Comparison of ML- and MM-estimations of period duration of dead time in modulated synchronous double stochastic flow of events

One considers the modulated synchronous twice stochastic flow of events, which rate is piecewise constant random process X(t) with two states: X₁, X₂ (X₁ > X₂ > 0) . The time when process X(t) is staying in i-state has exponential probability distribution function with the parameter a_i, i = 1,2 . During the time random interval when X(t) = X_i, there is the Poisson flow with the rate X_i, i = 1,2 . A state transition of process X(t) occurs in the moment of the Poisson flow event arrival, 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 state and parameter estimation of the flow. After each registered event, there is the time of fixed duration T (dead time), during which another flow events is inaccessible for observation. Now consider the nonextendable dead time, that is all events, which happen during dead time interval without its prolongation. At the end of the period duration of the dead time, the first happened event creates 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 dead time interval (events which cause its approach). For the flow which functions in dead time absence all events are observable. One of confounding factor during the flow state and parameter estimation is dead time of recording device, which is generated by observable flow event occurrence. All other events occurred during the dead time interval are not accessible for observation. The main purpose of the research work is to obtain dead time period estimator T using the maximum likelihood method and method of moments (ML-estimators and MM-estimators respectively) and to compare these estimators. During the investigation it was obtained that ML-estimator of dead time period duration takes a value T = %_m = min %_k, k = 1, n . It means that the solution of optimization problem is the dead time estimate T = x_m . Also, it was designed an algorithm of the single MM-estimator obtaining. On the base of these methods it was realized a computing program and it was made a statistical experiment during which for different values of input flow parameters 1, 2, P, q, T there was computed the values of ML-estimates TjMj and MM-estimates T, j = 1, N (N is a number of experiment steps). For the values obtained there are calculated the values of ^ N ^ ^ N ^ _ sample variances V_Mn = (1/N)^(T - T), TV_MM = (1/N)^ ОмМ - T) , j = 1, N . Then it was chosen which of estimates is j=1 j=1 better: if V_Mn < V_MM then ML-estimate is better than MM-estimate, otherwise MM-estimate is better than ML-estimate.

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