k

Index

Almost surely, 8, 473 Characteristic function (c.f.), 435 Alternative hypothesis, 62 Chernoff’s bound, 3, 468 Anderson–Darling statistic, 58 Coefficient of variation, 107, 329 k Auxiliary model, 401 COMFORT, 232 k Completeness, 286 Bartlett correction, 118 Conditional variance formula, 434 Bayesian information criterion, 119 Confidence Bayes’ rule, 424 coefficient, 13 Behrens–Fisher problem, 322 set, 13 Bessel function , 13, 313 modified, 422 actual coverage probability, 15 Beta function, 421 nominal level, 15, 313 incomplete, 422 pointwise, 49 Bias reduction, 293 simultaneous, 49 Bias–variance tradeoff, 348 Wald, 106 Binomial coefficient, 420 Consistency Binomial theorem, 421 weak, 5 Bootstrap Contrast, 315 double, 24 Convergence nonparametric,COPYRIGHTED 18, 334 almostMATERIAL surely, 10, 475 parametric, 17 complete, 12, 476 percentile, 20 continuity theorem for c.f., 14, 479 Borel–Cantelli lemmas, 7, 472 continuity theorem for m.g.f., 14, 479 Continuous mapping theorem, 14, 18, 478, Capital asset pricing model, 444 482 Central limit theorem, 455 Cramér–Wold device, 14, 479

Fundamental Statistical Inference: A Computational Approach, First Edition. Marc S. Paolella. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

561

k k

562 INDEX

Convergence (continued) Poisson, 307, 332, 333, 426, 430 in distribution, 13, 477 skew normal, 35, 36, 39, 441, 500, 503 Helly–Bray, 14, 478, 479 stable Paretian, 18, 69, 144, 202, 344, 358, in probability, 8, 473 368, 409, 483 in r-, 12, 477 Student’s t, 430, 446, 465 Scheffé’s lemma, 14, 478 testing Slutsky’s theorem, 14, 478 composite, 66 Convolution, 454 composite normality, 224 Correlation, 428 composite normality; Ghosh, 237 Countable additivity, 423 composite normality; Jarque–Bera, 234 Countable subadditivity, 5, 469 composite normality; Pearson, 240 Covariance, 428 composite normality; Stengos–Wu, 237 Coverage probability simple, 65 actual, 15 Torabi–Montazeri–Grané, 238 nominal, 15 uniform, 430 Cramér–Rao lower bound, 107 Weibull, 156, 205, 216, 305, 337, 430 Cross-validation, 76 DJIA, 233, 344, 366, 378 Cumulant generating function (c.g.f.), 435 Effect size, 80 Data-generating process, 114 Efficient estimator, 10 Delta method, 105 Elastic net, 76 de Moivre–Jordan theorem, 423 Elicitability, 440 De Morgan’s law, 6, 471 Empirical c.d.f., 38 Digamma function, 422 Entire function, 36, 501 Distribution Error function, 36, 501 k asymmetric double Weibull, 33, 498 complementary, 36, 501 k beta, 204, 430 imaginary, 36, 501 binomial, 425 Expectation-maximization algorithm, 169 Cauchy, 430 Expected shortfall, 437 contaminated normal, 160, 205 , 110, 425 discrete mixed normal, 157 Extreme value theory, 346 discrete mixture of normals, 447 double Weibull, 33, 498 Factorization, 272 F, 431 False discovery rate, 80 gamma, 155, 205, 307, 317, 430 Fast Fourier transform, 361 generalized asymmetric t (GAt), 142, 235, Fisher’s variance-stabilizing transformation, 446 328 generalized , 447 generalized hyperbolic, 457 Gamma function, 421 geometric, 289, 426 incomplete, 421 hyperbolic, 458 Generalized hypergeometric function, 422 hypergeometric, 426 German tank problem, 275 inverse hyperbolic sine (IHS), 396 Glivenko–Cantelli theorem, 38 Kolmogorov, 40 Goodness of fit, 37 Laplace, 205, 403, 430 Lévy, 23, 454, 488 Hill estimator, 347 multivariate normal, 462 Hypothesis test, 74 NIG, 458 asymptotically most powerful unbiased, 253 noncentral, 465 combined, 247 noncentral t, 35, 40, 351, 398, 500, 504 consistent, 68 Pareto, 431 most powerful, 252

