Estimation of the parameter of unextendable dead time random duration in the Poisson flow of events

We consider a Poisson flow of intensity events X. After each event registered at time tk, a dead time period of random duration occurs during which other events of the flow are unavailable to the observation and do not cause it to be prolonged (unextendable dead time). At the end of the dead time period, the first event that occurred again creates a period of dead time of random duration, etc. It is assumed that the dead time duration is distributed exponentially with the parameter a. The stationary mode of operation of the observed event flow is studied, therefore we neglect the transient processes on the observation half-line (t₀, t], where t₀ is the beginning of the observations, t is the end of the observations, then without loss of generality we can set t₀ = 0. It is assumed that the parameters of the observed flow are unknown. It is necessary, at the end of the observations (at time t), based on the sample t_b t₂, .., t_n the observed moments of the onset of the flow events, to make the maximum likelihood method and the method of moments constructing the estimates X , a and to compare the quality of the estimates obtained. The likelihood function is written out L(a, X|T1,...,Tn) = VT ПИ -e^-Xlk) V ^{X - a} J k=1 and likelihood equations na ^ e^alk_n nX ^ e^Xlk --+ 7 -= 0,--7 ;-= 0, X(X - a) k==1 e ^k - e ^k a(X - a) _k=₁ e ^k - e ^k where, i_k = t_k+₁ - tk , k = 1,2,..., is the sequence of the values of the lengths of intervals between adjacent events of the flow measured during the observation of the flow during the observation interval (0, t]. It is shown that the maximum likelihood method, regardless of the ratio of the unknown parameters X and a , provides estimates X and a in the form a = X = 2/1, i.e. its performance only applies to the option a = X . For the method of moments it is proved that the equations of moments ¹ Va² aX X²J ² relatively unknown parameters and have no solution. For the case of a known parameter X , an unknown parameter a is estimated both by the maximum likelihood method and by the moment method. In this situation, the numerical results show the advantage of the maximum likelihood method.

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