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The Asymptotic Distribution of REML Estimators

Author

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  • Cressie, N.
  • Lahiri, S. N.

Abstract

Restricted maximum likelihood (REML) estimation is a method employed to estimate variance-covariance parameters from data that follow a Gaussian linear model. In applications, it has either been conjectured or assumed that REML estimators are asymptotically Gaussian with zero mean and variance matrix equal to the inverse of the restricted information matrix. In this article, we give conditions under which the conjecture is true and apply our results to variance-components models. An important application of variance components is to census undercount; a simulation is carried out to verify REML's properties for a typical census undercount model.

Suggested Citation

  • Cressie, N. & Lahiri, S. N., 1993. "The Asymptotic Distribution of REML Estimators," Journal of Multivariate Analysis, Elsevier, vol. 45(2), pages 217-233, May.
    • Handle: RePEc:eee:jmvana:v:45:y:1993:i:2:p:217-233
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    Cited by:

    1. François Bachoc & Emile Contal & Hassan Maatouk & Didier Rullière, 2017. "Gaussian processes for computer experiments," Post-Print hal-01665936, HAL.
    2. Tingjin Chu & Jialuo Liu & Jun Zhu & Haonan Wang, 2022. "Spatio-Temporal Expanding Distance Asymptotic Framework for Locally Stationary Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 689-713, August.
    3. Yu, Dalei & Bai, Peng & Ding, Chang, 2015. "Adjusted quasi-maximum likelihood estimator for mixed regressive, spatial autoregressive model and its small sample bias," Computational Statistics & Data Analysis, Elsevier, vol. 87(C), pages 116-135.
    4. Tatsuya Kubokawa, 2010. "On Measuring Uncertainty of Small Area Estimators with Higher Order Accuracy," CIRJE F-Series CIRJE-F-754, CIRJE, Faculty of Economics, University of Tokyo.
    5. Tatsuya Kubokawa, 2009. "A Review of Linear Mixed Models and Small Area Estimation," CIRJE F-Series CIRJE-F-702, CIRJE, Faculty of Economics, University of Tokyo.
    6. Sandy Burden & Noel Cressie & David G. Steel, 2015. "The SAR Model for Very Large Datasets: A Reduced Rank Approach," Econometrics, MDPI, vol. 3(2), pages 1-22, May.
    7. Matt Higham & Michael Dumelle & Carly Hammond & Jay Hoef & Jeff Wells, 2024. "An Application of Spatio-Temporal Modeling to Finite Population Abundance Prediction," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 29(3), pages 491-515, September.
    8. Ghosh, Debashis, 2014. "An asymptotically minimax kernel machine," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 33-38.
    9. Bachoc, François & Lagnoux, Agnès & Nguyen, Thi Mong Ngoc, 2017. "Cross-validation estimation of covariance parameters under fixed-domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 42-67.
    10. Kramlinger, Peter & Schneider, Ulrike & Krivobokova, Tatyana, 2023. "Uniformly valid inference based on the Lasso in linear mixed models," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    11. Ganggang Xu & Marc G. Genton, 2017. "Tukey -and- Random Fields," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1236-1249, July.
    12. Tsung-Ta David Hsu & Danlin Yu & Meiyin Wu, 2023. "Predicting Fecal Indicator Bacteria Using Spatial Stream Network Models in A Mixed-Land-Use Suburban Watershed in New Jersey, USA," IJERPH, MDPI, vol. 20(6), pages 1-17, March.
    13. Liu, Jialuo & Chu, Tingjin & Zhu, Jun & Wang, Haonan, 2021. "Semiparametric method and theory for continuously indexed spatio-temporal processes," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    14. Bachoc, François, 2014. "Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 1-35.
    15. Deo, Rohit S., 2012. "Improved forecasting of autoregressive series by weighted least squares approximate REML estimation," International Journal of Forecasting, Elsevier, vol. 28(1), pages 39-43.

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