Numerical Approximation of Some Infinite Gaussian Series and Integrals

Abstract

The paper deals with numerical computation of the asymptotic variance of the so-called increment ratio (IR) statistic and its modifications. The IR statistic is useful for estimation and hypothesis testing on fractional parameter H ∈ (0, 1) of random process (time series), see Surgailis et al. [1], Bardet and Surgailis [2]. The asymptotic variance of the IR statistic is given by an infinite integral (or infinite series) of 4-dimensional Gaussian integrals which depend on parameter H. Our method can be useful for numerical computation of other similar slowly convergent Gaussian integrals/series. Graphs and tables of approximate values of the variances σp2(H) and σˆp2(H), p = 1, 2 are included.