Name
Abstract | ||
|---|---|---|
Recently, He and Owen (J R Stat Soc Ser B 78(4):917–931, 2016) proposed the use of Hilbert’s space filling curve (HSFC) in numerical integration as a way of reducing the dimension from \(d>1\) to \(d=1\). This paper studies the asymptotic normality of the HSFC-based estimate when using one-dimensional stratification inputs. In particular, we are interested in using scrambled van der Corput sequence in any base \(b\ge 2\) with sample sizes of the form \(n=b^m\), for which the sampling scheme is extensible in the sense of multiplying the sample size by a factor of b. We show that the estimate has an asymptotic normal distribution for functions in \(C^1([0,1]^d)\), excluding the trivial case of constant functions. The asymptotic normality also holds for discontinuous functions under mild conditions. Previously, it was only known that scrambled (0, m, d)-net quadratures enjoy the asymptotic normality for smooth enough functions, whose mixed partial gradients satisfy a Hölder condition. As a by-product, we find lower bounds for the variance of the HSFC-based estimate. Particularly, for non-trivial functions in \(C^1([0,1]^d)\), the lower bound is of order \(n^{-1-2/d}\), which matches the rate of the upper bound established in He and Owen (2016). |
Year | DOI | Venue |
|---|---|---|
2019 | 10.1007/s11222-017-9794-y | Statistics and Computing |
Keywords | Field | DocType |
Asymptotic normality, Hilbert’s space filling curve, Van der Corput sequence, Randomized quasi-Monte Carlo, Extensible grid sampling | Mathematical optimization,Normal distribution,Combinatorics,Upper and lower bounds,Van der Corput sequence,Numerical integration,Constant function,Hölder condition,Space-filling curve,Statistics,Mathematics,Asymptotic distribution | Journal |
Volume | Issue | ISSN |
29 | 1 | 1573-1375 |
Citations | PageRank | References |
0 | 0.34 | 7 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Zhijian He | 1 | 13 | 2.94 |
| Lingjiong Zhu | 2 | 19 | 7.41 |