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| Khuri A.I. — Advanced calculus with applications in statistics |
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| Предметный указатель |
Heteroscedasticity 118
Holder’s Inequality 230—231
Homogeneous function 317
Homogeneous function, Euler’s theorem for 317
Homoscedasticity 118
Hotelling’s  statistic 49
Householder matrix 38
Hypergeometric distribution 463
Hypergeometric probability 462
Hypergeometric series 157
Idempotent matrix 38
Ill conditioning 46
Image of a set 5
Implicit function 273
Implicit function theorem 282
Importance sampling 537
Improper integral 220
Improper integral, absolutely convergent 222 226
Improper integral, conditionally convergent 222 226
Improper integral, of the first kind 220 528
Improper integral, of the second kind 221 225
Indefinite integral 217
Indeterminate form 107
Inequality, Cauchy — Schwarz 25 229
Inequality, Chebyshev’s 19 184 251
Inequality, Holder’s 230 231
Inequality, Jensen’s 84 233
Inequality, Markov’s 19 185
Inequality, Minkowski’s 232
Infimum 9 73
Inner product 23
Input variables 339 351
Integral test 153 224
Integral, improper 220
Integral, indefinite 217
Integral, Riemann 206 293
Integral, Riemann — Stieltjes 234
Interaction 43
Interior point 262
Intermediate-value theorem 71 217
Interpolation 410
Interpolation, error 414 416 423
Interpolation, Lagrange 411 419 422
Interpolation, points 411—412 414 423
Intersection 2
Interval of convergence 174 190
Inverse function 6 76 102
Inverse function theorem 280 305
Inverse of a matrix 34
Inverse regression 242
Irrational number 9
Jacobi polynomials 443
Jacobi polynomials, applications of 462
Jacobian, determinant 268
Jacobian, matrix 268
Jensen’s Inequality 84 233
Jordan content 298
Jordan measurable 298
Kernel of a linear transformation 26
Khinchine’s theorem 192
Knots 418 431
Kolmogorov’s theorem 192
Kronecker product 30
Kummer’s test 154
Lack of fit 342 399
Lagrange interpolating polynomial 411 416 423 518 521
Lagrange interpolation 413 419 422
Lagrange interpolation, accuracy of 413
Lagrange multipliers 288 312 329 341 344 352 383
Laguerre polynomials 451
Laguerre polynomials, applications of 462
Laplace, approximation 531 546 548
Laplace, method of 531 534 546
Law of Large Numbers 84
Law of large numbers, Bernoulli’s 204
Law of Large Numbers, strong 192
Law of large numbers, weak 192
Least upper bound 9 73
Least-squares estimate 46 344
Least-squares polynomial approximation 453—455
Legendre polynomials 440 442
Leibniz’s formula 127 443 449 452
level of significance 15
Likelihood, equations 308 373
Likelihood, function 308 373
LIMIT 57
Limit point 11 262
Limit, left sided 59
Limit, lower 136
Limit, of a multivariable function 262
Limit, of a sequence 133
Limit, one sided 58
Limit, right sided 59
Limit, subsequential 136
Limit, two sided 58
Limit, upper 136
Linear model 43 46 121 195 355 367 382
Linear model, balanced 43
Linear model, first order 340
Linear model, second order 343
Linear span 23
Linear transformation 25
Linear transformation, kernel of 26
Linear transformation, one-to-one 27
Linearly dependent 22
Linearly independent 23
Lipschitz condition 75
Lipschitz continuous function 75 409
Logarithmic series distribution 193
Logistic distribution 241
Loss function 86
Lower limit 136
Lower semicontinuous function 90
Lower sum 206 294
L’Hospital’s Rule 103 120 254
Maclaurin’s integral test 153
Maclaurin’s series 111
Mapping 5
Markov chain Monte Carlo 549
Markov’s inequality 19 185
Matrix 28
Matrix, diagonal 28
Matrix, full column rank 34
Matrix, full rank 34
Matrix, full row rank 34
Matrix, Hessian 275 285 374
Matrix, Householder 38
Matrix, idempotent 38
Matrix, identity 28
Matrix, inverse of 34
Matrix, Jacobian 268
Matrix, moment 363
Matrix, orthogonal 38 351 357
Matrix, partitioned 30
Matrix, rank of 33
Matrix, singular 32 37
Matrix, skew symmetric 29
Matrix, square 28
Matrix, symmetric 29
Matrix, trace of 29
Matrix, transpose of 29
Maximum likelihood estimate 308 372
Maximum, absolute 113 283 346
Maximum, local 113 283
Maximum, relative 113
Mean 117 239
Mean response 343 359 367 423
Mean response, maximum of 343
Mean squared error 360 431
Mean squared error, integrated 360 431
| Mean value theorem 99—100 271
Mean Value Theorem for Integrals 217
Median 124
Mertens’s theorem 163
Method of least squares 121
Minimax estimator 86
Minimization of prediction variance 356
