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The likelihood ratio method allows estimation of the expected performance and associated derivatives of a stochastic system for various values of the parameters from a single simulation run. In this method the choice of the reference value V. of the parameter vector, which is employed in the generated sample, is crucial. In this paper we discuss how to choose the reference vector V. in order to minimize the variance of the considered likelihood ratio estimators. We show that for exponential families of distributions this variance is a convex function of V.. We also show that for some exponential families (e.g. Gamma distributions) there is an order relation between the involved parameters and their optimal reference values. 1 INTRODUCTION Let l(v) = IEV{L(Y)} = / qY)f(Y,'v)~Y (1) be the expected performance of a stochastic system with L(Y) being a sample performance driven by an input vector Y having a probability distribution function f(~, v). Assume that l(v) is too complex, and must be evaluated by simulation. To do so, one generates a random sample Y1 ,..., YN from ~(y,-u) for a particular value v from a permissible parameter set V and then estimates J?(v) by the sample mean ijv(v) = N-'' &(Yi). (2) 2'=1 Of course, such a procedure requires generation of a new sample every time t(v) should be estimated for a new value of the parameter vector v, The likelihood ratio method on the other hand allows estimation of l(v) for various values of v from a single simulation run, That is, let us choose a particular V. c V and consider the likelihood ratio function Pv(y, v) = f (Y, v) f(Y>vo) Then it is not difficult to show that l(v) = B3&{qz)w(z,v)}, (3) where Z * f (z, Vo). We shall call V. the reference parameter vector. By generating now a sample 21,. .. . Z~ from f (z, Vo) we can estimate-t(v) by the corresponding sample mean f?jv(v)= r-ifL(zi)w'(zi, v). (4) i=l Moreover, under suitable regularity conditions ensuring interchangeability of the differentiation and integration operators in the right hand side of (3) we can estimate the gradient V/(W) of the expected performance by VZN(V) = fv-l '&(zi)vw(zi, v). (5) i=l The above approach can be extended to dynamic systems as follows. Consider a Discrete Event Dynamic System (DEDS) driven by the input sequence of independent identically distributed random vectors YI, Y2,. .. . with the common …

On optimal choice of reference parameters in the likelihood ratio method