Control-variate estimators

Let $(\mathsf{X},\mathcal{X},P)$ be a probability space, let $q\ge 1$ and $n\ge 1$, and let $f:\mathsf{X}\to \mathbb{R}$ and $h=(h_1,\dots ,h_q):\mathsf{X}\to {\mathbb{R}}^q$ be measurable with $\int \mathrm{\mid }f\mathrm{\mid }dP<\mathrm{\infty }$ and $\int \mathrm{\mid }{h}_j\mathrm{\mid }dP<\mathrm{\infty }$ for every $j$; these are integrals in Expectation of a nonnegative or integrable random variable. For any integrable $g$ write $Pg:={\int }_{\mathsf{X}}gdP$. Write $\mu =Pf$ and $\theta =Ph\in {\mathbb{R}}^q$, where $\theta$ is known, and use Moments, variance, and covariance on a probability space for variance and covariance. Given an IID sample $X_1,\dots ,X_n\sim P$ as in Identical distribution and IID families and a fixed coefficient $\beta \in {\mathbb{R}}^q$, the control-variate estimator is

${\widehat{\mu }}_{\beta }=\frac{1}{n}{\sum }_{i=1}^n[f(X_i)-{\beta }^{\mathsf{T}}(h(X_i)-\theta )]\mathrm{.}$

For one scalar control this is $n^{-1}{\sum }_i[f(X_i)-\beta (h(X_i)-\theta )]$. The coefficient is fixed independently of the production sample in this definition; a coefficient fitted from that same sample is treated separately.