The Annals of Statistics

Statistical inference for spatial statistics defined in the Fourier domain

Suhasini Subba Rao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A class of Fourier based statistics for irregular spaced spatial data is introduced. Examples include the Whittle likelihood, a parametric estimator of the covariance function based on the $L_{2}$-contrast function and a simple nonparametric estimator of the spatial autocovariance which is a nonnegative function. The Fourier based statistic is a quadratic form of a discrete Fourier-type transform of the spatial data. Evaluation of the statistic is computationally tractable, requiring $O(nb^{})$ operations, where $b$ are the number of Fourier frequencies used in the definition of the statistic and $n$ is the sample size. The asymptotic sampling properties of the statistic are derived using both increasing domain and fixed-domain spatial asymptotics. These results are used to construct a statistic which is asymptotically pivotal.

Article information

Ann. Statist., Volume 46, Number 2 (2018), 469-499.

Received: February 2015
Revised: January 2017
First available in Project Euclid: 3 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M30: Spatial processes
Secondary: 62M15: Spectral analysis

Fixed and increasing domain asymptotics irregular spaced locations quadratic forms spatial spectral density function stationary spatial random fields


Subba Rao, Suhasini. Statistical inference for spatial statistics defined in the Fourier domain. Ann. Statist. 46 (2018), no. 2, 469--499. doi:10.1214/17-AOS1556.

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. (2015). A frequency domain empirical likelihood method for irregularly spaced spatial data. Ann. Statist. 43 519–545.
  • [3] Bandyopadhyay, S. and Subba Rao, S. (2017). A test for stationarity for irregularly spaced spatial data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 95–123.
  • [4] Beutler, F. J. (1970). Alias-free randomly timed sampling of stochastic processes. IEEE Trans. Inform. Theory 16 147–152.
  • [5] Bickel, P. J. and Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhya, Ser. A 50 381–393.
  • [6] Brillinger, D. (1969). The calculation of cumulants via conditioning. Ann. Inst. Statist. Math. 21 215–218.
  • [7] Brillinger, D. R. (1981). Time Series: Data Analysis and Theory, 2nd ed. Holden-Day, Inc., Oakland, CA.
  • [8] Can, S. U., Mikosch, T. and Samorodnitsky, G. (2010). Weak convergence of the function-indexed integrated periodogram for infinite variance processes. Bernoulli 16 995–1015.
  • [9] Chen, W. W., Hurvich, C. M. and Lu, Y. (2006). On the correlation matrix of the discrete Fourier transform and the fast solution of large Toeplitz systems for long-memory time series. J. Amer. Statist. Assoc. 101 812–822.
  • [10] Cressie, N. and Huang, H.-C. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. J. Amer. Statist. Assoc. 94 1330–1340.
  • [11] Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York. Revised reprint of the 1991 edition.
  • [12] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749–1766.
  • [13] Dahlhaus, R. and Janas, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis. Ann. Statist. 24 1934–1963.
  • [14] Deo, R. S. and Chen, W. W. (2000). On the integral of the squared periodogram. Stochastic Process. Appl. 85 159–176.
  • [15] Dunsmuir, W. (1979). A central limit theorem for parameter estimation in stationary vector time series and its application to models for a signal observed with noise. Ann. Statist. 7 490–506.
  • [16] Fox, R. and Taqqu, M. S. (1987). Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Related Fields 74 213–240.
  • [17] Fuentes, M. (2007). Approximate likelihood for large irregularly spaced spatial data. J. Amer. Statist. Assoc. 102 321–331.
  • [18] Giné, E. and Nickl, R. (2008). A simple adaptive estimator of the integrated square of a density. Bernoulli 14 47–61.
  • [19] Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probab. Theory Related Fields 86 87–104.
  • [20] Hall, P., Fisher, N. I. and Hoffmann, B. (1994). On the nonparametric estimation of covariance functions. Ann. Statist. 22 2115–2134.
  • [21] Hall, P. and Patil, P. (1994). Properties of nonparametric estimators of autocovariance for stationary random fields. Probab. Theory Related Fields 99 399–424.
  • [22] Hannan, E. J. (1971). Non-linear time series regression. J. Appl. Probab. 8 767–780.
  • [23] Kawata, T. (1959). Some convergence theorems for stationary stochastic processes. Ann. Math. Stat. 30 1192–1214.
  • [24] 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.
  • [25] Laurent, B. (1996). Efficient estimation of integral functionals of a density. Ann. Statist. 24 659–681.
  • [26] Masry, E. (1978). Poisson sampling and spectral estimation of continuous-time processes. IEEE Trans. Inform. Theory 24 173–183.
  • [27] Matsuda, Y. and Yajima, Y. (2009). Fourier analysis of irregularly spaced data on $\mathbb{R}^{d}$. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 191–217.
  • [28] Neibuhr, T. and Kreiss, J.-P. (2014). Asymptotics for autocovariances and integrated periodograms for linear processes observed at lower frequencies. Int. Stat. Rev. 82 123–140.
  • [29] Parzen, E. (1957). On consistent estimates of the spectrum of a stationary time series. Ann. Math. Stat. 28 329–348.
  • [30] Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation. Bocconi & Springer Series 1. Springer, Milan; Bocconi Univ. Press, Milan.
  • [31] Rice, J. (1979). On the estimation of the parameters of a power spectrum. J. Multivariate Anal. 9 378–392.
  • [32] Shapiro, H. S. and Silverman, R. A. (1960). Alias-free sampling of random noise. J. Soc. Ind. Appl. Math. 8 225–248.
  • [33] Stein, M. L. (1995). Fixed-domain asymptotics for spatial periodograms. J. Amer. Statist. Assoc. 90 1277–1288.
  • [34] Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
  • [35] Stein, M. L., Chi, Z. and Welty, L. J. (2004). Approximating likelihoods for large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 275–296.
  • [36] Subba Rao, S. (2018). Supplement to “Statistical inference for spatial statistics defined in the Fourier domain.” DOI:10.1214/17-AOS1556SUPP.
  • [37] Subba Rao, S. (2017). Orthogonal samples for estimators in time series. Under revision.
  • [38] Vecchia, A. V. (1988). Estimation and model identification for continuous spatial processes. J. R. Stat. Soc., B 50 297–312.
  • [39] Walker, A. M. (1964). Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time-series. J. Aust. Math. Soc. 4 363–384.
  • [40] Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99 250–261.
  • [41] Zhang, H. and Zimmerman, D. L. (2005). Towards reconciling two asymptotic frameworks in spatial statistics. Biometrika 92 921–936.

Supplemental materials

  • Supplement to “Statistical inference for spatial statistics defined in the Fourier domain”. The supplement contains the proofs for all the results of the main article and some related results.