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Equalities between OLSE, BLUE and BLUP in the linear model. (English) Zbl 1334.62110
Summary: We consider equalities between the ordinary least squares estimator (OLSE), the best linear unbiased estimator (BLUE) and the best linear unbiased predictor (BLUP) in the general linear model $$\{\mathbf y,\mathbf X\boldsymbol\beta,\mathbf V\}$$ extended with the new unobservable future value $$\mathbf y_\ast$$ of the response whose expectation is $$\mathbf X_\ast\boldsymbol\beta$$. Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the BLUP and provide new results giving upper bounds for the Euclidean norm of the difference between the $$\mathrm{BLUP}(\mathbf y_\ast)$$ and $$\mathrm{BLUE}(\mathbf X_\ast\boldsymbol\beta)$$ and between the $$\mathrm{BLUP}(\mathbf y_\ast)$$ and $$\mathrm{OLSE}(\mathbf X_\ast\mathbf\beta)$$. A remark is made on the application to small area estimation.

##### MSC:
 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis 62H20 Measures of association (correlation, canonical correlation, etc.)
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##### References:
 [1] Alderman, H; Babita, M; Demombynes, G; Makhatha, N; Özler, B, How low can you go? combining census and survey data for mapping poverty in south africa, J Afr Econ, 11, 169-200, (2003) [2] Amarasinghe U, Madar S, Anputhas M (2005) Spatial clustering of rural poverty and food insecurity in Sri Lanka. Food Policy 30:493-509 · Zbl 1153.62334 [3] Baksalary, JK; Kala, R, A new bound for the Euclidean norm of the difference between the least squares and the best linear unbiased estimators, Ann Stat, 8, 679-681, (1980) · Zbl 0464.62055 [4] Baksalary, JK; Kala, R, Simple least squares estimation versus best linear unbiased prediction, J Stat Plan Inference, 5, 147-151, (1981) · Zbl 0476.62057 [5] Baksalary, JK; Mathew, T, Rank invariance criterion and its application to the unified theory of least squares, Linear Algebra Appl, 127, 393-401, (1990) · Zbl 0694.15003 [6] Baksalary, JK; Puntanen, S; Styan, GPH, A property of the dispersion matrix of the best linear unbiased estimator in the general Gauss-Markov model, Sankhyā Ser A, 52, 279-296, (1990) · Zbl 0727.62072 [7] Baksalary OM, Trenkler G (2009) A projector oriented approach to the best linear unbiased estimator. Stat Pap 50:721-733 · Zbl 1247.62165 [8] Baksalary, OM; Trenkler, G, Between OLSE and BLUE, Aust NZ J Stat, 53, 289-303, (2011) · Zbl 1334.62106 [9] Ben-Israel A, Greville TNE (2003) Generalized inverses: theory and applications, 2nd edn. Springer, New York · Zbl 1026.15004 [10] Christensen R (2011) Plane answers to complex questions: the theory of linear models, 4th edn. Springer, New York · Zbl 1266.62043 [11] Davidson R, MacKinnon JG (2004) Econometric theory and methods. Oxford University Press, New York [12] Elbers, C; Lanjouw, JO; Lanjouw, P, Micro-level estimation of poverty and inequality, Econometrica, 71, 355-364, (2003) · Zbl 1184.91163 [13] Elbers C, Lanjouw JO, Lanjouw P, Leite PG (2004) Poverty and inequality in Brazil: new estimates from combined PPV-PNAD data. The World Bank. Vrije Universiteit, Amsterdam · Zbl 0171.17103 [14] Elian, SN, Simple forms of the best linear unbiased predictor in the general linear regression model, Am Stat, 54, 25-28, (2000) [15] Farrow, A; Larrea, C; Hyman, G; Lema, G, Exploring the spatial variation of food poverty in ecuador, Food Policy, 30, 510-531, (2005) [16] Goldberger, AS, Best linear unbiased prediction in the generalized linear regression model, J Am Stat Assoc, 58, 369-375, (1962) · Zbl 0124.35502 [17] Groß J (2004) The general Gauss-Markov model with possibly singular dispersion matrix. Stat Pap 45: 311-336 · Zbl 1048.62064 [18] Harville DA (1997) Matrix algebra from a statistician’s perspective. Springer, New York · Zbl 0881.15001 [19] Haslett, SJ; Isidro, MC; Jones, G, Comparison of survey regression techniques in the context of small area estimation of poverty, Surv Methodol, 36, 157-170, (2010) [20] Haslett SJ, Jones G (2006) Small area estimation of poverty, caloric intake and malnutrition in Nepal. Nepal Central Bureau of Statistics / World Food Programme, United