Linear and nonlinear models: fixed effects, random effects, and mixed models
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
de Gruyter
2006
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 752 S. graph. Darst. |
| ISBN: | 3110162164 9783110162165 |
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MARC
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| 100 | 1 | |a Grafarend, Erik W. |d 1939-2020 |e Verfasser |0 (DE-588)121959368 |4 aut | |
| 245 | 1 | 0 | |a Linear and nonlinear models |b fixed effects, random effects, and mixed models |c Erik W. Grafarend |
| 264 | 1 | |a Berlin [u.a.] |b de Gruyter |c 2006 | |
| 300 | |a XX, 752 S. |b graph. Darst. | ||
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| adam_text | Contents
1 The first problem of algebraic regression
consistent system of linear observational equations
underdetermined system of linear equations:
{Ax = y|AelR i ,ye^(A)~rkA = n,« = dimY} 1
1 1 Introduction 3
1 11 The front page example 4
1 12 The front page example in matrix algebra 5
1 13 Minimum norm solution of the front page example by
means of horizontal rank partitioning 7
1 14 The range 1Z(f) and the kernel 7V(A) 9
1 15 Interpretation of MINOS by three partitionings 12
1 2 The minimum norm solution: MINOS 17
1 21 A discussion of the metric of the parameter space X 23
1 22 Alternative choice of the metric of the parameter space X 24
1 23 Gx MINOS and its generalized inverse 25
1 24 Eigenvalue decomposition of Gx MINOS:
canonical MINOS 26
1 3 Case study:
Orthogonal functions, Fourier series versus Fourier Legendre
series, circular harmonic versus spherical harmonic regression 40
1 31 Fourier series 41
1 32 Fourier Legendre series 52
1 4 Special nonlinear models 68
1 41 Taylor polynomials, generalized Newton iteration 68
1 42 Linearized models with datum defect 74
1 5 Notes 82
2 The first problem of probabilistic regression special Gauss
Markov model with datum defect Setup of the linear uniformly
minimum bias estimator of type LUMBE for fixed effects 85
2 1 Setup of the linear uniformly minimum bias estimator of type
LUMBE 86
2 2 The Equivalence Theorem of Gs MINOS and S LUMBE 90
2 3 Examples 91
3 The second problem of algebraic regression
inconsistent system of linear observational equations
overdetermined system of linear equations:
{Ax + i = y I A e R m ,y € ft (A) ~ rk A = m,m = dimX} 95
3 1 Introduction 97
3 11 The front page example 91
xiv Contents
3 12 The front page example in matrix algebra 98
3 13 Least squares solution of the front page example by means
of vertical rank partitioning 100
3 14 The range U(f) and the kernel Af(f), interpretation of the
least squares solution by three partitionings 103
3 2 The least squares solution: LESS 111
3 21 A discussion of the metric of the parameter space X 118
3 22 Alternative choices of the metric of the
observation space Y 119
3 221 Optimal choice of weight matrix: SOD 120
3 222 The Taylor Karman criterion matrix 124
3 223 Optimal choice of the weight matrix: 125
The space TI(a) and fc(A)1
3 224 Fuzzy sets 129
3 23 Gx LESS and its generalized inverse 129
3 24 Eigenvalue decomposition of Gy LESS: canonical LESS 131
3 3 Case study
Partial redundancies, latent conditions, high leverage points
versus break points, direct and inverse Grassmann coordinates,
Pliicker coordinates 143
3 31 Canonical analysis of the hat matrix,
partial redundancies, high leverage points 143
3 32 Multilinear algebra, join and meet ,
the Hodge star operator 152
3 33 From A to B: latent restrictions, Grassmann coordinates,
Pliicker coordinates 158
3 34 From B to A: latent parametric equations,
dual Grassmann coordinates, dual Plücker coordinates 172
3 35 Breakpoints 176
3 4 Special linear and nonlinear models
A family of means for direct observations 184
3 5 A historical note on C. F. Gauss, A. M. Legendre and the
invention of Least Squares and its generalization 185
4 The second problem of probabilistic regression
special Gauss Markov model without datum defect
Setup of BLUUE for the moments of first order and of BIQUUE
for the central moment of second order 187
4 1 Introduction 190
4 11 The front page example 191
4 12 Estimators of type BLUUE and BIQUUE of the
front page example 192
4 13 BLUUE and BIQUUE of the front page example, sample
