In discriminating stochastic processes there arises a need of observation data reduction concerning the length of the realization to be considered as well as the variety (alphabet) of the instantaneous process states to be identified. In the paper a method for such data compression is given based on the theory of asymptotic discernibility of two stationary random processes as developed by the author for processes with memory.

Data compression in discriminating stochastic processes

In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.

https://projecteuclid.org/download/pdf_1/euclid.aoms/1177729330

A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

1. Summary. Simple hypotheses HP and HQ, specifying two distinct positive transition densities p(x I y) and q(x I y) and initial densities p(x) and q(x) with respect to a finite Lebesgue-Stieltjes measure, are assumed for a discrete time parameter Markov process. Let Rn be the likelihood ratio static based on the first n + 1 observations of the process, and consider the class of sequences of likelihood ratio tests T(a, a) = {[Rn > naa]: n = 0, 1, 2, *.} generated by letting a and a vary over the real numbers. Under certain regularity assumptions on Kt(x, y) = p'-t(x y)qt(x I y) and the initial densities p and q, the subclass of consistent sequences is determined, and the limiting rates at which the error probabilities tend to zero for tests in this subclass are found. A definition of the best asymptotic rate for distinguishing between HP and HQ is made for the class of consistent tests. This "asymptotic rate of discrimination" is evaluated and is shown to be attained by a certain subclass of these tests. Some applications and extensions of the theory to infinite Lebesgue-Stieltjes measures are given. 2. Introduction. This paper is primarily a study of the asymptotic properties of the tail probabilities for the likelihood ratio statistic as applied to testing simple hypotheses for discrete time parameter Markov processes. Similar investigations have been carried out by Cram6r [2], Chernoff [1], and Thomasian 16] for sums of independent random variables and from more general points of view. In the special case that the Markov process reduces to a sequence of independent, identically distributed random variables, most of the results proved herein may be obtained from these papers. Let SC be the real line, 6 the Borel sets, and ,u a Lebesgue-Stieltjes measure defined on (SC, 6). Attention is restricted to the situation in which the distribution of a Markov process f = {Xn: n = 0, 1, 2, * * } is determined by a transition density p(x I y) (measurable in y for fixed x and a probability density with respect to It in x for fixed y) and an initial density p(x) with respect to 1t. Let Hp and HQ be simple hypotheses for W specifying transition densities p(x I y) and q(x I y) and initial densities p and q. The likelihood ratio statistic Rn based on the first n + 1 observations is then

/pdf/asymptotic-rate-of-discrimination-for-markov-processes-56cr8knxn8.pdf

Data compression in discriminating stochastic processes

References

A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

Asymptotic Rate of Discrimination for Markov Processes

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