×

zbMATH — the first resource for mathematics

Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics. (English) Zbl 1418.62365
Summary: We study estimation and prediction of Gaussian random fields with covariance models belonging to the generalized Wendland (GW) class, under fixed domain asymptotics. As for the Matérn case, this class allows for a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divided into three parts: first, we characterize the equivalence of two Gaussian measures with GW covariance function, and we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn and GW covariance functions. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GW covariance model, under fixed domain asymptotics. The third part elucidates the consequences of our results in terms of (misspecified) best linear unbiased predictor, under fixed domain asymptotics. Our findings are illustrated through a simulation study: the former compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter with the given asymptotic distribution. The latter compares the finite-sample behavior of the prediction and its associated mean square error when using two equivalent Gaussian measures with Matérn and GW covariance models, using covariance tapering as benchmark.

MSC:
62M30 Inference from spatial processes
62F12 Asymptotic properties of parametric estimators
60G25 Prediction theory (aspects of stochastic processes)
62M20 Inference from stochastic processes and prediction
Software:
spam; George
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Abramowitz, M. and Stegun, I. A., eds. (1965). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series55. Superintendent of Documents, U.S. Government Printing Office, Washington, DC. · Zbl 0171.38503
[2] Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester. · Zbl 0478.60059
[3] Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W. and O’Neil, M. (2016). Fast Direct Methods for Gaussian Processes. IEEE Trans. Pattern Anal. Mach. Intell.38 252–265.
[4] Anderes, E. (2010). On the consistent separation of scale and variance for Gaussian random fields. Ann. Statist.38 870–893. · Zbl 1204.60041
[5] Askey, R. (1973). Radial characteristic functions. Technical Report, Research Center, Univ. Wisconsin-Madison, Madison, WI. · Zbl 0253.33009
[6] Bevilacqua, M., Faouzi, T., Furrer, R. and Porcu, E. (2019). Supplement to “Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics.” DOI:10.1214/17-AOS1652SUPP.
[7] Chernih, A. and Hubbert, S. (2014). Closed form representations and properties of the generalised Wendland functions. J. Approx. Theory177 17–33.
[8] Chernih, A., Sloan, I. H. and Womersley, R. S. (2014). Wendland functions with increasing smoothness converge to a Gaussian. Adv. Comput. Math.40 17–33. · Zbl 1298.41002
[9] Du, J., Zhang, H. and Mandrekar, V. S. (2009). Fixed-domain asymptotic properties of tapered maximum likelihood estimators. Ann. Statist.37 3330–3361. · Zbl 1369.62248
[10] Furrer, R., Bachoc, F. and Du, J. (2016). Asymptotic properties of multivariate tapering for estimation and prediction. J. Multivariate Anal.149 177–191. · Zbl 1341.62263
[11] Furrer, R., Genton, M. G. and Nychka, D. (2006). Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Statist.15 502–523.
[12] Furrer, R. and Sain, S. R. (2010). spam: A sparse matrix R package with emphasis on MCMC methods for Gaussian Markov random fields. J. Stat. Softw.36 1–25.
[13] Gneiting, T. (2002a). Compactly supported correlation functions. J. Multivariate Anal.83 493–508. · Zbl 1011.60015
[14] Gneiting, T. (2002b). Nonseparable, stationary covariance functions for space-time data. J. Amer. Statist. Assoc.97 590–600. · Zbl 1073.62593
[15] Golubov, B. I. (1981). On Abel–Poisson type and Riesz means. Anal. Math.7 161–184. · Zbl 0484.42004