median, median absolute deviation 201
Contents xv
4 14 Alternative estimation Maximum Likelihood (MALE) 205
4 2 Setup of the best linear uniformly unbiased estimators of type
BLUUE for the moments of first order 208
4 21 The best linear uniformly unbiased estimation
Ç of I : Ly BLUUE 208
4 22 The Equivalence Theorem of Gy LESS and E„ BLUUE 216
4 3 Setup of the best invariant quadratic uniform
by unbiased estimator of type BIQUUE for the
central moments of second order 217
4 31 Block partitioning of the dispersion matrix and linear
space generated by variance covariance components 218
4 32 Invariant quadratic estimation of variance covariance
components of type IQE 223
4 33 Invariant quadratic uniformly unbiased estimations of
variance covariance components of type IQUUE 226
4 34 Invariant quadratic uniformly unbiased estimations
of one variance component (IQUUE) from
Ly BLUUE: HIQUUE 230
4 35 Invariant quadratic uniformly unbiased estimators of
variance covariance components of Helmert type:
HIQUUE versus HIQE 232
4 36 Best quadratic uniformly unbiased estimations of one
variance component: BIQUUE 236
5 The third problem of algebraic regression
inconsistent system of linear observational equations
with datum defect overdetermined underdermined system of linear
equations: {Ax + i = y | A e R xm, y € U(A) ~ rk A mm{m,n } 243
5 1 Introduction 245
5 11 The front page example 246
5 12 The front page example in matrix algebra 246
5 13 Minimum norm least squares solution of the front page
example by means of additive rank partitioning 248
5 14 Minimum norm least squares solution of the front page
example by means of multiplicative rank partitioning: 252
5 15 The range 1Z(f) and the kernel M(f) interpretation of
MINOLESS by three partitionings 256
5 2 MINOLESS and related solutions like weighted minimum norm
weighted least squares solutions 263
5 21 The minimum norm least squares solution: MINOLESS 263
5 22 (G^G,.) MINOS and its generalized inverse 273
5 23 Eigenvalue decomposition of (Gx, Gv ) MINOLESS 277
5 24 Notes 282
xvi Contents
5 3 The hybrid approximation solution: a HAPS and Tykhonov
Phillips regularization 282
6 The third problem of probabilistic regression
special Gauss Markov model with datum problem
Setup of BLUMBE and BLE for the moments of first order and
of BIQUUE and BIQE for the central moment of second order 285
6 1 Setup of the best linear minimum bias estimator
of type BLUMBE 287
6 11 Definitions, lemmas and theorems 289
6 12 The first example: BLUMBE versus BLE, BIQUUE
versus BIQE, triangular leveling network 296
6 121 The first example: I3,13 BLUMBE 297
6 122 The first example: V, S BLUMBE 301
6 123 The first example: I3,13 BLE 306
6 124 The first example: V, S BLE 308
6 2 Setup of the best linear estimators of type hom BLE,
hom S BLE and hom a BLE for fixed effects 312
7 A spherical problem of algebraic representation
Inconsistent system of directional observational equations
overdetermined system of nonlinear equations on curved
manifolds 327
7 1 Introduction 328
7 2 Minimal geodesic distance: MINGEODISC 331
7 3 Special models: from the circular normal distribution to the
oblique normal distribution 335
7 31 A historical note of the von Mises distribution 335
7 32 Oblique map projection 337
7 33 A note on the angular metric 340
7 4 Case study 341
8 The fourth problem of probabilistic regression
special Gauss Markov model with random effects
Setup of BLIP and VIP for the moments of first order 347
8 1 The random effect model 348
8 2 Examples 362
9 The fifth problem of algebraic regression the system of
conditional equations: homogeneous and inhomogeneous equations
{By = Bi versus c + By = Bi} 373
9 1 Gy LESS of system of inconsistent homogeneous
conditional equations 374
9 2 Solving a system of inconsistent
inhomogeneous conditional equations 376
Contents xvjj
9 3 Examples 377
10 The fifth problem of probabilistic regression
general Gauss Markov model with mixed effects
Setup of BLUUE for the moments of first order
(Kolmogorov Wiener prediction) 379
10 1 Inhomogeneous general linear Gauss Markov model
(fixed effects and random effects) 380
10 2 Explicit representations of errors in the general
Gauss Markov model with mixed effects 385
10 3 An example for collocation 386
10 4 Comments 397
