Monte carlo integration using simpson rule

Abstract

This work considers Monte Carlo method for approximating the integral of any four times differentiable function f over a unit interval [0,1]. Whereas earlier Monte Carlo schemes have yielded on $n^{-1},\;, n^{-3},\; n^{-4}$, or $n^{-5}$ convergence rate for the expected square error, this thesis shows that by allowing nonlinear operations on the random samples ${(U_i,f(U_i)}\;}_{i=1}^n$ much more rapid convergence can be achieved. Specifically, the new scheme attains the rate of convergence $n^{-8}$ by the Simpson rule based on an ordered random sample.