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a how many resamples 140–141 Agresti–Coull interval 217–218 parametric 437–441 Alternative hypothesis 48 percentile interval 118–119 ANOVA 419–421 vs. permutation test 122 permutation test approach 428 ratio of 129 regression 332 b relative risk 132 Barchart 21 sample rows of the data 332 k Bayes 391 single population 106 k Bayes theorem 392 smoothed 433–437 computational issues 446–452 107 importance 461–467 stratified 445–446 Bellman, Richard 447 t confidence interval 219–226 487 t-test 246–247 504–505 two populations 119 as prior for binomial data 400 variation of bootstrap distribution Bias 12, 131–133, 164 135, 139 487–488 Z interval 238 Blinding 12 Boxplot 28–29 Bonferroni correction 270 Bootstrap 103 c accuracy 135–140 Case studies bias 114,COPYRIGHTED 131–133 beer MATERIAL and hot wings 9, 37–40, bias for parametric bootstrap 440 51–55, 61, 145, 352, 472 bootstrap distributions and sampling birth weights of babies 2, 29, 103, distributions 107 202, 223, 248, 354 bootstrap idea, the 104 black spruce seedlings 10, 43, 72, bootstrap sample 104 231, 287, 313–314 center, bias, spread, shape 107, 114 bushmeat 336–340 distribution too narrow 436 flight delays 1, 21, 29, 42, 71, 95, estimate cdf with ecdf 112–113 147, 231, 384, 430

Mathematical Statistics with Resampling and R, Second Edition. Laura M. Chihara and Tim C. Hesterberg. © 2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc.

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Case studies (cont’d.) for one (𝜎 unknown) General Social Survey (GSS) 7, 42, 192–196 216, 218, 251, 359–364, 384 for one proportion 214–218 Google mobile ads 12, 43, 236, 279, one-sided 212–214 290 properties 226–228 recidivism 4, 43, 67, 72, 94, 380 score 215 Verizon repair times 3, 58–61, t confidence interval 194, 200, 62–64, 124, 129–131, 224 222, 223 wind energy 158, 170, 437 Wald 217 Categorical variable see Variable z 191 156, 175, 274 Conjugate family 402 Cause and effect 11 Consistency 173–175 cdf see Cumulative distribution Consulting see Statistical practice function Contingency table 22, 359 Cell means model 421 Continuity correction 89–91, Census 8 245–246, 253 Center 22, 25–26 Yates 370 Central limit theorem 84–94 Control group 12 accuracy 91–92 Convergence in probability 173 binomial data 86–91 Correlation 301 k finite population 92–93 bootstrap 332 k Chebyshev’s Inequality 484 coefficient 301 𝜒2 m 365 permutation test of independence Chi-square 336 and likelihood ratio (see Likelihood) sample 303 test of goodness-of-fit 374–379 Covariance 297–301 test of homogeneity 372–374 Cramer–Rao Inequality 170 test of independence 365–372 Credible interval 403 test statistic 361, 367 Critical Chi-square distribution 322, 365, region 259, 266 499–502 value 259 sum of chi-square random Cumulative distribution function 478 variables 500 Curse of dimensionality 447 Conditioning 371–372 Confidence interval d Agresti–Coull 217–218 Data sets bootstrap tvs.formulat 224 Alelager 287, 355 for difference of two proportions Bangladesh 114, 144, 220, 235, 218–219 247, 472 for difference of two means BookPrices 147 198–204 Cereals 383 in general 204–212 Challenger 356 for one mean (𝜎 known) 187–192 ChiMarathonMen 46, 429