11 The sixth problem of probabilistic regression
the random effect model errors in variables 401
11 1 Solving the nonlinear system of the model
errors in variables 404
11 2 Example: The straight line fit 406
11 3 References 410
12 The sixth problem of generalized algebraic regression
the system of conditional equations with unknowns
(Gauss Helmert model) 411
12 1 Solving the system of homogeneous condition equations
with unknowns 414
12 11 W LESS 414
12 12 R, W MINOLESS 416
12 13 R, W HAPS 419
12 14 R, W MINOLESS against R, W HAPS 421
12 2 Examples for the generalized algebraic regression problem:
homogeneous conditional equations with unknowns 421
12 21 The first case: I LESS 422
12 22 The second case: I, I MINOLESS 422
12 23 The third case: I, I HAPS 423
12 24 The fourth case: R, W MINOLESS,
R positive semidefinite, Wpositive semidefinite 423
12 3 Solving the system of inhomogeneous condition equations
with unknowns 424
12 31 W LESS 424
12 32 R, W MINOLESS 426
12 33 R, W HAPS 427
12 34 R, W MINOLESS against R, W HAPS 428
12 4 Conditional equations with unknowns: from the algebraic
approach to the stochastic one 429
xviii Contents
12 41 Shift to the center 429
12 42 The condition of unbiased estimators 429
12 43 The first step: unbiased estimation of % and E{t} 430
12 44 The second step: unbiased estimation Kt and k2 430
13 The nonlinear problem of the 3d datum transformation and the
Procrustes Algorithm 431
13 1 The 3d datum transformation and the Procrustes Algorithm 433
13 2 The variance covariance matrix of the error matrix E 441
13 3 Case studies: The 3d datum transformation and the
Procrustes Algorithm 441
13 4 References 444
14 The seventh problem of generalized algebraic regression
revisited: The Grand Linear Model:
The split level model of conditional equations with unknowns
(general Gauss Helmert model) 445
14 1 Solutions of type W LESS 446
14 2 Solutions of type R, W MINOLESS 449
14 3 Solutions of type R, W HAPS 450
14 4 Review of the various models: the sixth problem 453
15 Special problems of algebraic regression and stochastic estimation:
multivariate Gauss Markov model, the n way classification model,
dynamical systems 455
15 1 The multivariate Gauss Markov model a special problem
of probabilistic regression 455
15 2 n way classification models 460
15 21 A first example: 1 way classification 460
15 22 A second example: 2 way classification
without interaction 464
15 23 A third example: 2 way classification with interaction 469
15 24 Higher classifications with interaction 474
15 3 Dynamical Systems 476
Appendix A: Matrix Algebra 485
Al Matrix Algebra 485
A2 Special Matrices 488
A3 Scalar Measures and Inverse Matrices 495
A4 Vectorvalued Matrix Forms 506
A5 Eigenvalues and Eigenvectors 509
A6 Generalized Inverses 513
Contents xix
Appendix B: Matrix Analysis 522
Bl Derivations of Scalar valued and Vector valued
Vector Functions 522
B2 Derivations of Trace Forms 523
B3 Derivations of Determinantal Forms 526
B4 Derivations of a Vector/Matrix Function of a Vector/Matrix 527
B5 Derivations of the Kronecker Zehfuß product 528
B6 Matrix valued Derivatives of Symmetric or
Antisymmetric Matrix Functions 528
B7 Higher order derivatives 530
Appendix C: Lagrange Multipliers 533
Cl A first way to solve the problem 533
Appendix D: Sampling distributions and their use:
Confidence Intervals and Confidence Regions 543
Dl A first vehicle: Transformation of random variables 543
D2 A second vehicle: Transformation of random variables 547
D3 A first confidence interval of Gauss Laplace normally
distributed observations: ju, j2 known, the Three Sigma Rule 553
D31 The forward computation of a first confidence interval of
Gauss Laplace normally distributed observations: /j, t2 known 557
D32 The backward computation of a first confidence interval of
Gauss Laplace normally distributed observations: fu,a2 known 564
D4 Sampling from the Gauss Laplace normal distribution:
a second confidence interval for the mean, variance known 567
D41 Sampling distributions of the sample mean /}, a2 known,
and of the sample variance â2 582
D42 The confidence interval for the sample mean, variance known 592
D5 Sampling from the Gauss Laplace normal distribution:
a third confidence interval for the mean, variance unknown 596
D51 Student s sampling distribution of the random variable (/j /u)/â 596
