We approximate the integral of a smooth function on [0,1], where values are only known at n random points (i.e., a random sample from the uniform-(0,1) distribution), and at 0 and 1. Our approximations are based on the trapezoidal rule and Simpson's rule (generalized to the non-equidistant case), respectively. In the first case, we obtain an n2-rate of convergence with a degenerate limiting distribution; in the second case, the rate of con-vergence is as fast as n3½, whereas the limiting distribution is Gaussian then.
|Journal||Journal of Applied Mathematics and Stochastic Analysis|
|Publication status||Published - 1998|
Einmahl, J. H. J., & Zuijlen, van, M. C. A. (1998). On the approximation of an integral by a sum of random variables. Journal of Applied Mathematics and Stochastic Analysis, 11(2), 107-114. https://doi.org/10.1155/S1048953398000100, https://doi.org/10.1007/s00041-004-4031-4