Stochastic Interpolatory Rules for $U_n$

It is useful to have methods for computing error estimates for integration rules that are used in practical computations. One traditional method for error estimation relies on the use of differences between successive values in the sequence $Q^{(1,n)}(f), Q^{(2,n)}(f), \ldots$. This method could be used with the family of rules described in the previous section, but it is sometimes unreliable, and sometimes infeasible for large values of $m$ and $n$. A second class of methods for error estimation uses randomization. There are two common methods for randomization of integration rules. The first method uses random copies of the complete rule, and the second method uses comparisons between the integrand at random points from the integration region and the polynomial model for the rule. The use of these two error estimation methods for the $Q^{(m,n)}$ rules is discussed in the following subsections.