k k

INDEX 563

power, 62 Likelihood principle, 80 power envelope, 247, 253 Likelihood ratio, 117, 253, 383 p-value, 59 UMP, 67 Mahalanobis distance, 99 UMPU, 68 robust, 101 unbiased, 68 Maximally existing moment, 342 Maximum likelihood Identifiability, 158 estimate, 6 Identifiable, 92 estimator, 6, 274 Indirect inference, 401 invariance, 89 Inequality singularity, 162 Bonferroni, 423 Mean squared error, 5, 286, 305 Boole’s, 5, 469 -unbiased estimation, 296 Cantelli’s, 3, 468 Method of moments estimator, 185 Cauchy–Schwarz, 428 Mid-p-values, 17 Chebyshev’s, 3, 468 Minimum covariance determinant (MCD), 100, Chebyshev’s order, 4, 469 232 Chernoff’s, 3, 468 Minimum description length, 119 Minimum variance bound estimator, 109 Cramér–Rao, 107 adjusted estimator, 297 DKW, 40 Model averaging, 164 Hölder’s,1,467 Model misspecification, 114 information, 107 Model selection, 114 Jensen, 427 Moment generating function (m.g.f.), 434 Kolmogorov’s other, 4, 469 Moment plots, 342 Lyapunov’s, 1, 467 k Multinomial theorem, 421 k Markov’s,2,468 Minkowski’s, 2, 467 NASDAQ, 133, 343 one-sided Chebyshev, 3, 468 Noncentral distribution, 465 triangle, 1, 467 Null bands Information pointwise, 212 Fisher, 91 simultaneous, 215 Kullback–Leibler, 102 matrix, 92 Optimization observed (Fisher), 87 BFGS, 139 Information inequality, 107 box constraints, 140, 164 Inversion formula, 436 CMAES, 149 differential evolution, 146 Jackknife, 302 evolutionary algorithm, 145 Jacobian transformation, 451 method of scoring, 137 method of steepest descent, 138 Kolmogorov–Smirnov distance, 57 Newton–Raphson, 137 Kullback–Leibler information, 102 Order statistics, 460 Kurtosis, 427 Outliers, 100, 159

Lasso, 174 Pitman closeness, 296 Law of likelihood, 80, 118 Pivot, 315 Law of the iterated expectation, 434 asymptotic, 318 Leading principle minor, 111 exact, 315 Leibniz’s rule, 317 Poincaré’s theorem, 423 Likelihood, 6, 85 Pooled variance estimator, 322

k k

564 INDEX

P-P plot, 210 Shrinkage estimation, 174, 200 Pre-test estimation, 323 , 427 Probability integral transform, 44, 58, 61, 327 Stein’s lemma, 443 Problem of coincidences, 424 Stop-loss premium, 438 Strong law of large numbers, 12, 477 Q-Q plot, 210 Subadditivity, 437 Quantile, 425 Sufficiency, 269 Quantile function, 448 minimal, 276 Quantile regression, 192, 257 Survivorship bias, 350 Quasi-Bayesian estimation, 176 Quasi-log-likelihood, 104 Tail estimation, 346 Taylor series, 420 Random effects models, 325 Tolerance parameters, 123 Randomized response technique, 403 Tower property of expectation, 434 Rao–Blackwell theorem, 283 Trimmed mean, 125 Regularity conditions, 86 Rényi’s representation, 347, 461 Unbiasedness Ridge regression, 174 asymptotic, 9 Robust estimation, 100 mean, 4 breakdown point, 101 Uniformly minimum variance unbiased masking, 100 estimator, 109, 286 Rosenblatt transformation, 242 Value-at-risk, 438 Saddlepoint approximation, 441, 456 Variance component, 315 Sandwich estimator, 105 Volatility clustering, 378 k Score function, 87 k Semi-parametric estimator, 346 Zero–one law, 8, 473

k