the mathematical expectation of this function. The results obtained can be applied to the estimation of instants of changes of the success probability in Bernoulli trials.
We consider a tuple of states of an (s – 1)-order Markov chain whose transition probabilities
depend on a small part of s – 1 preceding states. We obtain limit distributions of certain
χ2-statistics X and Y based on frequencies of tuples of states of the Markov chain. For the statistic X, frequencies of tuples of only those states are used on which the transition probabilities depend, and for the statistic Y, frequencies of s-tuples without gaps. The statistical test with statistic X which distinguishes the hypotheses H1 (a high-order Markov chain) and H0 (an independent equiprobable sequence) appears to be more powerful than the test with statistic Y . The statistic Z of the Neyman–Pearson test, as well as X, depends only on frequencies of tuples with gaps. The statistics X and Y are calculated without use of distribution parameters under the hypothesis H1, and their probabilities of errors of the first and second kinds depend only on the non-centrality parameter, which is a function of transition probabilities. Thus, for these statistics the hypothesis H1 can be considered as composite.