Nations / World Bank, Kathmandu, p 184 ISBN 999337018-5 [21] Haslett, SJ; Jones, G, Small area estimation of poverty: the aid industry standard and its alternatives, Aust NZ J Stat, 52, 341-362, (2010) · Zbl 1373.62047 [22] Haslett, SJ; Puntanen, S, A note on the equality of the BLUPs for new observations under two linear models, Acta et Commentationes Universitatis Tartuensis de Mathematica, 14, 27-33, (2010) · Zbl 1229.15023 [23] Haslett, SJ; Puntanen, S, Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Stat Pap, 51, 465-475, (2010) · Zbl 1247.62167 [24] Haslett, SJ; Puntanen, S, On the equality of the BLUPs under two linear mixed models, Metrika, 74, 381-395, (2011) · Zbl 1226.62066 [25] Hauke, J; Markiewicz, A; Puntanen, S, Comparing the BLUEs under two linear models, Commun Stat Theory Methods, 41, 2405-2418, (2012) · Zbl 1319.62138 [26] Horn, RA; Olkin, I, When does $$A^{*}A=B^{*}B$$ and why does one want to know?, Am Math Mon, 103, 470-482, (1996) · Zbl 0861.15011 [27] Isotalo, J; Puntanen, S, Linear prediction sufficiency for new observations in the general Gauss-Markov model, Commun Stat Theory Methods, 35, 1011-1023, (2006) · Zbl 1102.62072 [28] Isotalo, J; Puntanen, S; Styan, GPH, A useful matrix decomposition and its statistical applications in linear regression, Commun Stat Theory Methods, 37, 1436-1457, (2008) · Zbl 1163.62051 [29] Kam, S-P; Hossain, M; Bose, ML; Villano, LS, Spatial patterns of rural poverty and their relationship with welfare-influencing factors in bangladesh, Food Policy, 30, 551-567, (2005) [30] Kristjanson, P; Radeny, M; Baltenweck, I; Ogutu, J; Notenbaert, A, Livelihood mapping and poverty correlates at a meso-level in kenya, Food Policy, 30, 568-583, (2005) [31] Liu, Y, On equality of ordinary least squares estimator, best linear unbiased estimator and best linear unbiased predictor in the general linear model, J Stat Plan Inf, 139, 1522-1529, (2009) · Zbl 1153.62334 [32] Mäkinen, J, Bounds for the difference between a linear unbiased estimate and the best linear unbiased estimate, Phys Chem Earth Part A, 25, 693-698, (2000) [33] Mäkinen, J, A bound for the Euclidean norm of the difference between the best linear unbiased estimator and a linear unbiased estimator, J Geodesy, 76, 317-322, (2002) · Zbl 1158.86335 [34] McElroy, FW, A necessary and sufficient condition that ordinary least-squares estimators be best linear unbiased, J Am Stat Assoc, 62, 1302-1304, (1967) · Zbl 0153.48102 [35] Minot, N; Baulch, B, Spatial patterns of poverty in Vietnam and their implications for policy, Food Policy, 30, 461-475, (2005) [36] Molina, I; Rao, JNK, Small area estimation of poverty indicators, Can J Stat, 38, 369-385, (2010) · Zbl 1235.62140 [37] Pordzik, PR, A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model, Stat Pap, 53, 299-304, (2012) · Zbl 1440.62087 [38] Puntanen S, Styan GPH (1989) The equality of the ordinary least squares estimator and the best linear unbiased estimator (with discussion). Am Stat 43:151-161 (Commented by Kempthorne O on pp 161-162 and by Searle SR on pp 162-163, reply by the authors on p 164) [39] Puntanen S, Styan GPH, Isotalo J (2011) Matrix tricks for linear statistical models: our personal top twenty. Springer, Heidelberg · Zbl 1291.62014 [40] Puntanen S, Styan GPH, Werner HJ (2000) A comment on the article “Simple forms of the best linear unbiased predictor in the general linear regression model” by SN Elian (Am Stat 54:25-28). Lett Editor Am Stat 54:327-328 [41] Rao, CR; Cam, LM (ed.); Neyman, J (ed.), Least squares theory using an estimated dispersion matrix and its application to measurement of signals, No. 1, 355-372, (1967), Berkeley [42] Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York · Zbl 0236.15004 [43] Sengupta D, Jammalamadaka SR (2003) Linear models: an integrated approach. World Scientific, River Edge · Zbl 1049.62080 [44] Watson, GS, Prediction and the efficiency of least squares, Biometrika, 59, 91-98, (1972) · Zbl 0232.62032 [45] Zyskind, G, On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models, Ann Math Stat, 38, 1092-1109, (1967) · Zbl 0171.17103
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