Let the unknown
parameter of interest be \mu, and assume we have a
statistic m such that the
expected value of
m is μ: \mathbb{E}\left[m\right]=\mu, i.e.
m is an
unbiased estimator for μ. Suppose we calculate another statistic t such that \mathbb{E}\left[t\right]=\tau is a known value. Then :m^\star = m + c\left(t-\tau\right) \, is also an unbiased estimator for \mu for any choice of the coefficient c. The
variance of the resulting estimator m^{\star} is :\textrm{Var}\left(m^{\star}\right)=\textrm{Var}\left(m\right) + c^2\,\textrm{Var}\left(t\right) + 2c\,\textrm{Cov}\left(m,t\right). By differentiating the above expression with respect to c, it can be shown that choosing the optimal coefficient :c^\star = - \frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)} minimizes the variance of m^{\star}. (Note that this coefficient is the same as the coefficient obtained from a
linear regression.) With this choice, :\begin{align} \textrm{Var}\left(m^{\star}\right) & =\textrm{Var}\left(m\right) - \frac{\left[\textrm{Cov}\left(m,t\right)\right]^2}{\textrm{Var}\left(t\right)} \\ & = \left(1-\rho_{m,t}^2\right)\textrm{Var}\left(m\right) \end{align} where :\rho_{m,t}=\textrm{Corr}\left(m,t\right) \, is the
correlation coefficient of m and t. The greater the value of \vert\rho_{m,t}\vert, the greater the
variance reduction achieved. In the case that \textrm{Cov}\left(m,t\right), \textrm{Var}\left(t\right), and/or \rho_{m,t}\; are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain
least squares system; therefore this technique is also known as
regression sampling. When the expectation of the control variable, \mathbb{E}\left[t\right]=\tau, is not known analytically, it is still possible to increase the precision in estimating \mu (for a given fixed simulation budget), provided that the two conditions are met: 1) evaluating t is significantly cheaper than computing m; 2) the magnitude of the correlation coefficient |\rho_{m,t}| is close to unity. ==Example==