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On estimation of the error of Monte-Carlo technique caused by imperfections of the distribution of random numbers
G. A. Kozlov Moscow
Abstract:
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.
Received: 24.03.1970
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