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Fixed-domain asymptotic properties of maximum composite likelihood estimators for Gaussian processes. (English) Zbl 1441.62191
Summary: We consider the estimation of the variance and spatial scale parameters of the covariance function of a one-dimensional Gaussian process with fixed smoothness parameter \(s\). We study the fixed-domain asymptotic properties of composite likelihood estimators. As an improvement of previous references, we allow for any fixed number of neighbor observation points, both on the left and on the right sides, for the composite likelihood. First, we examine the case where only the variance parameter is unknown. We prove that for small values of \(s\), the composite likelihood estimator converges at a sub-optimal rate and we provide its non-Gaussian asymptotic distribution. For large values of \(s\), the estimator converges at the optimal rate. Second, we consider the case where the variance and the spatial scale are jointly estimated. We obtain the same conclusion as for the first case for the estimation of the microergodic parameter. The theoretical results are confirmed in numerical simulations.

62J10 Analysis of variance and covariance (ANOVA)
60G15 Gaussian processes
Full Text: DOI
[1] Andrianakis, I.; Challenor, P. G., The effect of the nugget on Gaussian process emulators of computer models, Comput. Statist. Data Anal., 56, 4215-4228 (2012) · Zbl 1255.62306
[2] Azaïs, J.-M.; Bachoc, F.; Klein, T.; Lagnoux, A.; Nguyen, T. M.N., Semi-parametric estimation of the variogram of a Gaussian process with stationary increments (2018), arXiv:1806.03135
[3] Bachoc, F., Cross validation and maximum likelihood estimations of hyper-parameters of Gaussian processes with model mispecification, Comput. Statist. Data Anal., 66, 55-69 (2013) · Zbl 06958972
[4] Bachoc, F., Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes, J. Multivariate Anal., 125, 1-35 (2014) · Zbl 1280.62100
[5] Bachoc, F., Asymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified case, Bernoulli, 24, 2, 1531-1575 (2018) · Zbl 1429.60035
[6] Bachoc, F.; Ammar, K.; Martinez, J., Improvement of code behavior in a design of experiments by metamodeling, Nucl. Sci. Eng., 183, 3, 387-406 (2016)
[7] Bachoc, F.; Bevilacqua, M.; Velandia, D., Composite likelihood estimation for a Gaussian process under fixed domain asymptotics, J. Multivariate Anal., 174 (2019) · Zbl 1428.62419
[8] Bachoc, F.; Lagnoux, A.; Nguyen, T. M.N., Cross-validation estimation of covariance parameters under fixed-domain asymptotics, J. Multivariate Anal., 160, 42-67 (2017) · Zbl 1378.62096
[9] Cao, Y., Fleet, D.J., 2014. Generalized product of experts for automatic and principled fusion of Gaussian process predictions. In: Modern Nonparametrics 3: Automating the Learning Pipeline Workshop at NIPS, Montreal, arXiv preprint arXiv:1410.7827.
[10] Chang, C.-H.; Huang, H.-C.; Ing, C.-K., Mixed domain asymptotics for a stochastic process model with time trend and measurement error, Bernoulli, 23, 1, 159-190 (2017) · Zbl 1359.62353
[11] Chen, H.-S.; Simpson, D.; Ying, Z., Infill asymptotics for a stochastic process model with measurement error, Statist. Sinica, 10, 141-156 (2000) · Zbl 0970.62061
[12] Cressie, N., Statistics for Spatial Data (1993), J. Wiley
[13] Cressie, N.; Lahiri, S., Asymptotics for REML estimation of spatial covariance parameters, J. Statist. Plann. Inference, 50, 327-341 (1996) · Zbl 0847.62044
[14] Datta, A.; Banerjee, S.; Finley, A. O.; Gelfand, A. E., Hierarchical nearest-neighbor Gaussian process models for large geostatistical datasets, J. Amer. Statist. Assoc., 111, 514, 800-812 (2016)
[15] Deisenroth, M.P., Ng, J.W., 2015. Distributed Gaussian processes. In: Proceedings of the 32nd International Conference on Machine Learning, Lille, France. JMLR: W&CP vol. 37.
