On the distribution of integration error by randomly-shifted lattice rules
On the distribution of integration error by randomly-shifted lattice rules
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DOI：
10.1214/10-ejs574
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发表时间：
2010
影响因子：
1.1
通讯作者：
P. L'Ecuyer;D. Munger;B. Tuffin
中科院分区：
文献类型：
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作者：
P. L'Ecuyer;D. Munger;B. Tuffin
随机移格的格律估计一个数学期望，写成一个在s维单位超立方体上的积分，通过在单位超立方体内移格的n个点上对被积函数进行n次求值的平均值。该平均值提供了积分的无偏估计，并且在被积函数上适当的平滑条件下，它作为n的函数收敛速度比在n个独立随机点（标准蒙特卡洛估计）的平均值更快。在本文中，我们研究了估计误差作为随机位移的函数的行为，以及随机位移在不同设置下的分布。众所周知，当n !时蒙特卡罗估计量服从中心极限定理。由于函数求值之间有很强的依赖性，所以随机支配的格律就不能。我们证明了对于一维被积函数的简单情况，如果被积函数是非周期的，其极限误差分布在有界区间上是均匀的，如果被积函数是周期的，其极限误差分布在有界区间上是平方根形式。我们发现，在高维中，在一般情况下，用一种计算置信区间的有用方法来精确表征极限分布的希望很小。然而，我们从不同的角度和不同的例子来研究这个误差作为随机位移的函数是如何表现的。我们还指出了当维度趋于无穷时经典中心极限定理成立的一种情况，我们提供了误差分布不应离正态太远的指导方针，并且我们在实际应用中启发的示例中检查了误差分布离正态有多远。
A lattice rule with a randomly-shifted lattice estimates a math- ematical expectation, written as an integral over the s-dimensional unit hy- percube, by the average of n evaluations of the integrand, at the n points of the shifted lattice that lie inside the unit hypercube. This average provides an unbiased estimator of the integral and, under appropriate smoothness conditions on the integrand, it has been shown to converge faster as a func- tion of n than the average at n independent random points (the standard Monte Carlo estimator). In this paper, we study the behavior of the esti- mation error as a function of the random shift, as well as its distribution for a random shift, under various settings. While it is well known that the Monte Carlo estimator obeys a central limit theorem when n ! 1, the ran- domized lattice rule does not, due to the strong dependence between the function evaluations. We show that for the simple case of one-dimensional integrands, the limiting error distribution is uniform over a bounded in- terval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. We find that in higher dimen- sions, there is little hope to precisely characterize the limiting distribution in a useful way for computing confidence intervals in the general case. We nevertheless examine how this error behaves as a function of the random shift from different perspectives and on various examples. We also point out a situation where a classical central-limit theorem holds when the dimen- sion goes to infinity, we provide guidelines on when the error distribution should not be too far from normal, and we examine how far from normal is the error distribution in examples inspired from real-life applications.