Integral Evaluation Using the Δ2-distribution. Simulation and Illustration

Abstract. The ∆2-distribution is a multivariate distribution, which plays an important role in variance reduction of Monte Carlo integral evaluation. Selecting the nodes of random cubature formulae according to ∆2 ensures an unbiased and efficient estimate of the studied integral regardless of the region it is solved over. The ∆2 distribution is also relevant in problems such as separating errors in regression analysis and constructing D-optimal designs in multidimensional regions. Inefficient simulation of ∆2 prevented the application of the underlying theory in real problems. Ermakov and Missov, [3], proposed an algorithm which combines all rejection, inversion, and mixture techniques. Its complexity allows simulating ∆2 vectors of big lengths. Moreover, it works in the most general settings of the problem of integral evaluation. This article presents a modification of the simulation algorithm as well as its illustration for a popular integral in Reliability Theory.