[16] Furrer, R.; Genton, M. G.; Nychka, D., Covariance tapering for interpolation of large spatial datasets, J. Comput. Graph. Statist., 15, 3, 502-523 (2006)
[17] Gramacy, R. B.; Apley, D. W., Local Gaussian process approximation for large computer experiments, J. Comput. Graph. Statist., 24, 2, 561-578 (2015)
[18] Gramacy, R. B., laGP: large-scale spatial modeling via local approximate Gaussian processes in R, J. Stat. Softw., 72, 1, 1-46 (2016)
[19] Gu, M.; Berger, J. O., Parallel partial Gaussian process emulation for computer models with massive output, Ann. Appl. Stat., 10, 3, 1317-1347 (2016) · Zbl 1391.62184
[20] Hartikainen, J.; Särkkä, S., Kalman filtering and smoothing solutions to temporal Gaussian process regression models, (2010 IEEE International Workshop on Machine Learning for Signal Processing (2010), IEEE), 379-384
[21] Hensman, J.; Fusi, N., Gaussian Processes for big data, (Uncertainty in Artificial Intelligence (2013)), 282-290
[22] Hinton, G. E., Training products of experts by minimizing contrastive divergence, Neural Comput., 14, 8, 1771-1800 (2002) · Zbl 1010.68111
[23] Ibragimov, I.; Rozanov, Y., Gaussian Random Processes (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0392.60037
[24] Istas, J.; Lang, G., Quadratic variations and estimation of the local Hölder index of a Gaussian process, Ann. Inst. Henri Poincaré, 33, 407-436 (1997) · Zbl 0882.60032
[25] Kaufman, C. G.; Schervish, M. J.; Nychka, D. W., Covariance tapering for likelihood-based estimation in large spatial data sets, J. Amer. Statist. Assoc., 103, 484, 1545-1555 (2008) · Zbl 1286.62072
[26] Kaufman, C.; Shaby, B., The role of the range parameter for estimation and prediction in geostatistics, Biometrika, 100, 473-484 (2013) · Zbl 1284.62590
[27] Li, W. V.; Linde, W., Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab., 27, 1556-1578 (1999) · Zbl 0983.60026
[28] Loh, W.-L., Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations, Ann. Statist., 43, 6, 2766-2794 (2015) · Zbl 1327.62482
[29] López-Lopera, A.; Bachoc, F.; Durrande, N.; Roustand, O., Finite-dimensional Gaussian approximation with linear inequality constraints, SIAM/ASA J. Uncertain. Quant., 6, 3, 1224-1255 (2018) · Zbl 1405.60047
[30] Mardia, K.; Marshall, R., Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika, 71, 135-146 (1984) · Zbl 0542.62079
[31] Mateu, J.; Porcu, E.; Christakos, G.; Bevilacqua, M., Fitting negative spatial covariances to geothermal field temperatures in Nea Kessani (Greece), Environmetrics, 18, 7, 759-773 (2007)
[32] Pardo-Igúzquiza, E.; Dowd, P. A., AMLE3D: a computer program for the inference of spatial covariance parameters by approximate maximum likelihood estimation, Comput. Geosci., 23, 7, 793-805 (1997)
[33] Rasmussen, C.; Williams, C., Gaussian Processes for Machine Learning (2006), The MIT Press: The MIT Press Cambridge · Zbl 1177.68165
[34] Rue, H.; Held, L., Gaussian Markov Random Fields, Theory and Applications (2005), Chapman & Hall · Zbl 1093.60003
[35] Rullière, D.; Durrande, N.; Bachoc, F.; Chevalier, C., Nested Kriging predictions for datasets with a large number of observations, Stat. Comput., 28, 4, 849-867 (2018) · Zbl 1384.62246
[36] Santner, T.; Williams, B.; Notz, W., The Design and Analysis of Computer Experiments (2003), Springer: Springer New York · Zbl 1041.62068
[37] Shaby, B. A.; Ruppert, D., Tapered covariance: Bayesian estimation and asymptotics, J. Comput. Graph. Statist., 21, 2, 433-452 (2012)
[38] Slepian, D., On the zeros of Gaussian noise, (Proc. Sympos. Time Series Analysis (Brown Univ., 1962) (1963), Wiley: Wiley New York), 104-115
[39] Stein, M., Asymptotically efficient prediction of a random field with a misspecified covariance function, Ann. Statist., 16, 55-63 (1988) · Zbl 0637.62088
[40] Stein, M., Bounds on the efficiency of linear predictions using an incorrect covariance function, Ann. Statist., 18, 1116-1138 (1990) · Zbl 0749.62061
[41] Stein, M., Uniform asymptotic optimality of linear predictions of a random field using an incorrect second-order structure, Ann. Statist., 18, 850-872 (1990) · Zbl 0716.62099
[42] Stein, M., Interpolation of Spatial Data: Some Theory for Kriging (1999), Springer: Springer New York · Zbl 0924.62100
[43] Stein, M. L., Limitations on low rank approximations for covariance matrices of spatial data, Spat. Stat., 8, 1-19 (2014)
[44] Stein, M. L.; Chi, Z.; Welty, L. J., Approximating likelihoods for large spatial data sets, J. R. Stat. Soc. Ser. B Stat. Methodol., 66, 2, 275-296 (2004) · Zbl 1062.62094
[45] van Stein, B.; Wang, H.; Kowalczyk, W.; Bäck, T.; Emmerich, M., Optimally weighted cluster Kriging for big data regression, (International Symposium on Intelligent Data Analysis (2015), Springer), 310-321
[46] Tresp, V., A Bayesian committee machine, Neural Comput., 12, 11, 2719-2741 (2000)
[47] Varin, C.; Reid, N.; Firth, D., An overview of composite likelihood methods, Statist. Sinica, 21, 5-42 (2011) · Zbl 05849508
[48] Vecchia, A. V., Estimation and model identification for continuous spatial processes, J. R. Stat. Soc. Ser. B Stat. Methodol., 50, 2, 297-312 (1988)
[49] Ying, Z., Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process, J. Multivariate Anal., 36, 280-296 (1991) · Zbl 0733.62091
[50] Zhang, H., Inconsistent estimation and asymptotically equivalent interpolations in model-based geostatistics, J. Amer. Statist. Assoc., 99, 250-261 (2004) · Zbl 1089.62538
[51] Zhang, H.; Zimmerman, D., Towards reconciling two asymptotic frameworks in spatial statistics, Biometrika, 92, 921-936 (2005) · Zbl 1151.62348
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