Asymptotic variance of Newton–Cotes quadratures based on randomized sampling points

Abstract

We consider the problem of numerical integration when the sampling nodes form a stationary point process on the real line. In previous papers it was argued that a naïve Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. We show that this inflation can be avoided using a higher-order Newton–Cotes quadrature rule which exploits smoothness properties of the integrand. Under mild assumptions, the resulting estimator is unbiased and its variance asymptotically obeys a power law as a function of the mean point distance. If the Newton–Cotes rule is of sufficiently high order, the exponent of this law turns out to only depend on the point process through its mean point distance. We illustrate our findings with the stereological estimation of the volume of a compact object, suggesting alternatives to the well-established Cavalieri estimator.

Improving Cavalieri volume estimation based on non‐equidistant planar sections: The trapezoidal estimator

“…Cavalieri estimation is thus solving the problem of estimating the integral $$Q=\int _{\mathbb {R}} f(x) dx$$ from finitely many values of f at random sampling points. This mathematical formulation of the problem is used in the papers 1–3, which we will repeatedly refer to.…”

“…4, 9, 10 and Ref. 2 of ( m , 1)‐piecewise smoothness, which additionally requires that the derivative of order $$m+1$$ must in fact have finite jumps. However, in Ref.…”

“…The following result provides a basic characterization of the variance of the trapezoidal estimator based on slices with an average thickness T ; see [Ref. 2, Proposition 6.1] for details. In principle, the statement about the remainder is only valid for certain models of section positions [Ref.…”

“…In principle, the statement about the remainder is only valid for certain models of section positions [Ref. 2, Definition 2.1], however, the models considered in this paper fulfill this requirement. In particular, the result also covers the case of the Cavalieri estimator () if the sections are equidistant with a distance T apart.…”

“…We also describe how the variance of the trapezoidal estimator can be estimated in different realistic scenarios and that it even can be used if dropouts occur. Part of this paper serves as a more practitioner‐oriented description of the mathematically rigorous results of papers 1 , 2 , 3 , whereas in particular the formulas for variance estimation are novel.…”