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Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling | | Fred J. Hickernell
; Lan Jiang
; Yuewei Liu
; Art Owen
; | | Date: |
21 Aug 2012 | | Abstract: | Monte Carlo methods are used to approximate the means, $mu$, of random
variables $Y$, whose distributions are not known explicitly. The key idea is
that the average of a random sample, $Y_1,..., Y_n$, tends to $mu$ as $n$
tends to infinity. This article explores how one can reliably construct a
confidence interval for $mu$ with a prescribed half-width (or error tolerance)
$varepsilon$. Our proposed two stage algorithm assumes that the
emph{kurtosis} of $Y$ does not exceed some user-specified bound. An initial
independent and identically distributed (IID) sample is used to confidently
estimate the variance of $Y$. A Berry-Esseen inequality then makes it possible
to determine the size of the IID sample required to construct the desired
confidence interval for $mu$. We discuss the important case where $Y=f(vX)$
and $vX$ is a random $d$-vector with probability density $
ho$. In this case
$mu$ can be interpreted as the integral $int_{
eals^d} f(vx)
ho(vx) ,
dif vx$, and the Monte Carlo method becomes a method for multidimensional
cubature. | | Source: | arXiv, 1208.4318 | | Services: | Forum | Review | PDF | Favorites | |
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