Some New Perspectives on the Method of Control Variates
The method of control variates is one of the most widely used variance reduction techniques associated with Monte Carlo simulation. This paper studies the method of control variates from several different viewpoints, and establishes new connections betwee
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Introduction
Suppose that a simulation analyst wishes to compute a quantity 0: that can be expressed as the expectation of a real-valued random variable (rv) X, so that 0: = EX . The conventional sampling-based algorithm for computing 0: involves simulating n independent and identically distributed (iid) copies of the rv X , denoted X 1> . •• ,Xn- The corresponding estimator for 0: is then just the sample mean Xn = n- l L~=l Xi . However, in many situations, the analyst can take advantage of existing problem structure so as to create a more efficient means of computing 0: . One powerful approach to exploiting problem structure is the method of control variates. Specifically, suppose that there exists a random variable Y, jointly distributed with X, for which EY is known. Then, the control variate C = Y - EY is guaranteed to be a "mean zero" random variable, so that X('x) = X -'xC is an estimator for 0: . Consequently, this creates the possibility of computing 0: by generating iid copies (Xl, C I ) , · · · ,(Xn, Cn) of the pair (X, C) and estimating 0: via Xn('x) = 'xCn, where C n = n- l L~=l C i and ,X E lR. is an arbitrary scalar. By choosing the "control coefficient" ,X judiciously, one can therefore obtain a variance reduction relative to the conventional estimator
x; -
x;
In particular, it is natural to choose ,X so as to minimize the variance of X('x) . The variance-minimizing choice is just ,X*
= cov(X,C)jvarC.
K.‒T. Fang et al. (eds.)., Monte Carlo and Quasi-Monte Carlo Methods 2000 © Springer-Verlag Berlin Heidelberg 2002
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Since A* involves moment quantities that are generally unknown to the analyst, it must be estimated somehow. Fortunately, one of the great strengths of Monte Carlo sampling-based methodology is its ability to internally estimate such problem-dependent parameters by using the sample moments of the (Xi, Gi)'S to compute the population moments of (X, G). In particular, A* can be estimated via
thereby suggesting the estimator O:l(n) = X n -A1(n)Cn . This estimator for 0: can easily be implemented in many practical problem settings. In particular, relative to the conventional estimator Xn , the estimator 0:1 (n) requires only that the simulation code collect the control variate outcomes G1 , · · · ,Gn , compute A1(n), and form O:l(n). Note that this variance-reduction method is "non-invasive", in the sense that it requires only that additional statistics be collected during the simulation run, and it does not require that the simulation analyst modify the code that is used to generate the Xi'S themselves. Thus, control variates can be implemented (for example) in the background, while some visualization involving the Xi'S proceeds in the foreground. Because of the relative ease with which control variates can be implemented, it is perhaps the most widely applied of all variance reduction techniques. It is therefore of some interest and importance to understand this method in greater depth. This paper is intended to provide new insights into the foundations of the method of contr
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