We propose a new definition of effective sample size. Although the recent works of Griffith (2005, 2008) and Vallejos and Osorio (2014) provide a theoretical framework to address the reduction of information in a spatial sample due to spatial autocorrelation, the asymptotic properties of the estimations have not been studied in those studies or in previously ones. In addition, the concept of effective sample size has been developed primarily for spatial regression processes with a constant mean. This paper introduces a new definition of effective sample size for general spatial regression models that is coherent with previous definitions. The asymptotic normality of the maximum likelihood estimation is obtained under an increasing domain framework. In particular, the conditions for which the limiting distribution holds are established for the Matérn covariance family. Illustrative examples accompany the discussion of the limiting results, including some cases where the asymptotic variance has a closed form. The asymptotic normality leads to an approximate hypothesis testing that establishes whether there is redundant information in the sample. Simulation results support the theoretical findings and provide information about the behavior of the power of the suggested test. A real dataset in which a transect sampling scheme has been used is analyzed to estimate the effective sample size when a spatial linear regression model is assumed.
Electron. J. Statist.
12(2):
3147-3180
(2018).
DOI: 10.1214/18-EJS1460
ACOSTA, J., OSORIO, F. and VALLEJOS, R. (2016). Effective sample size for line transect sampling models with an application to marine macroalgae., Journal of Agricultural, Biological and Environmental Statistics, 21, 407–425. 1347.62236 10.1007/s13253-016-0252-7ACOSTA, J., OSORIO, F. and VALLEJOS, R. (2016). Effective sample size for line transect sampling models with an application to marine macroalgae., Journal of Agricultural, Biological and Environmental Statistics, 21, 407–425. 1347.62236 10.1007/s13253-016-0252-7
BANERJEE, S., CARLIN B. and GELFAND A. (2004), Hierarchical Modeling and Analysis for Spatial Data, Boca Raton: Chapman Hall/CRC. 1053.62105BANERJEE, S., CARLIN B. and GELFAND A. (2004), Hierarchical Modeling and Analysis for Spatial Data, Boca Raton: Chapman Hall/CRC. 1053.62105
BERGER, J., BAYARRI, M. J. and PERICCHI, L. R. (2014). The effective sample size., Econometric Letters 33 197–2014. MR3170846 10.1080/07474938.2013.807157BERGER, J., BAYARRI, M. J. and PERICCHI, L. R. (2014). The effective sample size., Econometric Letters 33 197–2014. MR3170846 10.1080/07474938.2013.807157
BOX, G.E.P. (1954a). Some theorems on quadratic forms applied in the study of analysis of variance problems: I. Effect of inequality of variance in the one way classification., Ann. Math. Statist., 25, 290–302. 0055.37305 10.1214/aoms/1177728786 euclid.aoms/1177728786BOX, G.E.P. (1954a). Some theorems on quadratic forms applied in the study of analysis of variance problems: I. Effect of inequality of variance in the one way classification., Ann. Math. Statist., 25, 290–302. 0055.37305 10.1214/aoms/1177728786 euclid.aoms/1177728786
BOX, G.E.P. (1954b). Some theorems on quadratic forms applied in the study of analysis of variance problems: II. Effects of inequality of variance and of correlation between errors in the two way classification., Ann. Math. Statist., 25, 484–498. 0056.36604 10.1214/aoms/1177728717 euclid.aoms/1177728717BOX, G.E.P. (1954b). Some theorems on quadratic forms applied in the study of analysis of variance problems: II. Effects of inequality of variance and of correlation between errors in the two way classification., Ann. Math. Statist., 25, 484–498. 0056.36604 10.1214/aoms/1177728717 euclid.aoms/1177728717
BOX, G. E. P. and Cox, D. R. (1964). An analysis of transformations., Journal of the Royal Statistical Society. Series B, 26, 211–252. 0156.40104 10.1111/j.2517-6161.1964.tb00553.xBOX, G. E. P. and Cox, D. R. (1964). An analysis of transformations., Journal of the Royal Statistical Society. Series B, 26, 211–252. 0156.40104 10.1111/j.2517-6161.1964.tb00553.x
CLIFFORD, P., RICHARDSON, S. and HÉMON, D. (1989). Assessing the significance of the correlation between two spatial processes., Biometrics, 45, 123–134.CLIFFORD, P., RICHARDSON, S. and HÉMON, D. (1989). Assessing the significance of the correlation between two spatial processes., Biometrics, 45, 123–134.
