We consider the semi-parametric estimation of the scale parameter of the variogram of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based both on quadratic variations and the moment method. We provide asymptotic approximations of the mean and variance of this estimator, together with asymptotic normality results, for a large class of Gaussian processes. We allow for general mean functions, provide minimax upper bounds and study the aggregation of several estimators based on various variation sequences. In extensive simulation studies, we show that the asymptotic results accurately depict the finite-sample situations already for small to moderate sample sizes. We also compare various variation sequences and highlight the efficiency of the aggregation procedure.
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DOI : 10.1051/ps/2020021
Mots-clés : Gaussian processes, semi-parametric estimation, quadratic variations, scale covariance parameter, asymptotic normality, moment method, minimax upper bounds, aggregation of estimators
@article{PS_2020__24_1_842_0,
author = {Aza{\"\i}s, Jean-Marc and Bachoc, Fran\c{c}ois and Lagnoux, Agn\`es and Nguyen, Thi Mong Ngoc},
title = {Semi-parametric estimation of the variogram scale parameter of a {Gaussian} process with stationary increments},
journal = {ESAIM: Probability and Statistics},
pages = {842--882},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2020021},
mrnumber = {4178368},
zbl = {1461.60019},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2020021/}
}
TY - JOUR AU - Azaïs, Jean-Marc AU - Bachoc, François AU - Lagnoux, Agnès AU - Nguyen, Thi Mong Ngoc TI - Semi-parametric estimation of the variogram scale parameter of a Gaussian process with stationary increments JO - ESAIM: Probability and Statistics PY - 2020 SP - 842 EP - 882 VL - 24 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2020021/ DO - 10.1051/ps/2020021 LA - en ID - PS_2020__24_1_842_0 ER -
%0 Journal Article %A Azaïs, Jean-Marc %A Bachoc, François %A Lagnoux, Agnès %A Nguyen, Thi Mong Ngoc %T Semi-parametric estimation of the variogram scale parameter of a Gaussian process with stationary increments %J ESAIM: Probability and Statistics %D 2020 %P 842-882 %V 24 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2020021/ %R 10.1051/ps/2020021 %G en %F PS_2020__24_1_842_0
Azaïs, Jean-Marc; Bachoc, François; Lagnoux, Agnès; Nguyen, Thi Mong Ngoc. Semi-parametric estimation of the variogram scale parameter of a Gaussian process with stationary increments. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 842-882. doi : 10.1051/ps/2020021. http://www.numdam.org/articles/10.1051/ps/2020021/
[1] and , Uniform quadratic variation for Gaussian processes. Stochastic Process. Appl. 48 (1993) 191–209 | DOI | MR | Zbl
[2] and , Level sets and Extrema of Random Processes and Fields. John Wiley & Sons, Inc., Hoboken, NJ (2009) | DOI | MR | Zbl
[3] , Cross validation and maximum likelihood estimations of hyper-parameters of Gaussian processes with model misspecification. Computat. Stat. Data Anal. 66 (2013) 55–69 | DOI | MR | Zbl
[4] , Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes. J. Multivariate Anal. 125 (2014) 1–35 | DOI | MR | Zbl
[5] , Asymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified case. Bernoulli 24 (2018) 1531–1575 | DOI | MR | Zbl
[6] and , Fixed-domain asymptotic properties of maximum composite likelihood estimators for Gaussian processes. J. Stat. Plann. Inference 209 (2020) 62–75 | DOI | MR | Zbl
[7] and , The combination of forecasts. J. Oper. Res. Soc. 20 (1969) 451–468 | DOI
[8] , A strong limit theorem for Gaussian processes. Proc. Am. Math. Soc. 7 (1956) 522–527 | DOI | MR | Zbl
[9] and , 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. Preprint (2014) | arXiv
[10] , Practical Geostatistics, Vol. 3. Applied Science Publishers, London (1979)
[11] , Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199–227 | DOI | MR | Zbl
[12] and , Fractional Fields and Applications, With a foreword by Stéphane Jaffard. Vol. 73 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2013) | DOI | MR | Zbl
[13] , Statistics for Spatial Data. John Wiley (1993) | DOI | MR | Zbl
[14] and , Robust estimation of the variogram: I. J. Int. Assoc. Math. Geol. 12 (1980) 115–125 | DOI | MR
[15] , Efficient parameter estimation for self-similar processes. Ann. Stat. (1989) 1749–1766 | MR | Zbl
[16] , , , and , Hierarchical nearest-neighbor Gaussian process models for large geostatistical datasets. J. Am. Stat. Assoc. 111 (2016) 800–812 | DOI | MR
[17] , Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909–996 | DOI | MR | Zbl
[18] , Geostatistical Ore Reserve Estimation. Elsevier (2012)
[19] and , Distributed Gaussian processes, In Proceedings of the 32nd International Conference on Machine Learning, Lille, France. JMLR: W&CP volume 37 (2015)
[20] , and , Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Stat. 15 (2006) 502–523 | DOI | MR
[21] , A new limit theorem for stochastic processes with Gaussian increments. Teor. Verojatnost. Primenen. 6 (1961) 57–66 | MR | Zbl
[22] , Abstract inference. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York (1981) | MR | Zbl
[23] and , Convergence en loi des H-variations d’un processus gaussien stationnaire sur R. Ann. Inst. Henri Poincaré Probab. Statist. 25 (1989) 265–282 | Numdam | MR | Zbl
