Index 245

model, 86

linear span-space, 87

Lipschitz condition, 102

Lipschitz function, 102

local asymptotic normality

(LAN) condition, 29

local linear estimator of regression

function, 105

local polynomial approximation, 115

local polynomial estimator, 118

location parameter, 4

log-likelihood function, 4

log-likelihood ratio, 24

loss function, 11

absolute, 11

bounded, 11

quadratic, 11

sup-norm, 102

lower bound

asymptotically minimax, 23

Cram´ er-Rao, 7

minimax, 18

Markov stopping time, see

stopping time, 66

maximum likelihood estimator

(MLE), 4, 33

maximum normalized risk, 16, 103

mean integrated squared error (MISE),

90

mean squared error (MSE), 90

mean squared risk at a point, see mean

squared error (MSE), 90

measurable event, see

Ft-measurable event, 65

up to random time, 68

minimax estimator, 16

minimax lower bound, 18

minimax risk, 16

of detection, 69

minimax risk of detection, 69

minimax separation rate, 229

more eﬃcient estimator, 12

multiple regression model, 193

Nadaraya-Watson kernel

estimator, 106

non-linear functional, 188

non-linear integral functional, 188

nonparametric hypotheses testing, 228

nonparametric regression model, 101

normal equations, 88

normalized quadratic risk, 12

normalized risk, 11

maximum, 16

normalized risk function, see

normalized risk, 11

null hypothesis, 227

on-line detection problem, 69

on-line detector, 69

on-line estimation, 78

on-line estimator, 78

optimal bandwidth, 108, 118

optimal smoothing kernel

estimator, 109

orthogonal series, see projection

estimator, 147

orthonormal basis, 141

parametric hypotheses testing, 227

parametric regression model, 85

linear, 86

random error in, 86

partition of unity, 153

pixel, 208

point estimator, see estimator, 4

polynomial regression, 86

posterior density, 14

weighted, 14

posterior mean, 14

non-weighted, 14

weighted, 14

power spline, 156

predicted, see fitted response

vector, 87

predictor variable, see explanatory

variable, 85

prior density, 13

prior distribution, 13

conjugate, 15

projection, see orthogonal series

estimator, 147

quadratic loss function, 11

random error, 85

random regular design, 95

random time, 68

random walk,

two-sided Gaussian, 52

rate of convergence, 23

regression coeﬃcient, 85

least-squares estimator of, 88

regression equation, 85, 101