Fixed-domain asymptotic properties of tapered maximum likelihood estimators

When the spatial sample size is extremely large, which occurs in many environmental and ecological studies, operations on the large covariance matrix are a numerical challenge. Covariance tapering is a technique to alleviate the numerical challenges. Under the assumption that data are collected along a line in a bounded region, we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator (MLE) for the microergodic parameter in the Matern covariance function by establishing the fixed-domain asymptotic distribution of the exact MLE and that of the tapered MLE. Our results imply that, under some conditions on the taper, the tapered MLE is asymptotically as efficient as the true MLE for the microergodic parameter in the Matern model.

[1]  D. Sengupta.  Linear models , 2003 .

[2]  Michael L. Stein,et al.  Bounds on the Efficiency of Linear Predictions Using an Incorrect Covariance Function , 1990 .

[3]  Michael L. Stein,et al.  Asymptotically Efficient Prediction of a Random Field with a Misspecified Covariance Function , 1988 .

[4]  W. Rudin.  Real and complex analysis , 1968 .

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

[6]  Zhe Jiang,et al.  Spatial Statistics , 2013 .

[7]  M. Stein.  Predicting random fields with increasing dense observations , 1999 .

[8]  Jackson B. Lackey,et al.  Errata: Handbook of mathematical functions with formulas, graphs, and mathematical tables (Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964) by Milton Abramowitz and Irene A. Stegun , 1977 .

[9]  K. Mardia,et al.  Maximum likelihood estimation of models for residual covariance in spatial regression , 1984 .

[10]  Z. Ying.  Maximum likelihood estimation of parameters under a spatial sampling scheme , 1993 .

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

[12]  Holger Wendland,et al.  Error Estimates for Interpolation by Compactly Supported Radial Basis Functions of Minimal Degree , 1998 .

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

[14]  Zongmin Wu,et al.  Compactly supported positive definite radial functions , 1995 .

[15]  J. Schmee.  Matrices with Applications in Statistics , 1982 .

[16]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[17]  Sergio Pissanetzky,et al.  Sparse Matrix Technology , 1984 .

[18]  John R. Gilbert,et al.  Sparse Matrices in MATLAB: Design and Implementation , 1992, SIAM J. Matrix Anal. Appl..

[19]  Steven E. Koch,et al.  An Overview of the International H2O Project (IHOP_2002) and Some Preliminary Highlights , 2004 .

[20]  Wei-Liem Loh,et al.  Fixed-domain asymptotics for a subclass of Matern-type Gaussian random fields , 2005, math/0602302.

[21]  D. Nychka,et al.  Covariance Tapering for Interpolation of Large Spatial Datasets , 2006 .

[22]  Michael L. Stein,et al.  Uniform Asymptotic Optimality of Linear Predictions of a Random Field Using an Incorrect Second-Order Structure , 1990 .

[23]  Steven Kay,et al.  Gaussian Random Processes , 1978 .

[24]  F. Graybill,et al.  Matrices with Applications in Statistics. , 1984 .

[25]  Nathaniel E. Helwig,et al.  An Introduction to Linear Algebra , 2006 .

[26]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[27]  Tilmann Gneiting,et al.  Radial Positive Definite Functions Generated by Euclid's Hat , 1999 .

[28]  T. Gneiting.  Compactly Supported Correlation Functions , 2002 .

[29]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[30]  D. Zimmerman,et al.  Towards reconciling two asymptotic frameworks in spatial statistics , 2005 .

[31]  Zhiliang Ying,et al.  Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process , 1991 .

[32]  Michael L. Stein,et al.  A Comparison of Generalized Cross Validation and Modified Maximum Likelihood for Estimating the Parameters of a Stochastic Process , 1990 .