Joint asymptotics for estimating the fractal indices of bivariate Gaussian processes

Multivariate (or vector-valued) processes are important for modeling multiple variables. The fractal indices of the components of the underlying multivariate process play a key role in characterizing the dependence structures and statistical properties of the multivariate process. In this paper, under the infill asymptotics framework, we establish joint asymptotic results for the increment-based estimators of bivariate fractal indices. Our main results quantitatively describe the effect of the cross-dependence structure on the performance of the estimators.

[1]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

[2]  F. T. Wright,et al.  A Bound on Tail Probabilities for Quadratic Forms in Independent Random Variables , 1971 .

[3]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[4]  A. Wood,et al.  Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields , 2004, math/0406525.

[5]  P. Hall,et al.  Characterizing surface smoothness via estimation of effective fractal dimension , 1994 .

[6]  F. PASCUAL,et al.  ESTIMATION OF LINEAR CORRELATION COEFFICIENT OF TWO CORRELATED SPATIAL PROCESSES , 2006 .

[7]  Douglas W. Nychka,et al.  Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets , 2008 .

[8]  Hao Zhang,et al.  When Doesn't Cokriging Outperform Kriging? , 2015, 1507.08403.

[9]  Tatiyana V. Apanasovich,et al.  Cross-covariance functions for multivariate random fields based on latent dimensions , 2010 .

[10]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

[11]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[12]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[13]  Emilio Porcu,et al.  Classes of compactly supported covariance functions for multivariate random fields , 2015, Stochastic Environmental Research and Risk Assessment.

[14]  H. Wackernagle,et al.  Multivariate geostatistics: an introduction with applications , 1998 .

[15]  Andrew T. A. Wood,et al.  INCREMENT-BASED ESTIMATORS OF FRACTAL DIMENSION FOR TWO-DIMENSIONAL SURFACE DATA , 2000 .

[16]  A. Gelfand,et al.  Handbook of spatial statistics , 2010 .

[17]  Douglas W. Nychka,et al.  Nonstationary modeling for multivariate spatial processes , 2012, J. Multivar. Anal..

[18]  Andrew T. A. Wood,et al.  On the performance of box-counting estimators of fractal dimension , 1993 .

[19]  Chunsheng Ma,et al.  Vector random fields with compactly supported covariance matrix functions , 2013 .

[20]  John T. Kent,et al.  Estimating the Fractal Dimension of a Locally Self-similar Gaussian Process by using Increments , 1997 .

[21]  Marc G. Genton,et al.  Cross-Covariance Functions for Multivariate Geostatistics , 2015, 1507.08017.

[22]  D. Surgailis,et al.  Measuring the roughness of random paths by increment ratios , 2008, 0802.0489.

[23]  Zhengyuan Zhu,et al.  PARAMETER ESTIMATION FOR FRACTIONAL BROWNIAN SURFACES , 2003 .

[24]  Ying Sun,et al.  A Valid Matérn Class of Cross-Covariance Functions for Multivariate Random Fields With Any Number of Components , 2012 .

[25]  E. J. G. Pitman.  On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin , 1968 .

[26]  Yimin Xiao.  Recent Developments on Fractal Properties of Gaussian Random Fields , 2013 .

[27]  J. Coeurjolly,et al.  HURST EXPONENT ESTIMATION OF LOCALLY SELF-SIMILAR GAUSSIAN PROCESSES USING SAMPLE QUANTILES , 2005, math/0506290.

[28]  Pierre-Olivier Amblard,et al.  Identification of the Multivariate Fractional Brownian Motion , 2011, IEEE Transactions on Signal Processing.

[29]  A. Yaglom.  Correlation Theory of Stationary and Related Random Functions I: Basic Results , 1987 .

[30]  Juan Du,et al.  Asymptotic properties of multivariate tapering for estimation and prediction , 2015, J. Multivar. Anal..

[31]  T. Gneiting,et al.  Matérn Cross-Covariance Functions for Multivariate Random Fields , 2010 .

[32]  Dimension Results for Gaussian Vector Fields and Index-$\alpha$ Stable Fields , 1995 .

[33]  Chae Young Lim,et al.  Properties of spatial cross-periodograms using fixed-domain asymptotics , 2008 .

[34]  Emilio Porcu,et al.  Radial basis functions with compact support for multivariate geostatistics , 2013, Stochastic Environmental Research and Risk Assessment.

[35]  V. Mandrekar,et al.  Fixed-domain asymptotic properties of tapered maximum likelihood estimators , 2009, 0909.0359.

[36]  R. Adler.  The Geometry of Random Fields , 2009 .

[37]  Wei-Liem Loh.  Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations , 2015, 1510.08699.

[38]  Haotian Hang,et al.  Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics , 2004 .

[39]  T. Gneiting,et al.  Estimators of Fractal Dimension : Assessing the Roughness of Time Series and Spatial Data , 2010 .

[40]  M. D. Ruiz-Medina,et al.  Equivalence of Gaussian measures of multivariate random fields , 2015, Stochastic Environmental Research and Risk Assessment.

[41]  Zhiliang Ying,et al.  INFILL ASYMPTOTICS FOR A STOCHASTIC PROCESS MODEL WITH MEASUREMENT ERROR , 2000 .