Electronic Journal of Statistics

Asymptotic properties of parameter estimates for random fields with tapered data

Huda Mohammed Alomari, M. Pilar Frías, Nikolai N. Leonenko, María D. Ruiz-Medina, Lyudmyla Sakhno, and Antoni Torres

Full-text: Open access


In this paper we present novel results on the asymptotic behavior of the so-called Ibragimov minimum contrast estimates. The case of tapered data for various models of Gaussian random fields is investigated. The CLT for quadratic forms with tapered data is presented.

Article information

Electron. J. Statist., Volume 11, Number 2 (2017), 3332-3367.

Received: December 2016
First available in Project Euclid: 2 October 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62M30: Spatial processes
Secondary: 60G60: Random fields

Gaussian random field Ibragimov minimum contrast tapered data consistency and asymptotic normality

Creative Commons Attribution 4.0 International License.


Alomari, Huda Mohammed; Frías, M. Pilar; Leonenko, Nikolai N.; Ruiz-Medina, María D.; Sakhno, Lyudmyla; Torres, Antoni. Asymptotic properties of parameter estimates for random fields with tapered data. Electron. J. Statist. 11 (2017), no. 2, 3332--3367. doi:10.1214/17-EJS1315. https://projecteuclid.org/euclid.ejs/1506931551

