Statistics and Its Interface

Volume 10 (2017)

Number 2

Box dimension estimation of multi-dimensional random fields via wavelet shrinkage

Pages: 207 – 216



Ali Reza Taheriyoun (Department of Statistics, Shahid Beheshti University, Evin, Tehran, Iran)

Yazhen Wang (Department of Statistics, Medical Science Center, University of Wisconsin, Madison, Wi., U.S.A.)


Computation of the box dimension for high-dimensional surfaces is much more complicated than the one-dimensional case. To obtain a fast computation, we employ the relationship between the box dimension and the wavelet coefficients of a surface through the local oscillation. This approach gives an appropriate consistent estimator of box dimension for noisy paths. The behavior of convergence is also studied under Hölder continuity assumption for the family of index-$\beta$ Gaussian fields. We show that the precision of the estimation procedure is not affected by the growth of dimension of the sample path. We finally examine the properties of the proposed estimator using a simulation study and a real dataset on equlibration of a particular solution.


box dimension, Hölder continuity, index-$\beta$ Gaussian field, spatial adaptation, wavelet

2010 Mathematics Subject Classification

Primary 60Dxx, 60G60. Secondary 65Txx.

Published 31 October 2016