[16] Hirano, T. and Yajima, Y. (2013). Covariance tapering for prediction of large spatial data sets in transformed random fields. Ann. Inst. Statist. Math.65 913–939. · Zbl 1273.62233
[17] Ibragimov, I. d. A. and Rozanov, Y. A. (1978). Gaussian Random Processes. Applications of Mathematics9. Springer, New York.
[18] Kaufman, C. G., Schervish, M. J. and Nychka, D. W. (2008). Covariance tapering for likelihood-based estimation in large spatial data sets. J. Amer. Statist. Assoc.103 1545–1555. · Zbl 1286.62072
[19] Kaufman, C. G. and Shaby, B. A. (2013). The role of the range parameter for estimation and prediction in geostatistics. Biometrika100 473–484. · Zbl 1284.62590
[20] Mardia, K. V. and Marshall, J. (1984). Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika71 135–146. · Zbl 0542.62079
[21] Matérn, B. (1960). Spatial Variation. Meddelanden Från Statens Skogsforskningsinstitut, Band 49, Nr 5., Stockholm.
[22] Porcu, E., Zastavnyi, V. P. and Bevilacqua, M. (2017). Buhmann covariance functions, their compact supports, and their smoothness. Dolomites Res. Notes Approx.10 33–42.
[23] Putter, H. and Young, G. A. (2001). On the effect of covariance function estimation on the accuracy of kriging predictors. Bernoulli7 421–438. · Zbl 0987.62061
[24] Schaback, R. (2011). The missing Wendland functions. Adv. Comput. Math.34 67–81. · Zbl 1229.41020
[25] Schoenberg, I. J. (1938). Metric spaces and completely monotone functions. Ann. of Math. (2) 39 811–841. · JFM 64.0617.03
[26] Skorokhod, A. V. and Yadrenko, M. I. (1973). On absolute continuity of measures corresponding to homogeneous Gaussian fields. Theory Probab. Appl.18 27–40. · Zbl 0282.60026
[27] Stein, M. (1988). Asymptotically efficient prediction of a random field with a misspecified covariance function. Ann. Statist.16 55–63. · Zbl 0637.62088
[28] Stein, M. (1990). Uniform asymptotic optimality of linear predictions of a random field using an incorrect second-order structure. Ann. Statist.18 850–872. · Zbl 0716.62099
[29] Stein, M. L. (1993). A simple condition for asymptotic optimality of linear predictions of random fields. Statist. Probab. Lett.17 399–404. · Zbl 0779.62093
[30] Stein, M. L. (1999a). Predicting random fields with increasing dense observations. Ann. Appl. Probab.9 242–273. · Zbl 0955.62095
[31] Stein, M. L. (1999b). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York. · Zbl 0924.62100
[32] Stein, M. L. (2004). Equivalence of Gaussian measures for some nonstationary random fields. J. Statist. Plann. Inference123 1–11. · Zbl 1057.60034
[33] Stein, M. L. (2013). Statistical properties of covariance tapers. J. Comput. Graph. Statist.22 866–885.
[34] Wackernagel, H. (2003). Multivariate Geostatistics: An Introduction with Applications, 3rd ed. Springer, New York. · Zbl 1015.62128
[35] Wang, D. and Loh, W.-L. (2011). On fixed-domain asymptotics and covariance tapering in Gaussian random field models. Electron. J. Stat.5 238–269. · Zbl 1274.62643
[36] Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math.4 389–396. · Zbl 0838.41014
[37] Wu, Z. (1995). Compactly supported positive definite radial functions. Adv. Comput. Math.18 283–292. · Zbl 0837.41016
[38] Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results. Springer, New York. · Zbl 0685.62077
[39] Zastavnyi, V. P. (2006). On some properties of Buhmann functions. Ukrainian Math. J.58 1184–1208. · Zbl 1116.42002
[40] Zhang, H. (2004). Inconsistent estimation and asymptotically equivalent interpolations in model-based geostatistics. J. Amer. Statist. Assoc.99 250–261. · Zbl 1089.62538
[41] Zhang, H. and Zimmerman, D. L. (2005). Towards reconciling two asymptotic frameworks in spatial statistics. Biometrika92 921–936. · Zbl 1151.62348
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.