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Cuckoos 429 e Diving2017 68–70, 124, 146, 204 ecdf see Empirical distribution Fatalities 340, 348–349 function FishMercury 144 EDA see Exploratory data analysis Girls2004 146, 231 Edgeworth approximation 92 Groceries 73, 146, 232 Effect size 265 IceCream 73, 146, 232 Efficiency 167–171 ILBoys 419, 425 relative 455, 456 Illiteracy 352, 355 Efron, Bradley 141 Lottery 387 EM algorithm 467–472 MathAnxiety 145, 355, 356 Empirical cumulative distribution Maunaloa 354 function 35 MnGroundwater 235 Error sum of squares see Sum of Nasdaq 285 squares NBA1617 351 Estimate 152 Olympics2012 230, 232, 350 Estimator 113, 152 Phillies2009 72, 357, 386 European stock option 454–458 Quakes 182, 473 Exhaustive calculation 54, 105 Quetzal 287, 352 Expectation-Maximization see EM RangersTwins2016 350 Expected value 478 k Salaries 287 Experiment 11 k Service 182 Experimental units 11 Skateboard 120 Exploratory data analysis 21 Skating2010 318, 327, 495–496 334, 336 rate parameter 496 Starcraft 430 scale parameter 495 Titanic 358 TV 145 f TXBirths2004 235 F Volleyball2009 353 distribution 425, 505–507 wafers 426, 430 statistic 425 Walleye 286, 355 test (see ANOVA) Watertable 357 Fisher, Ronald, A. 258, Degrees of freedom 267, 365 chi-square 365 Fisher information 170, 177 Welch’s approximation 199, 238 Fisher’s exact test 370–372 Delta method 441–445 Fitted value 306 Density estimate, kernel 435 Five-number summary 27 Design distribution see Importance Flat prior see Prior distribution see Specific distributions g Dot plot 24 G statistic 380 Double-blind see Blinding Galton, Francis 308

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Gamma distribution 497–499 Indicator function 459 sum of gamma random variables inference 8 499 Interquartile range (IQR) 26 490 German tank problem 207, 238 j Goodness-of-fit test Joint density 479 all parameters known 374–377 some parameters estimated k 377–379 Kernel density estimate see Density Google Analytics Content Experiments estimate 270, 411 Kurtosis 40

h l Histogram 23 Law of averages 483 Homogeneity, test of 372–374 Law of total probability 477 Hypergeometric distribution 370, Least-squares regression, see 492–493 Regression Hypothesis Likelihood composite 271 Bayesian 392 simple 271 complete data 468 k Hypothesis test 48 function 150, 151, 153 k one sample mean t-test 242–244 generalized ratio test 275–279 one sample mean z test 241–242 incomplete data 468 one sample proportion z test ratio and chi-square 379–380 244–246 ratio test 271–274 one-sided 48 ratio test statistic 271, 275 two sample means t-test Linear model 248–251 assumptions 320, 330–332 two sample proportions z test inference for response 326–329 251–255 inference for slope and intercept two-sided 48 322–326 Location family 208 i Location parameter 208 I see Indicator function Location-scale family 212 iid 482 Logistic regression 340–349 Importance sampling 452–467 inference 346 design distribution 453 importance function 453 m target distribution 453 MAD see absolute deviation Improper prior 398 Margin of error 191 Independence Matched pairs 68–70, 124–127, 204 chi-square test of 365–372 median 146 permutation test of 361–364 paired t-test 250 of random variables 479 proportions test 254–255

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Maximum likelihood estimation 150, MLE for multiple parameters 151, 153 157–158 asymptotic bias 166 as prior for continuous data 407 asymptotic normality 177 sum of squared normal random continuous variables 153–157 variables 501 discrete variables 150–152 sumsofnormalrandomvariables multiple parameters 157–161 482–438 unbiased estimator 164–167 Normal quantile plot 31 McNemar’s test 255 Null distribution 50, 241 Mean 25 Null hypothesis 48 absolute deviation 26 midmean 25 o of a sample 481 Observation 1 square error 171–173, 425 Observational study 11 square for treatments 424 Odds 341 trimmed 25 Odds ratio 342, 371, 388 Median 25 Outlier 198, 244, 313 Method of moments 161–163 mgf see Moment generating function p MLE see Maximum likelihood estimate P-value 49, 241, 267 k k Moment 484 vs. critical region 266 central 40, 485 one-sided vs. two-sided 61 generating function 485 for permutation test 52 Monte Carlo two-sided 61 integration 448–452 Paired data see Matched pairs sampling 134 Paired t-test see Matched pairs Parameter 6 MSE see Mean square error pdf see Probability density function 366, Pearson, Karl 361, 365 488–490 Permutation distribution 54 Multiple testing 269 Permutation resample see Resample Permutation test 50, 122 n assumptions 64–67 Natural variability 22 correlation 335 Negative binomial distribution of independence 361–364 491–492 two sample 54 Neyman–Pearson Lemma 273 Placebo 12 Noncentral t distribution see t Plug-in principle 110 distribution pmf see Probability mass function Noninformative prior see Prior 152, distribution 493–495 480 sum of Poisson random central limit theorem 84 variables 494

