Further research on simple projection prediction under a general linear model

Abstract

Linear model is a kind of important model in statistics. The estimation and prediction of unknown parameters in the model is a basic problem in statistical inference. According to different evaluation criteria, different forms of predictors are obtained. The simple projection prediction (SPP) boasts simple form while the best linear unbiased prediction (BLUP) enjoys very excellent properties. Naturally, the relationship between the two predictors are considered. We give the necessary and sufficient conditions for the equality of SPP and BLUP. Finally, we study the robustness of SPP and BLUP with respect to covariance matrix and illustrate the application of equivalence conditions by use of error uniform correlation models.

Keywords

Linear model / SPP / BLUP / equivalence / robustness

References

[1] Alalouf I S, Styan G P H. Characterizations of estimability in the general linear model. Ann Statist 1979; 7(1): 194–200

[2] Bolfarine H, Rodrigues J. On the simple projection predictor in finite populations. Austral J Statist 1988; 30(3): 338–341

[3] Bolfarine H, Zacks S, Elian S N, Rodrigues J. Optimal prediction of the finite population regression coefficient. Sankhya Ser B 1994; 56(1): 1–10

[4] Drygas H. Estimation and prediction for linear models in general spaces. Math Operationsforsch Statist 1975; 6(2): 301–324

[5] Gan S J, Sun Y Q, Tian Y G. Equivalence of predictors under real and over-parameterized linear models. Comm Statist Theory Methods 2017; 46(11): 5368–5383

[6] Goldberger A S. Best linear unbiased prediction in the generalized linear regression models. J Amer Statist Assoc 1962; 57: 369–375

[7] Henderson C R. Best linear unbiased estimation and prediction under a selection model. Biometrics 1975; 31(2): 423–447

[8] Jiang J M. A derivation of BLUP—best linear unbiased predictor. Statist Probab Lett 1997; 32(3): 321–324

[9] Liu X Q, Rong J Y, Liu X Y. Best linear unbiased prediction for linear combinations in general mixed linear models. J Multivariate Anal 2008; 99(8): 1503–1517

[10] Lu C L, Gan S J, Tian Y G. Some remarks on general linear model with new regressors. Statist Probab Lett 2015; 97: 16–24

[11] Lu C L, Sun Y Q, Tian Y G. A comparison between two competing fixed parameter constrained general linear models with new regressors. Statistics 2018; 52(4): 769–781

[12] Marsaglia G, Styan G P H. Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 1974/1975; 2: 269–292

[13] Penrose R. A generalized inverse for matrices. Proc Cambridge Philos Soc 1955; 51: 406–413

[14] PuntanenSStyanG P HIsotaloJ. Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Heidelberg: Springer, 2011

[15] Rao C R. Unified theory of linear estimation. Sankhya Ser A 1971; 33: 371–394

[16] Rao C R. Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix. J Multivariate Anal 1973; 3: 276–292

[17] Rao C R. A lemma on optimization of a matrix function and a review of the unified theory of linear estimation, In: Statistical Data Analysis and Inference (Neuchatel, 1989). Amsterdam: North-Holland 1989; 397–417

[18] Särndal C-E, Wright R L. Cosmetic form of estimators in survey sampling. Scand J Statist 1984; 11(3): 146–156

[19] Tian Y G. A new derivation of BLUPs under random-effects model. Metrika 2015; 78(8): 905–918

[20] Tian Y G. A matrix handling of predictions under a general linear random-effects model with new observations. Electron J Linear Algebra 2015; 29: 30–45

[21] Tian Y G. Solutions of a constrained Hermitian matrix-valued function optimization problem with applications. Oper Matrices 2016; 10(4): 967–983

[22] Tian Y G, Jiang B. A new analysis of the relationships between a general linear model and its mis-specified forms. J Korean Statist Soc 2017; 46(2): 182–193

[23] Yu S H, He C Z. Comparison of general Gauss-Markov models in estimable subspace. Acta Math Appl Sinica 1997; 20(4): 580–586

[24] Yu S H, He C Z. Optimal prediction in finite populations. Appl Math J Chinese Univ Ser A 2000; 15(2): 199–205

[25] Yu S H, Liang X L. The simple projection predictor in finite populations with arbitrary rank. Mathematics in Economics 2001; 18(4): 49–52

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