Some large-sample results on a modified Monte Carlo integration method

The large-sample properties of the modified Monte Carlo integration method with locally antithetic variates proposed by Haber (Math. Comput. 21 (1967) 388) are provided under the mild assumption that the integrand is a C1 function. Moreover, the asymptotic distribution of the modified Monte Carlo estimator is obtained under the same condition. Furthermore, we propose a consistent variance estimation method, which avoids the replicated procedure considered by Haber (1967) which reduces the efficiency of this integration technique. On the basis of the achieved results, in addition to the integration framework, the method may be conveniently applied in the environmental sampling setting.