The Annals of Statistics

A frequency domain empirical likelihood method for irregularly spaced spatial data

Soutir Bandyopadhyay, Soumendra N. Lahiri, and Daniel J. Nordman

Full-text: Open access


This paper develops empirical likelihood methodology for irregularly spaced spatial data in the frequency domain. Unlike the frequency domain empirical likelihood (FDEL) methodology for time series (on a regular grid), the formulation of the spatial FDEL needs special care due to lack of the usual orthogonality properties of the discrete Fourier transform for irregularly spaced data and due to presence of nontrivial bias in the periodogram under different spatial asymptotic structures. A spatial FDEL is formulated in the paper taking into account the effects of these factors. The main results of the paper show that Wilks’ phenomenon holds for a scaled version of the logarithm of the proposed empirical likelihood ratio statistic in the sense that it is asymptotically distribution-free and has a chi-squared limit. As a result, the proposed spatial FDEL method can be used to build nonparametric, asymptotically correct confidence regions and tests for covariance parameters that are defined through spectral estimating equations, for irregularly spaced spatial data. In comparison to the more common studentization approach, a major advantage of our method is that it does not require explicit estimation of the standard error of an estimator, which is itself a very difficult problem as the asymptotic variances of many common estimators depend on intricate interactions among several population quantities, including the spectral density of the spatial process, the spatial sampling density and the spatial asymptotic structure. Results from a numerical study are also reported to illustrate the methodology and its finite sample properties.

Article information

Ann. Statist., Volume 43, Number 2 (2015), 519-545.

First available in Project Euclid: 24 February 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M30: Spatial processes
Secondary: 62E20: Asymptotic distribution theory

Confidence sets discrete Fourier transform estimating equations hypotheses testing periodogram spectral moment conditions stochastic design variogram Wilks’ theorem


Bandyopadhyay, Soutir; Lahiri, Soumendra N.; Nordman, Daniel J. A frequency domain empirical likelihood method for irregularly spaced spatial data. Ann. Statist. 43 (2015), no. 2, 519--545. doi:10.1214/14-AOS1291.

Export citation


  • [1] Bandyopadhyay, S. and Lahiri, S. N. (2009). Asymptotic properties of discrete Fourier transforms for spatial data. Sankhyā 71 221–259.
  • [2] Bandyopadhyay, S., Lahiri, S. N. and Nordman, D. J. (2013). A central limit theorem for periodogram based statistics for irregularly spaced spatial data and Whittle estimation. Preprint.
  • [3] Bandyopadhyay, S., Lahiri, S. N. and Nordman, D. J. (2015). Supplement to “A frequency domain empirical likelihood method for irregularly spaced spatial data.” DOI:10.1214/14-AOS1291SUPP.
  • [4] Bradley, R. C. (1989). A caution on mixing conditions for random fields. Statist. Probab. Lett. 8 489–491.
  • [5] Bradley, R. C. (1993). Equivalent mixing conditions for random fields. Ann. Probab. 21 1921–1926.
  • [6] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • [7] Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
  • [8] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer, New York.
  • [9] Du, J., Zhang, H. and Mandrekar, V. S. (2009). Fixed-domain asymptotic properties of tapered maximum likelihood estimators. Ann. Statist. 37 3330–3361.
  • [10] Fuentes, M. (2006). Testing for separability of spatial–temporal covariance functions. J. Statist. Plann. Inference 136 447–466.
  • [11] Fuentes, M. (2007). Approximate likelihood for large irregularly spaced spatial data. J. Amer. Statist. Assoc. 102 321–331.
  • [12] García-Soidán, P. (2007). Asymptotic normality of the Nadaraya–Watson semivariogram estimators. TEST 16 479–503.
  • [13] Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Int. Statist. Rev. 58 109–127.
  • [14] Hall, P. and Patil, P. (1994). Properties of nonparametric estimators of autocovariance for stationary random fields. Probab. Theory Related Fields 99 399–424.
  • [15] Im, H. K., Stein, M. L. and Zhu, Z. (2007). Semiparametric estimation of spectral density with irregular observations. J. Amer. Statist. Assoc. 102 726–735.
  • [16] Journel, A. G. and Huijbregts, C. J. (1978). Mining Geostatistics. Academic Press, San Diego, CA.
  • [17] Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. Ann. Statist. 25 2084–2102.
  • [18] Lahiri, S. N. (2003). Central limit theorems for weighted sums of a spatial process under a class of stochastic and fixed designs. Sankhyā 65 356–388.
  • [19] Lahiri, S. N. (2003). A necessary and sufficient condition for asymptotic independence of discrete Fourier transforms under short- and long-range dependence. Ann. Statist. 31 613–641.
  • [20] Lahiri, S. N., Lee, Y. and Cressie, N. (2002). On asymptotic distribution and asymptotic efficiency of least squares estimators of spatial variogram parameters. J. Statist. Plann. Inference 103 65–85.
  • [21] Lahiri, S. N. and Mukherjee, K. (2004). Asymptotic distributions of $M$-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs. Ann. Inst. Statist. Math. 56 225–250.
  • [22] Loh, W.-L. (2005). Fixed-domain asymptotics for a subclass of Matérn-type Gaussian random fields. Ann. Statist. 33 2344–2394.
  • [23] Maity, A. and Sherman, M. (2012). Testing for spatial isotropy under general designs. J. Statist. Plann. Inference 142 1081–1091.
  • [24] Matsuda, Y. and Yajima, Y. (2009). Fourier analysis of irregularly spaced data on $\mathbbR^d$. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 191–217.
  • [25] Monti, A. C. (1997). Empirical likelihood confidence regions in time series models. Biometrika 84 395–405.
  • [26] Nordman, D. J. (2008). A blockwise empirical likelihood for spatial lattice data. Statist. Sinica 18 1111–1129.
  • [27] Nordman, D. J. (2008). An empirical likelihood method for spatial regression. Metrika 68 351–363.
  • [28] Nordman, D. J. and Caragea, P. C. (2008). Point and interval estimation of variogram models using spatial empirical likelihood. J. Amer. Statist. Assoc. 103 350–361.
  • [29] Nordman, D. J. and Lahiri, S. N. (2004). On optimal spatial subsample size for variance estimation. Ann. Statist. 32 1981–2027.
  • [30] Nordman, D. J. and Lahiri, S. N. (2006). A frequency domain empirical likelihood for short- and long-range dependence. Ann. Statist. 34 3019–3050.
  • [31] Nordman, D. J., Lahiri, S. N. and Fridley, B. L. (2007). Optimal block size for variance estimation by a spatial block bootstrap method. Sankhyā 69 468–493.
  • [32] Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90–120.
  • [33] Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237–249.
  • [34] Owen, A. B. (2001). Empirical Likelihood. CRC Press, Boca Raton, FL.
  • [35] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300–325.
  • [36] SAS (2008). SAS/STAT(R) 9.2 User’s Guide. SAS Institute, Cary, NC.
  • [37] Stein, M. (1989). Asymptotic distributions of minimum norm quadratic estimators of the covariance function of a Gaussian random field. Ann. Statist. 17 980–1000.
  • [38] Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S, 4th ed. Springer, New York.
  • [39] Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Statist. 9 60–62.
  • [40] Zhang, H. and Zimmerman, D. L. (2005). Towards reconciling two asymptotic frameworks in spatial statistics. Biometrika 92 921–936.

Supplemental materials