On estimation of the error of Monte-Carlo technique caused by imperfections of the distribution of random numbers

An approach to estimation of the Monte-Carlo technique error caused by imperfections of the distribution of random numbers is proposed. The approach is illustrated by an example of the simple integral $\overline\varphi=\int_0^1\varphi(x)\,dx$ calculation by the method of indeopendent tests. The error is estimated by
$$ S=\sup U(\varphi),\quad\varphi\in G,\quad U(\varphi)=\Bigl(\int_0^1\varphi(x)\,dF(x)-\overline\varphi\Bigr)\bigg/\sqrt{\int_0^1(\varphi(x)-\overline\varphi)^2\,dx}, $$
where $F$ is the distribution function of random numbers in the interval $[0,1]$, $G$ is the class of functions with finite “standartized variation”:
$$ G=\biggl\{\varphi\colon\bigvee_0^1\varphi\bigg/\sqrt{\int_0^1(\varphi(x)-\overline\varphi)^2\,dx}\le v\biggr\}. $$

It is shown that the problem of determining the value $S$ can be reduced to a variational problem of finding the function that minimizes the functional $U(\varphi)=\int_0^1\varphi\,dF$ under the following restrictions:
$$ \int_0^1\varphi\,dx=0,\quad\int_0^1\varphi^2\,dx=1\quad\text{and}\quad\bigvee_0^1\varphi\le v $$
A solution of this variational problem is given.