Stability in the Neumann-Pearson theorem and its applications

This paper considers the problem of determining the state of the unobservable process of alternating observations of the Poisson point flow controlled alternating process. To solve this problem, we use the Lemma of Neyman-Pearson with approximated distributions corresponding to different states of alternating process. This Lemma is proving in the following assumptions. Theorem 1. Suppose that the unperturbed pj (x) and perturbed p j (x) conditional probabilities for some n* < N and all xx) - 9кРк(^х) > 6, £ = 1,.. .дЬ j(x), \р.(_х)-р.(х)\ < s, < 8. ⁽Pj⁽x) " Pj^(x)) xeX (N )\Xj (N ) xeX (N ) Then the Bayesian decision rule, defined by the unperturbed distributions Pj (x) , but applied to the perturbed distributions p j (x), satisfies the inequality for a probability of mistakenly rejecting the correct hypothesis: m a(5) < a(5) + ^ pj(x) + 8. ^j=¹ xeX (N*) Theorem 1 is applied to the stationary discrete Markov process y(t), t > 0, with the set of states 1, 2 and the intensity of transition Yj, from state 1 to state 2 and the transition intensity y₂. back. Random process y(t), t > 0, with the probability q = y₂/(Yi+Y2 ) equals 1 and with the probability q₂ = Yi + y₂) ^ecl^ual^s 2 · It splits the time axis t > 0 into politiskies О = T₀ x,T_x 2,..., so that on politiskies [7J,,Z[), [T₂,T₃),... the process y(t) accepts the value ^(0) and on politiskies \т_х,т₂), \T₃,T₄),... it takes the value y(7]). We believe that on politiskies \T_t, ), i > 0, we have Poisson points flows with the intensities X_yfj). Thus, we define the point flow л_г on the half interval [0, T). Using the number n(T) of flow A_r points on this half interval, it is required to build the procedure for the assessment of alternating process y(T) states. Here, as the unperturbed and the perturbed distributions we take (Х/У _Pj(x) = P(n(T) = x/y(0) = j, T_X>T) = exp(-X./)^j-, _Pj(x) = P(n(T) = x/y(T) = Д x = 0,1,... We construct estimates of the proximity of the unperturbed pj (x) and perturbed p j (x) distributions and derive inequalities for the similarity measures corresponding to these probability distributions, probabilities to reject the correct hypothesis. Numerical examples of the application of the approximate Bayesian decision rules are built in such a way.

Keywords

Authors

References