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Population 5 Resample finite 5 bootstrap 104 infinite 5 permutation 52 26 Residual 312 variance 26 assumptions in regression 320, 331 Posterior distribution 392, 399 plot 312 Power 261–266 standard error S 322 Precision 408 𝜌(X, Y) 301 Predicted value 306 R-square (r2) 311 Prediction interval 328 Prior distribution 392 s conjugate family 402 s 26 flat 402 S 165, 322 improper 398 Sample 5 noninformative 402, 409 finite population 5 Probability density function 478 probability 8 Probability mass function 477 random 5 Proportion of variance explained see standard deviation (see Sample R-Square standard deviation) variance (see Sample standard k q deviation) k qp 30 with replacement 5 qq plot see Quantile-quantile plot without replacement 5 Quantile 30 Sample standard deviation 26 Quantile-quantile plot 31 as estimate of 𝜎 192 S2 is unbiased estimate of 𝜎2 165 r Sample variance see Sample standard r 303 deviation Random assignment 11 Sampling Random variable cluster 9 continuous 477 Monte Carlo (see Monte Carlo discrete 477 sampling) Range 26 multi-stage 9 Reference distribution 50, 241 stratified 9, 445–446 Regression Sampling distribution 75, 76 least-squares 297–317 by calculating 80–84 logistic 340–349 of the maximum of a sample 81–82 multiple 317 of the minimum of a sample 81–82 toward the mean 308–309 by simulation 78–79 Rejection region see Critical region Sampling frame 8 Relative efficiency see Efficiency Scale family 210 Relative risk 236 Scale parameter 210 bootstrap (see Bootstrap) Scatter plot 38

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Scatter plot smooth 314 Test statistic 49 Sequential data 409–412 choice of 56 Sidak correction 269 Total sum of squares see Sum of squares Signal-to-noise 6 Transformation invariance 175–177, Significance level 259 227, 445 Single-blind see Blinding Translated exponential distribution 40 457 and t confidence intervals Treatments 11 196–198, 201–202 Treatment sum of squares see Sum SLLN see Strong law of large numbers of squares Smoothing spline 314 t statistic Spread 26 one sample mean 193, 243

ssx, ssy, ssxy 306 two sample means 199, 250 Standard deviation t-test see Hypothesis test population (see Population standard Tukey, John 21 deviation) Type I error 255–261 of a random variable479 Type II error 255, 261–266 Standard error 75, 77 residual 322 u Statistic 6 Ulam, Stanislaw 134 k Statistical practice 12, 178–180, Unbiased estimator 164–167 k 279–285 Undercoverage 8 Statistical significance 22, 48 Uniform distribution 154–155, 495 lack of 269 density of the maximum 81 vs. practical importance 268 density of the minimum 81 searching for 269 Stratified sampling 9, 445–446 v Strong law of large numbers 484 Variable 1 Student’s t distribution see t categorical 21 distribution lurking 480 Subjects 11 Variance Sum of squares 306 pooled 202, 251 error 422 of a random variable 478 total 422 sample (see Sample standard treatment 421 deviation) Surveys 8 von Neumann, John 134

t w Target distribution see Importance 158, 437 sampling Weights 179–180 t distribution 193, 324, 502–504 Welch’s approximation 199, 238 noncentral 266, 294 Wilson interval 217

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