Minimum bias estimation 398
Minimum norm quadratic unbiased estimation 327 378
Minimum, absolute 113 283 336 346
Minimum, local 113 283
Minimum, relative 113
Minkowski’s inequality 232
Minor 31
Minor, leading principal 32 40 285
Minor, principal 32 292
Modal approximation 549
Modulus of continuity 407
Moment generating function 186 250 506
Moment matrix 363
Moments 182 505
Moments, central 182 240 457
Moments, design 362
Moments, Factorial 191
Moments, first negative 242
Moments, noncentral 182 240
Moments, region 361
Monotone function 76 101 210
Monotone sequence 133
Monte Carlo method 535 540
Monte Carlo method, error bound for 537
Monte Carlo method, variance reduction 537
Monte Carlo simulation 83
Multicollinearity 46 196 353 387
Multimodal function 336 338
Multinomial distribution 325
Multiresponse, experiment 370
Multiresponse, function 370
Multiresponse, optimization 370
Multivariable function 261
Multivariable function, composite 265 276
Multivariable function, continuous 264
Multivariable function, inverse of 280
Multivariable function, limit of 262
Multivariable function, optima of 283
Multivariable function, partial derivative of 267
Multivariable function, Riemann integral of 293
Multivariable function, Taylor’s theorem for 277
Multivariable function, uniformly continuous 265
Negative binomial 182 195
Neighborhood 10 262
Neighborhood, deleted 57 262
Newton — Cotes methods 523
Newton — Raphson method 331 373
Noncentral chi-squared 44
Noncentrality parameter 44
Nonlinear model 367
Nonlinear model, approximate linearization of 422
Nonlinear model, design for 367
Nonlinear parameter 367
Norm 179
Norm, Euclidean 24 42 179 261
Norm, minimum 380
Norm, of a partition 205
Norm, spectral 179
Normal distribution 120 306
Normal distribution, bivariate 311
Normal distribution, p-variate 400
Normal integral 460 469
Normal integral, approximation of 460 469
Normal random variable 245
Null space 26
o notation 66 117
One-to-one correspondence 5
One-to-one function 5
One-way classification model 387
Open set 10 262
Optimality of design, A 365
Optimality of design, D 364 367 431
Optimality of design, E 365
Optimality of design, G 364
optimization techniques 339
Optimum, absolute 113
Optimum, compromise 371
Optimum, conditions 370
Optimum, ideal 371
Optimum, local 113 284
Ordered n-tuple 4
Ordered pair 3
Orthogonal complement 25
Orthogonal design 358
Orthogonal functions 437
Orthogonal matrix 38 351 357
Orthogonal matrix, parameterization 49
Orthogonal polynomials 437 453
Orthogonal polynomials, Chebyshev 444
Orthogonal polynomials, Chebyshev, of the first kind 444
Orthogonal polynomials, Chebyshev, of the second kind 445
Orthogonal polynomials, Chebyshev, zeros of 444
Orthogonal polynomials, Hermite 447
Orthogonal polynomials, Hermite, applications of 456
Orthogonal polynomials, Jacobi 443
Orthogonal polynomials, Laguerre 451
Orthogonal polynomials, least-squares approximation with 453
Orthogonal polynomials, Legendre 440 442
Orthogonal polynomials, on a finite set 455
Orthogonal polynomials, sequence of 437
Orthogonal polynomials, zeros of 440
Orthogonal vectors 24
Orthonormal basis 24
Parameterization of orthogonal matrices, Parseval’s theorem 496
Partial derivative 267
Partial sum 140
Partially nonlinear model 369
Partition 205 294
Partition of a set 5
periodic extension 482
Periodic function 473 489 504
Periodogram 501
Piecewise continuous function 481 486 498
Poisson approximation 194
Poisson distribution 119 183 204
Poisson distribution, truncated 373
Poisson process 122 131
Poisson random variable 14 124 183
Polynomial, approximation 403
Polynomial, Bernstein 406 410
Polynomial, Chebyshev 444
Polynomial, Hermite 447 456 542
Polynomial, interpolation 410
Polynomial, Jacobi 443 462
Polynomial, Lagrange interpolating 411 416 423
Polynomial, Laguerre 451 462
Polynomial, Legendre 440 442
Polynomial, piecewise 418
Polynomial, trigonometric 495 500 504
Power series 174
Power series, distribution 194
Prediction equation 348
Prediction variance 347 350 352 356
Prediction variance, minimization of 356
Prediction variance, standardized 363—364
Principal components 328
Probability generating function 190
Probability generating function, continuity theorem for 192
Probability of an event 13
Probability space 13
Product of matrices 29
Projection of a vector 25
Proper subset 1
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