CRESSIE, N. (1993)., Statistics for Spatial Data. New York: Wiley. 1347.62005CRESSIE, N. (1993)., Statistics for Spatial Data. New York: Wiley. 1347.62005
CRESSIE, N. AND LAHIRI, S. N. (1993). Asymptotic distribution of REML estimators., Journal of Multivariate Analysis, 45, 217–233. 0772.62008 10.1006/jmva.1993.1034CRESSIE, N. AND LAHIRI, S. N. (1993). Asymptotic distribution of REML estimators., Journal of Multivariate Analysis, 45, 217–233. 0772.62008 10.1006/jmva.1993.1034
CRUJEIRAS, R. M. and VAN KEILEGOM, I. (2010). Least squares estimation of nonlinear spatial trends., Computational Statistics and Data Analysis, 54, 452–465. 05689602 10.1016/j.csda.2009.09.014CRUJEIRAS, R. M. and VAN KEILEGOM, I. (2010). Least squares estimation of nonlinear spatial trends., Computational Statistics and Data Analysis, 54, 452–465. 05689602 10.1016/j.csda.2009.09.014
DALE, M. R. T. and FORTIN, M-J. (2009). Spatial autocorrelation and statistical tests: some solutions., Journal of Agricultural, Biological, and Environmental Statistics, 14, 188–206. 1306.62263 10.1198/jabes.2009.0012DALE, M. R. T. and FORTIN, M-J. (2009). Spatial autocorrelation and statistical tests: some solutions., Journal of Agricultural, Biological, and Environmental Statistics, 14, 188–206. 1306.62263 10.1198/jabes.2009.0012
de GRUIJTER, J. J. and ter BRAAK, C. J. F. (1990). Model-free estimation from spatial samples: A reappraisal of classical sampling theory., Mathematical Geology, 22, 407–415. 0970.86520 10.1007/BF00890327de GRUIJTER, J. J. and ter BRAAK, C. J. F. (1990). Model-free estimation from spatial samples: A reappraisal of classical sampling theory., Mathematical Geology, 22, 407–415. 0970.86520 10.1007/BF00890327
DUTILLEUL, P., PELLETIER, B. and ALPARGU, G. (2008). Modified $F$ tests for assessing the multiple correlation between one spatial process and several others., Journal of Statistical Planning and Inference, 138, 1402–1415. 1133.62076 10.1016/j.jspi.2007.06.022DUTILLEUL, P., PELLETIER, B. and ALPARGU, G. (2008). Modified $F$ tests for assessing the multiple correlation between one spatial process and several others., Journal of Statistical Planning and Inference, 138, 1402–1415. 1133.62076 10.1016/j.jspi.2007.06.022
FAES, C., MOLENBERGHS, G., AERTS, M., VERBEKE, G. and KENWARD, M. (2009). The effective sample size and an alternative small-sample degrees-of-freedom method., The American Statistician 63 389–399. 1182.62098 10.1198/tast.2009.08196FAES, C., MOLENBERGHS, G., AERTS, M., VERBEKE, G. and KENWARD, M. (2009). The effective sample size and an alternative small-sample degrees-of-freedom method., The American Statistician 63 389–399. 1182.62098 10.1198/tast.2009.08196
GELFAND, A. E., DIGGLE, P. J., FUENTES, M. and GUTTORP, P. (2010)., Handbook of Spatial Statistics, Boca Raton, Fl: Chapman & Hall/CRC. 1188.62284GELFAND, A. E., DIGGLE, P. J., FUENTES, M. and GUTTORP, P. (2010)., Handbook of Spatial Statistics, Boca Raton, Fl: Chapman & Hall/CRC. 1188.62284
GOLUB, G. H. and PEREYRA, V. (1973). The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate., SIAM Journal of Numerical Analysis, 22, 413–431. 0258.65045 10.1137/0710036GOLUB, G. H. and PEREYRA, V. (1973). The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate., SIAM Journal of Numerical Analysis, 22, 413–431. 0258.65045 10.1137/0710036
GRIFFITH, D. (2008). Geographic sampling of urban soils for contaminant mapping: how many samples and from where., Environ. Geochem. Health. 30 495–509.GRIFFITH, D. (2008). Geographic sampling of urban soils for contaminant mapping: how many samples and from where., Environ. Geochem. Health. 30 495–509.