D52 The confidence interval for the sample mean, variance unknown 605
D53 The Uncertainty Principle 611
D6 Sampling from the Gauss Laplace normal distribution:
a fourth confidence interval for the variance 613
D61 The confidence interval for the variance 613
D62 The Uncertainty Principle 619
xx Contents
D7 Sampling from the multidimensional Gauss Laplace
normal distribution: the confidence region for the fixed
parameters in the linear Gauss Markov model 621
Appendix E: Statistical Notions 163
El Moments of a probability distribution, the Gauss Laplace
normal distribution and the quasi normal distribution 644
E2 Error propagation 648
E3 Useful identities 651
E4 The notions of identifiability and unbiasedness 652
Appendix F: Bibliographic Indexes 655
References 659
Index 745
|
| adam_txt |
Contents
1 The first problem of algebraic regression
consistent system of linear observational equations
underdetermined system of linear equations:
{Ax = y|AelR" i'",ye^(A)~rkA = n,« = dimY} 1
1 1 Introduction 3
1 11 The front page example 4
1 12 The front page example in matrix algebra 5
1 13 Minimum norm solution of the front page example by
means of horizontal rank partitioning 7
1 14 The range 1Z(f) and the kernel 7V(A) 9
1 15 Interpretation of "MINOS" by three partitionings 12
1 2 The minimum norm solution: "MINOS" 17
1 21 A discussion of the metric of the parameter space X 23
1 22 Alternative choice of the metric of the parameter space X 24
1 23 Gx MINOS and its generalized inverse 25
1 24 Eigenvalue decomposition of Gx MINOS:
canonical MINOS 26
1 3 Case study:
Orthogonal functions, Fourier series versus Fourier Legendre
series, circular harmonic versus spherical harmonic regression 40
1 31 Fourier series 41
1 32 Fourier Legendre series 52
1 4 Special nonlinear models 68
1 41 Taylor polynomials, generalized Newton iteration 68
1 42 Linearized models with datum defect 74
1 5 Notes 82
2 The first problem of probabilistic regression special Gauss
Markov model with datum defect Setup of the linear uniformly
minimum bias estimator of type LUMBE for fixed effects 85
2 1 Setup of the linear uniformly minimum bias estimator of type
LUMBE 86
2 2 The Equivalence Theorem of Gs MINOS and S LUMBE 90
2 3 Examples 91
3 The second problem of algebraic regression
inconsistent system of linear observational equations
overdetermined system of linear equations:
{Ax + i = y I A e R"'m ,y € ft (A) ~ rk A = m,m = dimX} 95
3 1 Introduction 97
3 11 The front page example 91
xiv Contents
3 12 The front page example in matrix algebra 98
3 13 Least squares solution of the front page example by means
of vertical rank partitioning 100
3 14 The range U(f) and the kernel Af(f), interpretation of the
least squares solution by three partitionings 103
3 2 The least squares solution: "LESS" 111
3 21 A discussion of the metric of the parameter space X 118
3 22 Alternative choices of the metric of the
observation space Y 119
3 221 Optimal choice of weight matrix: SOD 120
3 222 The Taylor Karman criterion matrix 124
3 223 Optimal choice of the weight matrix: 125
The space TI(a) and fc(A)1
3 224 Fuzzy sets 129
3 23 Gx LESS and its generalized inverse 129
3 24 Eigenvalue decomposition of Gy LESS: canonical LESS 131
3 3 Case study
Partial redundancies, latent conditions, high leverage points
versus break points, direct and inverse Grassmann coordinates,
Pliicker coordinates 143
3 31 Canonical analysis of the hat matrix,
partial redundancies, high leverage points 143
3 32 Multilinear algebra, "join" and "meet",
the Hodge star operator 152
3 33 From A to B: latent restrictions, Grassmann coordinates,
Pliicker coordinates 158
3 34 From B to A: latent parametric equations,
dual Grassmann coordinates, dual Plücker coordinates 172
3 35 Breakpoints 176
3 4 Special linear and nonlinear models
A family of means for direct observations 184
3 5 A historical note on C. F. Gauss, A. M. Legendre and the
invention of Least Squares and its generalization 185
4 The second problem of probabilistic regression
special Gauss Markov model without datum defect
Setup of BLUUE for the moments of first order and of BIQUUE
for the central moment of second order 187
4 1 Introduction 190
4 11 The front page example 191
4 12 Estimators of type BLUUE and BIQUUE of the
