Maxumum likelihood estimator of dead time duration in modulated synchronous twice stochastic flow of events

Consider the modulated synchronous twice stochastic flow of events, which rate is piecewise constant random process X(t) with two states: X_t, X₂ (X_t > X₂ ) . The time when process X(t) is staying in the i-th state has exponential probability density function with a parameter a_{, i = 1,2 . During the time random interval when X(t) = X _{ there is a Poisson flow with the rate X_{, i = 1,2 . A state transition of process X(t) occurs at the moment of arrival of a Poisson flow event, moreover, the passing from the first to the second state is realized with probability p, the passing from the second to the first state is realized with probability q. The flow functions in conditions of dead time, which is one of the falsifying factors of the state and the parameter estimation of the flow. After each registered event there is some time of fixed duration T (dead time), during which another flow events is inaccessible for observation. Consider non-extendable dead time. Such time is all events, which happen during the dead time interval without its prolongation. When the duration of the dead time period finishes, the first happened event creates the dead time period of duration T again, and etc. Note that for the flow, which functions in dead time conditions, events are observable if they did not get into the dead time interval. For the flow, which functions in dead time absence, all events are observable. One of confounding factor by the flow state and parameter estimation is the dead time of recording device, which is generated by the observable flow of events. All other events occurred during the dead time interval are not accessible for observation. The main purpose of this paper is to estimate the dead time T. On the base of the maximum likelihood function method, the solution of the problem is obtained. It is shown that the likelihood function L(T | т,...,xk)) reaches its maximum at the point T = x_m = min T_k (k = 1, n) . So, the solution of optimization problem is the dead time estimator T = x_m .

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