[24] , and , On the nonparametric estimation of covariance functions. Ann. Stat. 22 (1994) 2115–2134 | DOI | MR | Zbl
[25] and , Properties of nonparametric estimators of autocovariance for stationary random fields. Probab. Theory Related Fields 99 (1994) 399–424 | DOI | MR | Zbl
[26] and , Financial time series volatility analysis using Gaussian process state-space models, in 2015 IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE (2015) 358–362 | DOI
[27] and , Gaussian processes for big data. Uncertainty Artif. Intell. (2013) 282–290
[28] , Training products of experts by minimizing contrastive divergence. Neural Computat. 14 (2002) 1771–1800 | DOI | Zbl
[29] and , Gaussian Random Processes. Springer-Verlag, New York (1978) | DOI | MR | Zbl
[30] and , Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Statist. 33 (1997) 407–436 | DOI | Numdam | MR | Zbl
[31] and , Mining geostatistics, in Bureau De Recherches Geologiques Et Miniers, France Academic Pres Harcout Brace & Company, Publishers, London, San Diego, New York, Boston, Sidney, Toronto (1978)
[32] , , and , Covariance tapering for likelihood-based estimation in large spatial data sets. J. Am. Stat. Assoc. 103 (2008) 1545–1555 | DOI | MR | Zbl
[33] and , Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59 (1997) 679–699 | MR | Zbl
[34] , and , On asymptotic distribution and asymptotic efficiency of least squares estimators of spatial variogram parameters. J. Stat. Plann. Inference 103 (2002) 65–85 | DOI | MR | Zbl
[35] and , Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inference Stoch. Process. 4 (2001) 283–306 | DOI | MR | Zbl
[36] and A general procedure to combine estimators. Computat. Stat. Data Anal. 94 (2016) 175–192 | DOI | MR | Zbl
[37] , Le mouvement brownien plan. Am. J. Math. 62 (1940) 487–550 | DOI | JFM | MR
[38] , Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley (1979) | Zbl
[39] , , and , Fitting negative spatial covariances to geothermal field temperatures in Nea Kessani (Greece). Environmetrics 18 (2007) 759–773 | DOI | MR
[40] , Traité de géostatistique appliquée, Tome I, Vol. 14 of Editions Technip, Paris. Mémoires du Bureau de Recherches Géologiques et Minières (1962)
[41] and , AMLE3D: a computer program for the inference of spatial covariance parameters by approximate maximum likelihood estimation. Comput. Geosci. 23 (1997) 793–805 | DOI
[42] , Quadratic variation for Gaussian processes and application to time deformation. Stochastic Process. Appl. 82 (1999) 293–305 | DOI | MR | Zbl
[43] , Remarks on characteristic functions, in Proceedings of the First Berkeley Symposium on Mathematical Statistics and Probability, August 13–18, 1945 and January 27–29, 1946, edited by . Statistical Laboratory of the University of California, Berkeley. University of California Press, Berkeley, CA (1949) 115–123 | MR | Zbl
[44] and , Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2006) | MR | Zbl
[45] , Anisotropy of Hölder Gaussian random fields: characterization, estimation, and application to image textures. Stat. Comput. 28 (2018) 1155–1168 | DOI | MR | Zbl
[46] and , Statistical tests of anisotropy for fractional Brownian textures. application to full-field digital mammography. J. Math. Imaging Vis. 36 (2010) 227–240 | DOI
[47] , , , , and , ADNI: Simultaneous multiscale registration using large deformation diffeomorphic metric mapping. IEEE Trans. Med. Imaging 30 (2011) 1746–1759 | DOI
[48] , and , DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Softw. 51 (2012) | DOI
[49] and , Gaussian Markov Random Fields Theory and Applications. Chapman & Hall (2005) | DOI | MR | Zbl
[50] , , and , Nested Kriging predictions for datasets with a large number of observations. Stat. Comput. 28 (2018) 849–867 | DOI | MR | Zbl
[51] and , Non-Gaussian Stable Processes: Stochastic Models with Infinite Variance. Chapman ft Hall, London (1994) | MR | Zbl
[52] , and , The Design and Analysis of Computer Experiments. Springer Series in Statistics. Springer-Verlag, New York (2003) | MR | Zbl
[53] , On the zeros of Gaussian noise, in Proc. Sympos. Time Series Analysis (Brown Univ., 1962). Wiley, New York (1963) 104–115 | MR | Zbl
[54] , Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York (1999) | DOI | MR | Zbl
[55] , Limitations on low rank approximations for covariance matrices of spatial data. Spatial Stat. 8 (2014) 1–19 | DOI | MR
[56] , and , Approximating likelihoods for large spatial data sets. J. Roy. Stat. Soc.: Ser. B (Stat. Methodol.) 66 (2004) 275–296 | DOI | MR | Zbl
[57] , A Bayesian committee machine. Neural Computat. 12 (2000) 2719–2741 | DOI
[58] , , , and , Optimally weighted cluster Kriging for big data regression, in International Symposium on Intelligent Data Analysis . Springer (2015) 310–321
[59] , and , An overview of composite likelihood methods. Stat. Sinica 21 (2011) 5–42 | MR | Zbl
[60] , Estimation and model identification for continuous spatial processes. J. Roy. Stat. Soc.: Ser. B (Methodological) 50 (1988) 297–312 | MR
[61] , and , Gaussian process volatility model, in Advances in Neural Information Processing Systems 27, edited by , , , and . Curran Associates Inc. (2014) 1044–1052
[62] and , Kriging and cross-validation for massive spatial data. Environmetrics 21 (2010) 290–304 | DOI | MR
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