front page example 192
4 13 BLUUE and BIQUUE of the front page example, sample
median, median absolute deviation 201
Contents xv
4 14 Alternative estimation Maximum Likelihood (MALE) 205
4 2 Setup of the best linear uniformly unbiased estimators of type
BLUUE for the moments of first order 208
4 21 The best linear uniformly unbiased estimation
Ç of I : Ly BLUUE 208
4 22 The Equivalence Theorem of Gy LESS and E„ BLUUE 216
4 3 Setup of the best invariant quadratic uniform
by unbiased estimator of type BIQUUE for the
central moments of second order 217
4 31 Block partitioning of the dispersion matrix and linear
space generated by variance covariance components 218
4 32 Invariant quadratic estimation of variance covariance
components of type IQE 223
4 33 Invariant quadratic uniformly unbiased estimations of
variance covariance components of type IQUUE 226
4 34 Invariant quadratic uniformly unbiased estimations
of one variance component (IQUUE) from
Ly BLUUE: HIQUUE 230
4 35 Invariant quadratic uniformly unbiased estimators of
variance covariance components of Helmert type:
HIQUUE versus HIQE 232
4 36 Best quadratic uniformly unbiased estimations of one
variance component: BIQUUE 236
5 The third problem of algebraic regression
inconsistent system of linear observational equations
with datum defect overdetermined underdermined system of linear
equations: {Ax + i = y | A e R"xm, y € U(A) ~ rk A mm{m,n\} 243
5 1 Introduction 245
5 11 The front page example 246
5 12 The front page example in matrix algebra 246
5 13 Minimum norm least squares solution of the front page
example by means of additive rank partitioning 248
5 14 Minimum norm least squares solution of the front page
example by means of multiplicative rank partitioning: 252
5 15 The range 1Z(f) and the kernel M(f) interpretation of
"MINOLESS" by three partitionings 256
5 2 MINOLESS and related solutions like weighted minimum norm
weighted least squares solutions 263
5 21 The minimum norm least squares solution: "MINOLESS" 263
5 22 (G^G,.) MINOS and its generalized inverse 273
5 23 Eigenvalue decomposition of (Gx, Gv ) MINOLESS 277
5 24 Notes " 282
xvi Contents
5 3 The hybrid approximation solution: a HAPS and Tykhonov
Phillips regularization 282
6 The third problem of probabilistic regression
special Gauss Markov model with datum problem
Setup of BLUMBE and BLE for the moments of first order and
of BIQUUE and BIQE for the central moment of second order 285
6 1 Setup of the best linear minimum bias estimator
of type BLUMBE 287
6 11 Definitions, lemmas and theorems 289
6 12 The first example: BLUMBE versus BLE, BIQUUE
versus BIQE, triangular leveling network 296
6 121 The first example: I3,13 BLUMBE 297
6 122 The first example: V, S BLUMBE 301
6 123 The first example: I3,13 BLE 306
6 124 The first example: V, S BLE 308
6 2 Setup of the best linear estimators of type hom BLE,
hom S BLE and hom a BLE for fixed effects 312
7 A spherical problem of algebraic representation
Inconsistent system of directional observational equations
overdetermined system of nonlinear equations on curved
manifolds 327
7 1 Introduction 328
7 2 Minimal geodesic distance: MINGEODISC 331
7 3 Special models: from the circular normal distribution to the
oblique normal distribution 335
7 31 A historical note of the von Mises distribution 335
7 32 Oblique map projection 337
7 33 A note on the angular metric 340
7 4 Case study 341
8 The fourth problem of probabilistic regression
special Gauss Markov model with random effects
Setup of BLIP and VIP for the moments of first order 347
8 1 The random effect model 348
8 2 Examples 362
9 The fifth problem of algebraic regression the system of
conditional equations: homogeneous and inhomogeneous equations
{By = Bi versus c + By = Bi} 373
9 1 Gy LESS of system of inconsistent homogeneous
conditional equations 374
9 2 Solving a system of inconsistent
inhomogeneous conditional equations 376
Contents xvjj
9 3 Examples 377
10 The fifth problem of probabilistic regression
general Gauss Markov model with mixed effects
Setup of BLUUE for the moments of first order
(Kolmogorov Wiener prediction) 379
10 1 Inhomogeneous general linear Gauss Markov model
(fixed effects and random effects) 380
10 2 Explicit representations of errors in the general
Gauss Markov model with mixed effects 385
