Some properties of linear sufficiency and the BLUPs in the linear mixed model
Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Haslett, S. J. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017 |
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| Schlagwörter: |
Best linear unbiased estimator |
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| Anmerkung: |
© Springer-Verlag GmbH Germany 2017 |
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| Übergeordnetes Werk: |
Enthalten in: Statistical papers - Springer Berlin Heidelberg, 1988, 61(2017), 1 vom: 25. Aug., Seite 385-401 |
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| Übergeordnetes Werk: |
volume:61 ; year:2017 ; number:1 ; day:25 ; month:08 ; pages:385-401 |
| Links: |
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| DOI / URN: |
10.1007/s00362-017-0943-3 |
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OLC202502973X |
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| 520 | |a Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. | ||
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10.1007/s00362-017-0943-3 doi (DE-627)OLC202502973X (DE-He213)s00362-017-0943-3-p DE-627 ger DE-627 rakwb eng 300 330 510 VZ Haslett, S. J. verfasserin aut Some properties of linear sufficiency and the BLUPs in the linear mixed model 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany 2017 Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. Best linear unbiased estimator Best linear unbiased predictor Linear mixed model Linear model Misspecified model Liu, X. Q. aut Markiewicz, A. aut Puntanen, S. (orcid)0000-0002-6776-0173 aut Enthalten in Statistical papers Springer Berlin Heidelberg, 1988 61(2017), 1 vom: 25. Aug., Seite 385-401 (DE-627)129572292 (DE-600)227641-0 (DE-576)015069486 0932-5026 nnns volume:61 year:2017 number:1 day:25 month:08 pages:385-401 https://doi.org/10.1007/s00362-017-0943-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_11 GBV_ILN_26 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4046 GBV_ILN_4277 GBV_ILN_4319 GBV_ILN_4326 AR 61 2017 1 25 08 385-401 |
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10.1007/s00362-017-0943-3 doi (DE-627)OLC202502973X (DE-He213)s00362-017-0943-3-p DE-627 ger DE-627 rakwb eng 300 330 510 VZ Haslett, S. J. verfasserin aut Some properties of linear sufficiency and the BLUPs in the linear mixed model 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany 2017 Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. Best linear unbiased estimator Best linear unbiased predictor Linear mixed model Linear model Misspecified model Liu, X. Q. aut Markiewicz, A. aut Puntanen, S. (orcid)0000-0002-6776-0173 aut Enthalten in Statistical papers Springer Berlin Heidelberg, 1988 61(2017), 1 vom: 25. Aug., Seite 385-401 (DE-627)129572292 (DE-600)227641-0 (DE-576)015069486 0932-5026 nnns volume:61 year:2017 number:1 day:25 month:08 pages:385-401 https://doi.org/10.1007/s00362-017-0943-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_11 GBV_ILN_26 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4046 GBV_ILN_4277 GBV_ILN_4319 GBV_ILN_4326 AR 61 2017 1 25 08 385-401 |
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10.1007/s00362-017-0943-3 doi (DE-627)OLC202502973X (DE-He213)s00362-017-0943-3-p DE-627 ger DE-627 rakwb eng 300 330 510 VZ Haslett, S. J. verfasserin aut Some properties of linear sufficiency and the BLUPs in the linear mixed model 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany 2017 Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. Best linear unbiased estimator Best linear unbiased predictor Linear mixed model Linear model Misspecified model Liu, X. Q. aut Markiewicz, A. aut Puntanen, S. (orcid)0000-0002-6776-0173 aut Enthalten in Statistical papers Springer Berlin Heidelberg, 1988 61(2017), 1 vom: 25. Aug., Seite 385-401 (DE-627)129572292 (DE-600)227641-0 (DE-576)015069486 0932-5026 nnns volume:61 year:2017 number:1 day:25 month:08 pages:385-401 https://doi.org/10.1007/s00362-017-0943-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_11 GBV_ILN_26 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4046 GBV_ILN_4277 GBV_ILN_4319 GBV_ILN_4326 AR 61 2017 1 25 08 385-401 |
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10.1007/s00362-017-0943-3 doi (DE-627)OLC202502973X (DE-He213)s00362-017-0943-3-p DE-627 ger DE-627 rakwb eng 300 330 510 VZ Haslett, S. J. verfasserin aut Some properties of linear sufficiency and the BLUPs in the linear mixed model 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany 2017 Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. Best linear unbiased estimator Best linear unbiased predictor Linear mixed model Linear model Misspecified model Liu, X. Q. aut Markiewicz, A. aut Puntanen, S. (orcid)0000-0002-6776-0173 aut Enthalten in Statistical papers Springer Berlin Heidelberg, 1988 61(2017), 1 vom: 25. Aug., Seite 385-401 (DE-627)129572292 (DE-600)227641-0 (DE-576)015069486 0932-5026 nnns volume:61 year:2017 number:1 day:25 month:08 pages:385-401 https://doi.org/10.1007/s00362-017-0943-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_11 GBV_ILN_26 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4046 GBV_ILN_4277 GBV_ILN_4319 GBV_ILN_4326 AR 61 2017 1 25 08 385-401 |
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10.1007/s00362-017-0943-3 doi (DE-627)OLC202502973X (DE-He213)s00362-017-0943-3-p DE-627 ger DE-627 rakwb eng 300 330 510 VZ Haslett, S. J. verfasserin aut Some properties of linear sufficiency and the BLUPs in the linear mixed model 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany 2017 Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. Best linear unbiased estimator Best linear unbiased predictor Linear mixed model Linear model Misspecified model Liu, X. Q. aut Markiewicz, A. aut Puntanen, S. (orcid)0000-0002-6776-0173 aut Enthalten in Statistical papers Springer Berlin Heidelberg, 1988 61(2017), 1 vom: 25. Aug., Seite 385-401 (DE-627)129572292 (DE-600)227641-0 (DE-576)015069486 0932-5026 nnns volume:61 year:2017 number:1 day:25 month:08 pages:385-401 https://doi.org/10.1007/s00362-017-0943-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_11 GBV_ILN_26 GBV_ILN_70 GBV_ILN_130 GBV_ILN_2018 GBV_ILN_4046 GBV_ILN_4277 GBV_ILN_4319 GBV_ILN_4326 AR 61 2017 1 25 08 385-401 |
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some properties of linear sufficiency and the blups in the linear mixed model |
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Some properties of linear sufficiency and the BLUPs in the linear mixed model |
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Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. © Springer-Verlag GmbH Germany 2017 |
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Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. © Springer-Verlag GmbH Germany 2017 |
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Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ for $$\mathbf {X}\varvec{\beta }$$, for $$\mathbf {Z}\mathbf {u}$$ and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ of $$\mathbf {X}\varvec{\beta }$$ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ for some matrix $$\mathbf {A}$$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under one mixed model continue to be $${{\mathrm{BLUE}}}$$s and/or $${{\mathrm{BLUP}}}$$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach. © Springer-Verlag GmbH Germany 2017 |
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