Comparison of MM and ML estimation of dead time period in Modulated MAP

This paper considers the modulated MAP, which rate is a piecewise constant random process X(t) with two states: X(t) = X₁ and X(t) = X₂ (X₁ > X₁ > 0). The time when the process X(t) remains at the i-th state, i = 1,2, depends on two random values: 1) the first random value has the exponential distribution function F/- = 1 - e, i = 1,2, when the i-th state ends process X(t) transits with the probability equal one from the i-th state to the j-th one, i, j = 1,2 (i Ф j); 2) the second random value has the exponential distribution function F, = 1 - e, i = 1,2; when the i-th state ends, process X(t) transits with the probability P₁ (Xj | X,) from the i-th state to the j-th one (i Ф j) by a flow event occurs or X(t) transits with the probability P₀ (Xj | X,) from the i-th state to the j-th (i ^ j) by an event does not occur, or the process X(t) transits with the probability P₁ (X, | X,) from the i-th state to the i-th one by a flow event occurs. Here P₁ (Xj | X,) + P₀ (Xj | X,) + P₁ (X, | X,) = 1, i, j = 1,2, i Ф j. The block transition rate matrix for the process X(t) is as follows: - (1 +X1) a-1 +X1P0(X₂| X1) a₂ +X₂01 X22 +2 X1P1 (XJX1) X1P1 (X ₂ | X1) X2P (XJX2 ) X2P1 (X2 | X2 ) = 1D0 | DJ|. D = Occurring an event generates the period of time called dead time, during which the flow cannot be observed. After this period ends, a new event also generates dead time. Having only a sample (t_b..., t_n) of events moments, we need to estimate the dead time period applying the two methods: method of maximum likelihood and method of moments. The ML-estimator can be found by maximizing the following function: L(,...x) = i\p_T (xmax,Xm_in > T > 0. 1=1 The moments equation is as follows: ^ ZxkXk+1 k1 = ^Pe^+*f)(1 - f(T)))-L--L] П - 1 k=1 V П k=1 ) z₁ z ₂ ^ z₁ z ₂) where the sample covariance is on the left part of the equation and the theoretical covariance is on the right one. The estimator of the dead time period can be obtained by solving the equation above. To compare the results of dead time period estimation for the two methods and to find dependencies between estimates quality and process parameters, the numerical experiments was performed. The results show that MM-estimate is better than ML-estimate when the observability time is small enough (10-25 points) and the parameter X₂ is also small (X₂ < 0,5). But as observability time or parameter X₂ increases, ML-estimate becomes better than MM-estimator.

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