KYUNG, M. and GHOSH, S. K. (2010). Maximum likelihood estimation for directional conditionally autoregressive models., Journal of Statistical Planning and Inference 140 3160–3179. 1204.62164 10.1016/j.jspi.2010.04.012KYUNG, M. and GHOSH, S. K. (2010). Maximum likelihood estimation for directional conditionally autoregressive models., Journal of Statistical Planning and Inference 140 3160–3179. 1204.62164 10.1016/j.jspi.2010.04.012
LEDOIT, O. and WOLF, M. (2002). Some Hypothesis Tests for the covariance matrix when the dimension is large compared to the sample size., The Annals of Statistics, 30, 1081–1102. 1029.62049 10.1214/aos/1031689018 euclid.aos/1031689018LEDOIT, O. and WOLF, M. (2002). Some Hypothesis Tests for the covariance matrix when the dimension is large compared to the sample size., The Annals of Statistics, 30, 1081–1102. 1029.62049 10.1214/aos/1031689018 euclid.aos/1031689018
MARDIA, K. and MARSHALL, R. (1984). Maximum likelihood of models for residual covariance in spatial regression., Biometrika 71 135–146. 0542.62079 10.1093/biomet/71.1.135MARDIA, K. and MARSHALL, R. (1984). Maximum likelihood of models for residual covariance in spatial regression., Biometrika 71 135–146. 0542.62079 10.1093/biomet/71.1.135
RICHARDSON, S. (1990). Some remarks on the testing of association between spatial processes. In: Griffith, D. (Ed.), Spatial Statistics: Past, Present, and Future., Institute of Mathematical Geography, Ann Arbor, MI, 277?309.RICHARDSON, S. (1990). Some remarks on the testing of association between spatial processes. In: Griffith, D. (Ed.), Spatial Statistics: Past, Present, and Future., Institute of Mathematical Geography, Ann Arbor, MI, 277?309.
SCHABENBERGER, O. and GOTWAY, C. A. (2005)., Statistical Methods for Spatial Data Analysis. Boca Raton, FL: Chapman & Hall/CRC. 1068.62096SCHABENBERGER, O. and GOTWAY, C. A. (2005)., Statistical Methods for Spatial Data Analysis. Boca Raton, FL: Chapman & Hall/CRC. 1068.62096
SOO, YUH-WEN. and BATES, D. M. (1992). Loosely coupled nonlinear least squares., Computational Statistics and Data Analysis, 14, 249–259. 0875.62281 10.1016/0167-9473(92)90177-HSOO, YUH-WEN. and BATES, D. M. (1992). Loosely coupled nonlinear least squares., Computational Statistics and Data Analysis, 14, 249–259. 0875.62281 10.1016/0167-9473(92)90177-H
STEIN, M. (1999)., Interpolation of Spatial Data: Some Theory of Kriging. New York: Springer. MR1697409 0924.62100STEIN, M. (1999)., Interpolation of Spatial Data: Some Theory of Kriging. New York: Springer. MR1697409 0924.62100