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.
Highlights
Let X1, . . . , Xn be n observations of the discrete time process (X(t), t ∈ Z) and let [mτ ]Sm(τ ) := Xt, t=1 τ ∈ [0, 1], m ∈ N be the partial sums process
The increment ratio (IR) statistic is useful for estimation and hypothesis testing on fractional parameter H ∈ (0, 1) of random process, see Surgailis et al [1], Bardet and Surgailis [2]
Our method can be useful for numerical computation of other similar slowly convergent Gaussian integrals/series
Summary
Let X1, . . . , Xn be n observations of the discrete time process (X(t), t ∈ Z) and let [mτ ]Sm(τ ) := Xt, t=1 τ ∈ [0, 1], m ∈ N be the partial sums process. Our method can be useful for numerical computation of other similar slowly convergent Gaussian integrals/series. 0, σp2(H) , p = 1, 0 < H < 3/4, p = 2, 0 < H < 1, Numerical Approximation of Some Infinite Gaussian Series and Integrals where σp2(H) :=
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