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Basic symbols

AT A-t

A­tr(A)

det(A)

I diag (at, ... , ar )

.A ( A ), X'cva,( Ji1' )

Y N

m

fJ "J J, Jc , etc. Jw

1](fJ) J(fJ),H(fJ)

e int (9) e r

- an equality given by a definition - the transposition of the matrix A - the inverse of A - the generalized inverse of A (Section 1.1) - the trace of A - the determinant of A - the identity matrix - the r x r diagonal matrix with diagonal elements

at,.··, ar

- the column- and the kernel space of A (Section 1.1) - the error vector - the observed vector - usually, the number of observations (the dimension

ofy) - usually, the number of parameters (the dimension

of fJ) - the vector of unknown parameters - the true value of fJ - estimators of fJ - the Gauss-Markov (maximum likekelihood) estima-

tor (see (2.3.1» - the mean value of y when fJ is the parameter - the matrix of the first- and the array of the second-

order derivatives of 1]( fJ) (see Section 4.1) - the same in exponential families (Section 9.2) - the same for the derivatives of the canonical map-

ping (Section 9.2) - the parameter space - the interior of the set 9 - the closure of the set 9 - the space of the canonical parameter (Section 9.1)

256

P, Pw , P('I?),etc.

P" , P" Pr, Pr", etc. &

a~/. a.~/. 81/1(x) 'P, J'P' 8x

E( ), E,,( ), etc. Var( ) u2W

BASIC SYMBOLS

- projectors in the regression model (projection ma­trices) (see Section 1.1, (1.3.3)-(1.3.5), (2.2.5), (2.5.1), (3.1.1), etc.)

- projectors in exponential families (see Section 9.2) - probabilities - the expectation surface or the expectation plane

(see Section 3.1, Section 4.2, (9.2.1» - the tangent plane and the tangent space to C at

the point 'I? - the canonical surface (see (9.2.2»

- different symbols for derivatives (see Section 2.1 for different notations)

- mean values - the variance or the variance matrix - the covariance matrix of the error or of the observed

vector in regression models E( 'I?), E-y - the variance matrix of the sufficient statistic in ex­

ponential families (see (9.1.4» M,Mw,M('I?),Mw('I?) - the information matrix (for u = 1 in regression

Me, Mc('I?) ( , ), 1111 (a, b)e,

II lie Kint( 'I?)

Pint Cint ('I?), Cpar( 'I?) d('I?) q( ,11'1?)

q( ,11 'I?) , qe( ,11'1?) Q(,1, 'I?), Q(,1, 'I?)

D(b, ,1)

models) (see Section 1.6, Remark 2.2.1, (9.2.8» - formally like Mw, Mw('I?) but for W = C - a general inner product and norm - aTC-1b or aTC-b

- the norm correspponding to { , )e - the intrinsic curvature (see Proposition 3.1.1 and

Sections 4.2, 1.1, 9.2) - the parameter-effect curvature (see Sections 3.1,

4.2,9.2) - = [KintC'I?)]-l = the radius of curvature - curvatures of the canonical surface (Section 9.2) - the ancillary space (see Sections 3.2, 5.4, 7.2)

- the global approximation of the probability density of,1w (see (3.3.1»

- the same for other estimators (Chapters 7,9)

- the modified information matrices (see Proposition 3.3.1, Proposition 7.1.1 and Section 7.2)

- a matrix defined in Proposition 1.1.1

BASIC SYMBOLS

R(-Q)

Wi( -Q)

O(-Q) N( -Q), Z(-Q)

257

- the set of all (potentionally) regular and the set of singular points of int (0) (Section 4.1)

- the Riemannian curvature tensor (see (4.2.2), (7.6.2))

- the i th vector of the basis of the ancillary space (Sections 4.4, 7.1,9.4)

- = (Wt(-Q), ••• ,WN-m(-Q)) (see Section 6.2) - the intrinsic and the parameter-effect curvature ar-

rays (Sections 5.5,6.2) - the I-divergence (see (9.2.3), (9.2.5))

Subject index

affine connections 243 almost exact 69, 186

approximation 68 ancillary space 125, 173 approximate probability density 163,

174, 178, 179, 184, 186, 236 arc-length 57 asymptotic

normality 133, 201, 210 properties 131

bias 44, 140 bounded curvature 156

canonical mapping 224 surface 224

confidence interval 72, 74 region 27, 52, 53, 195, 197, 200

covering exponential family 218, 231 criterion of

A-optimality 29 D-optimality 29 E-optimality 30

curvature 57 arrays 128 vector 39

curve in the set e 38 on the surface CO 39

design 30, 209 measure 31

diffeomorphism 98 differentiable manifold 95

eigenvalue of a matrix 8 elliptically symmetrical distribution

182 entropy 76, 151 equivariant density 154 error vector 12, 14 estimable parameter function 24 estimators of cr 206 expectation

curve 56 mapping 224 plane 14 surface 224

experimental design 29, 209 explanatory variable 13 exponential families 215

first-order approximation 131, 140 flat models 92, 192

g-inverse matrices 9 Gauss-Markov estimator 16,41 Gauss-Newton method 116, 178 geodesic curve 39, 86,242 global approximations 154 gradient method 122

I-divergence 224 implicit function theorem 103, 143 information matrix 31,228,241 intrinsic curvature 58,60,88, 178,229

SUBJECT INDEX

intrinsically linear models 36, 90, 163 iterative methods 113

L2 estimator 15, 61, 101, 113 least-squares estimator 15 Levenberg-Marquardt method 124 likelihood ratio 203 linear

approximation 43 regression model 13

local approximations 131

maximal likelihood estimator 61,236, 66

mean-square error matrix 149 measure of information 76 metric tensor 241 moments of dw 74, 149

Newton's method 122 non-overlapping 157 normal equation 17,63, 145, 173 normed vector of curvature 63

orthogonal projector 12, 40, 53, 60, 228

orthonormal basis 125, 146, 179 overlapping 66

parameter effect curvature 61, 88, 229 space 36

pivotal variables 195 posterior density 177 potentially regular point 81 prior density 177, 211 projector 11,69,86,206

quantile 27 quasigradient methods 123

radius of curvature 57 Rao distance 241 regular

linear models 14 model 81 point 81 reparametrization 56, 89, 230

residual vector 49, 64, 152, 207 response variable 13

259

restricted sample space 156, 241 Riemannian curvature tensor 94, 185

saddle-point approximation 230 second-order approximation 140, 142 shift vector 173 singular

linear models 14 model 81, 94 point 81

square root of a matrix 8 step size 116, 118, 214 stopping rules 114 strongly consistent 132

tail product 132 tangent

plane 44, 97 space 53

three dimensional array 34 tube 66

unbiased estimator for (72 27 unbiased linear estimator 24