10 3 An example for collocation 386
10 4 Comments 397
11 The sixth problem of probabilistic regression
the random effect model "errors in variables" 401
11 1 Solving the nonlinear system of the model
"errors in variables" 404
11 2 Example: The straight line fit 406
11 3 References 410
12 The sixth problem of generalized algebraic regression
the system of conditional equations with unknowns
(Gauss Helmert model) 411
12 1 Solving the system of homogeneous condition equations
with unknowns 414
12 11 W LESS 414
12 12 R, W MINOLESS 416
12 13 R, W HAPS 419
12 14 R, W MINOLESS against R, W HAPS 421
12 2 Examples for the generalized algebraic regression problem:
homogeneous conditional equations with unknowns 421
12 21 The first case: I LESS 422
12 22 The second case: I, I MINOLESS 422
12 23 The third case: I, I HAPS 423
12 24 The fourth case: R, W MINOLESS,
R positive semidefinite, Wpositive semidefinite 423
12 3 Solving the system of inhomogeneous condition equations
with unknowns 424
12 31 W LESS 424
12 32 R, W MINOLESS 426
12 33 R, W HAPS 427
12 34 R, W MINOLESS against R, W HAPS 428
12 4 Conditional equations with unknowns: from the algebraic
approach to the stochastic one 429
xviii Contents
12 41 Shift to the center 429
12 42 The condition of unbiased estimators 429
12 43 The first step: unbiased estimation of % and E{t} 430
12 44 The second step: unbiased estimation Kt and k2 430
13 The nonlinear problem of the 3d datum transformation and the
Procrustes Algorithm 431
13 1 The 3d datum transformation and the Procrustes Algorithm 433
13 2 The variance covariance matrix of the error matrix E 441
13 3 Case studies: The 3d datum transformation and the
Procrustes Algorithm 441
13 4 References 444
14 The seventh problem of generalized algebraic regression
revisited: The Grand Linear Model:
The split level model of conditional equations with unknowns
(general Gauss Helmert model) 445
14 1 Solutions of type W LESS 446
14 2 Solutions of type R, W MINOLESS 449
14 3 Solutions of type R, W HAPS 450
14 4 Review of the various models: the sixth problem 453
15 Special problems of algebraic regression and stochastic estimation:
multivariate Gauss Markov model, the n way classification model,
dynamical systems 455
15 1 The multivariate Gauss Markov model a special problem
of probabilistic regression 455
15 2 n way classification models 460
15 21 A first example: 1 way classification 460
15 22 A second example: 2 way classification
without interaction 464
15 23 A third example: 2 way classification with interaction 469
15 24 Higher classifications with interaction 474
15 3 Dynamical Systems 476
Appendix A: Matrix Algebra 485
Al Matrix Algebra 485
A2 Special Matrices 488
A3 Scalar Measures and Inverse Matrices 495
A4 Vectorvalued Matrix Forms 506
A5 Eigenvalues and Eigenvectors 509
A6 Generalized Inverses 513
Contents xix
Appendix B: Matrix Analysis 522
Bl Derivations of Scalar valued and Vector valued
Vector Functions 522
B2 Derivations of Trace Forms 523
B3 Derivations of Determinantal Forms 526
B4 Derivations of a Vector/Matrix Function of a Vector/Matrix 527
B5 Derivations of the Kronecker Zehfuß product 528
B6 Matrix valued Derivatives of Symmetric or
Antisymmetric Matrix Functions 528
B7 Higher order derivatives 530
Appendix C: Lagrange Multipliers 533
Cl A first way to solve the problem 533
Appendix D: Sampling distributions and their use:
Confidence Intervals and Confidence Regions 543
Dl A first vehicle: Transformation of random variables 543
D2 A second vehicle: Transformation of random variables 547
D3 A first confidence interval of Gauss Laplace normally
distributed observations: ju, j2 known, the Three Sigma Rule 553
D31 The forward computation of a first confidence interval of
Gauss Laplace normally distributed observations: /j, t2 known 557
D32 The backward computation of a first confidence interval of
Gauss Laplace normally distributed observations: fu,a2 known 564
D4 Sampling from the Gauss Laplace normal distribution:
a second confidence interval for the mean, variance known 567
D41 Sampling distributions of the sample mean /}, a2 known,
and of the sample variance â2 582
D42 The confidence interval for the sample mean, variance known 592
D5 Sampling from the Gauss Laplace normal distribution:
a third confidence interval for the mean, variance unknown 596
D51 Student's sampling distribution of the random variable (/j /u)/â 596
D52 The confidence interval for the sample mean, variance unknown 605
D53 The Uncertainty Principle 611
D6 Sampling from the Gauss Laplace normal distribution:
a fourth confidence interval for the variance 613
D61 The confidence interval for the variance 613
D62 The Uncertainty Principle 619
xx Contents
D7 Sampling from the multidimensional Gauss Laplace
normal distribution: the confidence region for the fixed
parameters in the linear Gauss Markov model 621
Appendix E: Statistical Notions 163
El Moments of a probability distribution, the Gauss Laplace
normal distribution and the quasi normal distribution 644
E2 Error propagation 648
E3 Useful identities 651
E4 The notions of identifiability and unbiasedness 652
Appendix F: Bibliographic Indexes 655
References 659
Index 745 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Grafarend, Erik W. 1939-2020 |
| author_GND | (DE-588)121959368 |
| author_facet | Grafarend, Erik W. 1939-2020 |
| author_role | aut |
| author_sort | Grafarend, Erik W. 1939-2020 |
| author_variant | e w g ew ewg |
| building | Verbundindex |
| bvnumber | BV021700865 |
| callnumber-first | Q - Science |
| callnumber-label | QA278 |
| callnumber-raw | QA278.2 |
| callnumber-search | QA278.2 |
| callnumber-sort | QA 3278.2 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 840 |
| ctrlnum | (OCoLC)62742596 (DE-599)BVBBV021700865 |
| dewey-full | 519.5/36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5/36 |
| dewey-search | 519.5/36 |
| dewey-sort | 3519.5 236 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV021700865 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:17:13Z |
| indexdate | 2024-07-09T20:41:59Z |
| institution | BVB |
| isbn | 3110162164 9783110162165 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014914810 |
| oclc_num | 62742596 |
| open_access_boolean | |
| owner | DE-824 DE-706 DE-634 DE-83 DE-11 |
| owner_facet | DE-824 DE-706 DE-634 DE-83 DE-11 |
| physical | XX, 752 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | de Gruyter |
| record_format | marc |
| spelling | Grafarend, Erik W. 1939-2020 Verfasser (DE-588)121959368 aut Linear and nonlinear models fixed effects, random effects, and mixed models Erik W. Grafarend Berlin [u.a.] de Gruyter 2006 XX, 752 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematisches Modell Mathematical models Regression analysis Regressionsmodell (DE-588)4127980-3 gnd rswk-swf Regressionsmodell (DE-588)4127980-3 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014914810&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Grafarend, Erik W. 1939-2020 Linear and nonlinear models fixed effects, random effects, and mixed models Mathematisches Modell Mathematical models Regression analysis Regressionsmodell (DE-588)4127980-3 gnd |
| subject_GND | (DE-588)4127980-3 |
| title | Linear and nonlinear models fixed effects, random effects, and mixed models |
| title_auth | Linear and nonlinear models fixed effects, random effects, and mixed models |
| title_exact_search | Linear and nonlinear models fixed effects, random effects, and mixed models |
| title_exact_search_txtP | Linear and nonlinear models fixed effects, random effects, and mixed models |
| title_full | Linear and nonlinear models fixed effects, random effects, and mixed models Erik W. Grafarend |
| title_fullStr | Linear and nonlinear models fixed effects, random effects, and mixed models Erik W. Grafarend |
| title_full_unstemmed | Linear and nonlinear models fixed effects, random effects, and mixed models Erik W. Grafarend |
| title_short | Linear and nonlinear models |
| title_sort | linear and nonlinear models fixed effects random effects and mixed models |
| title_sub | fixed effects, random effects, and mixed models |
| topic | Mathematisches Modell Mathematical models Regression analysis Regressionsmodell (DE-588)4127980-3 gnd |
| topic_facet | Mathematisches Modell Mathematical models Regression analysis Regressionsmodell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014914810&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT grafarenderikw linearandnonlinearmodelsfixedeffectsrandomeffectsandmixedmodels |