On the consistent separation of scale and variance for Gaussian random fields. (English) Zbl 1204.60041

Anderes, Ethan

Ann. Stat. 38, No. 2, 870-893 (2010).

A common situation in spatial statistics is when one has observations on a single realization of a random field \(Y\) at a large number of spatial points \({\mathbf t}_1\), \({\mathbf t}_2,\dots\) within some bounded region \(\Omega\subset {\mathbb R}^d\). One is then faced with the problem of predicting some quantity that depends on \(Y\) at unobserved points in \(\Omega\). Typically, a fully nonparametric estimation of the covariance structure of \(Y\) is difficult since the observations are from one realization of the random field. In this case, it is common to consider a class of covariance structures indexed by a finite number of parameters which are then estimated using the observations.
Two common parameters found in many covariance models are an overall scale \(\alpha\) and an overall variance \(\sigma^2\). The simplest example of this model stipulates that the random field \(Y\) is a scale and amplitude chance by an unknown \(\alpha\) and \(\sigma\) of a known random field \(Z\):
\[ \{Y({\mathbf t}):~{\mathbf t}\in \Omega\}\doteq \{\sigma Z(\alpha {\mathbf t}):~{\mathbf t}\in \Omega\}, \]
where \(\doteq\) denotes equality of the finite-dimensional distributions.
The author presents fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a Gaussian random field with a geometric anisotropic Matérn autocovariance in dimension \(d>4\). When \(d<4\), this is impossible due to the mutual absolute continuity of Matérn Gaussian random fields with different scale and variance [see H. Zhang, J. Amer. Statist. Assoc. 99, No. 465, 250–261 (2004; Zbl 1089.62538)]. Informally, when \(d>4\), the author shows that one can estimate the coefficient on the principle irregular term accurately enough to get a consistent estimate of the coefficient on the second irregular term. These two coefficients can then be used to separate scale and variance. The author extends his results to the general problem of estimating variance and geometric anisotropy for more general autocovariance functions. These results illustrate the interaction space and the number of increments used for the estimation. As a corrolary, these results establish the orthogonality of Matérn Gaussian random fields with different parameters when \(d>4\). The case \(d=4\) is still open.

Reviewer: Anatoli Mogulskij (Novosibirsk)

MSC:

60G60	Random fields
62M30	Inference from spatial processes
62M40	Random fields; image analysis

Keywords:

Matérn class; quadratic variations; Gaussian random fields; infill asymptotics; covariance estimation; principle irregular term

Citations:

Zbl 1089.62538

References:

[1]	Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical Functions , 9th ed. Dover, New York. · Zbl 0171.38503
[2]	Adler, R. J. and Pyke, R. (1993). Uniform quadratic variation for Gaussian processes. Stochastic Process. Appl. 48 191-209. · Zbl 0783.60040 · doi:10.1016/0304-4149(93)90044-5
[3]	Anderes, E. (2005). Estimating Deformations of Isotropic Gaussian Random Fields . Ph.D. thesis, Univ. Chicago.
[4]	Anderes, E. and Chatterjee, S. (2009). Consistent estimates of deformed isotropic Gaussian random fields on the plane. Ann. Statist. 37 2324-2350. · Zbl 1171.62056 · doi:10.1214/08-AOS647
[5]	Baxter, G. (1956). A strong limit theorem for Gaussian processes. Proc. Amer. Math. Soc. 7 522-527. JSTOR: · Zbl 0070.36304 · doi:10.2307/2032765
[6]	Benassi, A., Cohen, S., Istas, J. and Jaffard, S. (1998). Identification of filtered white noises. Stochastic Process. Appl. 75 31-49. · Zbl 0932.60037 · doi:10.1016/S0304-4149(97)00123-3
[7]	Berman, S. M. (1967). A version of the Lévy-Baxter theorem for the increments of Brownian motion of several parameters. Proc. Amer. Math. Soc. 18 1051-1055. JSTOR: · Zbl 0178.20102 · doi:10.2307/2035793
[8]	Chilès, J. and Delfiner, P. (1999). Geostatistics: Modeling Spatial Uncertainty . Wiley, New York. · Zbl 0922.62098
[9]	Cohen, S., Guyon, X., Perrin, O. and Pontier, M. (2006). Identification of an isometric transformation of the standard Brownian sheet. J. Statist. Plann. Inference 136 1317-1330. · Zbl 1089.60047 · doi:10.1016/j.jspi.2004.09.012
[10]	Cohen, S., Guyon, X., Perrin, O. and Pontier, M. (2006). Singularity functions for fractional processes: Application to the fractional Brownian sheet. Ann. Inst. H. Poincaré Probab. Statist. 42 187-205. · Zbl 1095.60011 · doi:10.1016/j.anihpb.2005.03.002
[11]	Cressie, N. (1993). Statistics for Spatial Data . Wiley, New York. · Zbl 0799.62002
[12]	Du, J., Zhang, H. and Mandrekar, V. (2009). Fixed-domain asymptotic properties of tapered maximum likelihood estimators. Ann. Statist. 37 3330-3361. · Zbl 1369.62248 · doi:10.1214/08-AOS676
[13]	Dudley, R. M. (1973). Sample functions of the Gaussian process. Ann. Probab. 1 66-103. · Zbl 0261.60033 · doi:10.1214/aop/1176997026
[14]	Gladyshev, E. G. (1961). A new limit theorem for stochastic processes with Gaussian increments. Theory Probab. Appl. 6 52-61. · Zbl 0107.12601 · doi:10.1137/1106004
[15]	Guyon, X. and Leon, G. (1989). Convergence en loi des h-variations d’un processus gaussien stationnaire. Ann. Inst. H. Poincaré Probab. Statist. 25 265-282. · Zbl 0691.60017
[16]	Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic form in independent random variables. Ann. Math. Statist. 42 1079-1083. · Zbl 0216.22203 · doi:10.1214/aoms/1177693335
[17]	Horn, R. and Johnson, C. (2007). Matrix Analysis . Cambridge Univ. Press, Cambridge.
[18]	Ibragimov, I. A. and Rozanov, Y. A. (1978). Gaussian Random Processes . Springer, New York. · Zbl 0392.60037
[19]	Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 407-436. · Zbl 0882.60032 · doi:10.1016/S0246-0203(97)80099-4
[20]	Kaufman, C., Schervish, M. and Nychka, D. (2009). Covariance tapering for likelihood-based estimation in large spatial datasets. J. Amer. Statist. Assoc. 103 1545-1555. · Zbl 1286.62072 · doi:10.1198/016214508000000959
[21]	Klein, R. and Gine, E. (1975). On quadratic variation of processes with Gaussian increments. Ann. Probab. 3 716-721. · Zbl 0318.60031 · doi:10.1214/aop/1176996311
[22]	Lévy, P. (1940). Le mouvement Brownien plan. Amer. J. Math. 62 487-550. JSTOR: · Zbl 0024.13906 · doi:10.2307/2371467
[23]	Loh, W.-L. (2005). Fixed-domain asymptotics for a subclass of Matérn-type Gaussian random fields. Ann. Statist. 33 2344-2394. · Zbl 1086.62111 · doi:10.1214/009053605000000516
[24]	Loh, W.-L. and Lam, T.-K. (2000). Estimating structured correlation matrices in smooth Gaussian random field models. Ann. Statist. 28 880-904. · Zbl 1105.62376 · doi:10.1214/aos/1015952003
[25]	Stein, M. L. (1988). Asymptotically efficient prediction of a random field with a misspecified covariance function. Ann. Statist. 16 55-63. · Zbl 0637.62088 · doi:10.1214/aos/1176350690
[26]	Stein, M. L. (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 · doi:10.1214/aos/1176347629
[27]	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 · doi:10.1016/0167-7152(93)90261-G
[28]	Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging . Springer, New York. · Zbl 0924.62100
[29]	Stein, M. L. (2002). Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Statist. 11 587-599. JSTOR: · doi:10.1198/106186002466
[30]	Strait, P. T. (1969). On Berman’s version of the Lévy-Baxter theorem. Proc. Amer. Math. Soc. 23 91-93. · Zbl 0184.40901 · doi:10.2307/2037494
[31]	Wahba, G. (1990). Spline Models for Observational Data . SIAM, Philadelphia, PA. · Zbl 0813.62001
[32]	Ying, Z. (1991). Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process. J. Multivariate Anal. 36 280-296. · Zbl 0733.62091 · doi:10.1016/0047-259X(91)90062-7
[33]	Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99 250-261. · Zbl 1089.62538 · doi:10.1198/016214504000000241