Export citation


  • [1] Anh, V. V., Angulo, J. M. and Ruiz-Medina, M. D. (1999). Possible long-range dependence in fractional random fields., J. Stat. Plan. Infer. 80 95—110.
  • [2] Anh, V. V., Leonenko N. N. and Sakhno, L. M. (2004a). On a Class of Minimum Contrast Estimators for Fractional Stochastic Processes and Fields., J. Stat. Plan. Infer. 123 161–185.
  • [3] Anh, V. V., Leonenko N. N. and Sakhno L. M. (2004b). Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence. Stochastic methods and their applications., J. Appl. Probab. 41A 35–53.
  • [4] Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2007a). Statistical inference based on the information of the second and third order., J. Stat. Plan. Infer. 137 1302–1331.
  • [5] Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2007b). Evaluation of bias in higher-order spectral estimation., Theory Probab. Math. Stat. 80 1–14.
  • [6] Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2007c). Statistical inference using higher-order information., J. Multivariate Anal. 98(4) 706–742.
  • [7] Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2007d). Minimum contrast estimation of random processes based on information of second and third orders., J Stat Plan Infer 137(4) 1302–1331.
  • [8] Anh, V. V. and Lunney, K. E. (1995). Parameter estimation of random fields with longrange depedence., Math. Comput. Model. 21 67–77.
  • [9] Avram, F. (1988). On Bilinear Forms in Gaussian Random Variables and Toeplitz Matrices., Probab. Theory Relat. Field 79 37–45.
  • [10] Avram, F., Leonenko, N. N. and Sakhno, L. M. (2010a). On a Szegö Type Limit Theorem, the Hölder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields., ESAIM: PS. 14 210–255.
  • [11] Avram, F., Leonenko, N. N. and Sakhno, L. M. (2010b). Harmonic Analysis Tools for Statistical Inference in the Spectral domain., Lecture Notes in Statistics, Springer-Verlag. 200 59–70.
  • [12] Basu, S. and Reinsel, G. C. (1993). Properties of the spatial unilateral first-order ARMA model., Adv. Appl. Probab. 25(3) 631–648.
  • [13] Bentkus, R. (1972). The Error in Estimating the Spectral Function of a Stationary Process., Lietuvos Matematikos Rinkinys 12(1) 55–71.
  • [14] Bentkus, R. (1976). Cumulants of Estimates of the Spectrum of a Stationary Time Series., Litovskii Matematicheskii Sbornik 16(4) 37–61.
  • [15] Brillinger, D.R. (1981)., Time Series: Data Analysis and Theory. Holden Day, San Francisco.
  • [16] Boissy, Y., Bhattacharyya, B. B., Li, X. and Richardson, G. D. (2005). Parameter estimates for fractional autoregressive spatial process., Ann. Stat. 33 2553–2567.
  • [17] Chan, K. S. and Tsai, H. (2012). Inference of seasonal long-memory aggregate time series., Bernoulli 4(18) 1448–1464.
  • [18] Chung, C. F. (1996). Estimating a generalized long memory process., J. Econom. 73 237–259.
  • [19] Dahlhaus, R. (1983). Spectral Analysis with Tapered Data., J. Time Ser. Anal. 4(3) 163–175.
  • [20] Dahlhaus, R. and Künsch, H. (1987). Edge Effects and Efficient Parameter Estimation for Stationary Random Fields., Biometrika 73(4) 877–882.
  • [21] Doukhan, P., León, J. and Soulier, P. (1996). Central and non central limit theorems for strongly dependent stationary Gaussian field., Rebrape 10 205–223.
  • [22] Espejo, R., Leonenko, N. N., Olenko, A. and Ruiz-Medina, M. D. (2015). On a Class of Minimum Contrast Estimators for Gegenbauer Random Fields., TEST 24 657–680.
  • [23] Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series., Ann. Statist 14 517–532.
  • [24] Fox, R. and Taqqu, M. S. (1987). Central Limit Theorems for Quadratic Forms in Random Variables Having Long-Range Dependence., Probab. Theory Relat. Field 74 213–240.
  • [25] Ginovyan, M. S., Sahakyan, A. A. and Taqqu, M. S. (2014). The trace problem for Toeplitz matrices and operators and its impact in probability., Probab. Surv. 11 393–440.
  • [26] Guo, H., Lim, C. Y. and Meerschaert, M. M. (2009). Local Whittle estimator for anisotropic random field., J. Multivariate Anal. 100 993–1028.
  • [27] Guyon, X. (1982). Parameter Estimation for a Stationary Process on a d-Dimensional Lattice., Biometrika 69 95–105.
  • [28] Guyon, X. (1995)., Random Fields on a Network. Springer-Verlag, New York.
  • [29] Heyde, C. C. and Gay, R. (1993). Smoothed Periodogram Asymptotics and Estimation for Processes and Fields with Possible Long-Range Dependence., Stoch. Process. Their Appl. 45 169–182.
  • [30] Ibragimov, I. A. (1967). On Maximum Likelihood Estimation of Parameters of the Spectral Density of Stationary Time Series., Theory Probab. Appl. 12(1) 115–119.
  • [31] Ivanov, A.V. and Prikhodko, V. (2015). Asymptotic properties of Ibaragimov’s estimate of parameter of spectral density of noise in the nonlinear regression method., Theory Probab. Math.Stat. 93 50–66.
  • [32] Leonenko, N. N. and Sakhno, L. M. (2006). On the Whittle estimators for some classes of continuous-parameter random processes and fields., Stat. Probab. Lett. 76 781–795.
  • [33] Leonenko, N. N. and Taufer, E. (2013). Disaggregation of spatial autoregressive processes., Spatial Statistics 3 1–20.
  • [34] Li, W. K. and McLeod, A. I. (1986). Fractional time series modelling., Biometrika 73 217–221.
  • [35] Ludeña C. (2000). Parametric estimation for Gaussian long-range dependent processes based on the log-periodogram., Bernoulli 6 709–728.
  • [36] Ludeña, C. and Lavielle, M. (1999). The Whittle Estimator for Strongly Dependent Stationary Gaussian Fields., Board of the Foundation of the Scand. J. Stat. 26 433–450.
  • [37] Reisen, V., Rodrigues, A. L. and Palma, W. (2006). Estimation of seasonal fractionally integrated processes., Comput. Stat. Data Anal. 50 568–582.
  • [38] Robinson, P. M. (2007). Nonparametric spectrum estimation for spatial data., J. Stat. Plan. Infer. 137 1024–1034.
  • [39] Robinson, P. M. and Sanz, J. V. (2006). Modified Whittle estimation of multilateral models on a lattice., J. Multivariate Anal. 97 1090–1120.
  • [40] Sakhno, L. (2007). Bias control in the estimation of spectral functionals., Theory Stoch. Process. 13 225–233.
  • [41] Sakhno, L. (2014). Minimum Contrast Method for Parameter Estimation in the Spectral Domain. In: Modern Stochastics and Applications, Springer Optimization and Its Applications V. 90, 319 –, 336.
  • [42] Taniguchi, M. (1987). Minimum contrast estimation for spectral densities of stationary processes., J. R. Stat. Soc. Ser. B-Stat. Methodol. 49 315–325.
  • [43] WeiLin, X., WeiGuo, Z. and XiLi, Z. (2012). Minimum contrast estimator for fractional Ornstein-Uhlenbeck processes., Sci. China-Math. 55(7) 1497–1511.
  • [44] Yao, Q. and Brockwell. P. J. (2006). Gaussian maximum likelihood estimation for ARMA models II: Spatial processes., Bernoulli 12(3) 403–429.