Index

A

• adjusted coefficient of determination, 166–167
• Agresti–Coull confidence interval, 321–322
• Akaike's information criterion (AIC), 486–487, 520–521
• Akaike, Hirotugu, 486
• aligning, 262
• ANOVA table, 56–59, 92–93, 114–116, 166
• Anscombe's quartet, 93–95, 118, 120, 121, 130, 135, 206
• Anscombe, Francis, 93
• ar (autoregression) function, 515–517, 523, 524, 549, 550, 553, 559, 576, 577, 590, 593
• AR(p) models, 457, 561–592
  • as MA(∞) models, 562–564
  • comparison with MA(q) models, 621
  • confidence intervals, 578–583
  • definition, 561
  • forecasting, 588–591
  • model assessment, 583–588
  • parameter estimation, 573–583
    • least squares estimators, 576–577
    • maximum likelihood estimators, 577–583
    • method of moments, 573–576
  • population autocorrelation function, 564–567
  • population partial autocorrelation function, 567–569
  • residuals, 583–588
  • shifted, 569–570
  • simulation, 570–573
  • stationarity, 562
• AR(1) models, 493–526
  • as MA(∞) models, 495
  • confidence intervals, 515–517
  • definition, 493
  • forecasting, 521–525
  • model assessment, 517–520
  • model selection, 520–521
  • parameter estimation, 507–516
    • least squares estimators, 508–511
    • maximum likelihood estimators, 511–515
    • method of moments, 507–508
  • population autocorrelation function, 497
  • population partial autocorrelation function, 497–499
  • residuals, 517–520
  • shifted, 499
  • simulation, 499–503
  • stationarity, 494–497
• AR(2) models, 526–561
  • as MA(∞) models, 535–536
  • confidence intervals, 550–552
  • definition, 526
  • forecasting, 557–560
  • geometry, 532, 534, 546
  • model assessment, 552–555
  • model selection, 555–557
  • parameter estimation, 544–552
    • least squares estimators, 546–544
    • maximum likelihood estimators, 548–552
    • method of moments, 544–546
  • population autocorrelation function, 529–533
  • population partial autocorrelation function, 533–535
  • residuals, 552–555
  • shifted, 536–537
  • simulation, 537–541
  • stationarity, 527–529
• arima function, 520, 551, 556, 582, 583, 585, 593, 617, 622–626, 631
• ARIMA($p,d,q$) models, 631–635
  • definition, 632
• ARMA($p,q$) models, 456–465, 621–627
  • forecasting, 626–627
  • model assessment, 623–626
  • shifted, 463–465, 622
• asymptotic properties, 256–258, 551–552
• autocorrelation function, see population autocorrelation function, sample autocorrelation function
• autocovariance function, see population autocovariance function, sample autocovariance function
• autoregressive models, 492–594

B

• backshift operator, 425, 448–451, 495–496
• baseline hazard function, 242, 290–306
• bathtub-shaped hazard function, 217–218
• Bayesian information criterion (BIC), 487
• bivariate normal distribution, 537–538
• Box, George, 5, 142, 631
• Box–Cox transformation, 142
• Box–Pierce test, 482–483, 518–519, 553, 623–624
• Brown, Robert, 378
• Brown–Forsythe test, 205
• Brownian motion, 378

C

• categorical independent variables, 162–163
• Cauchy–Schwartz inequality, 12, 68
• causality, 451–456, 494–495
• censoring, 259–265
  • interval censoring, 260
  • left censoring, 260
  • right censoring, 259–265
    • likelihood function, 263
    • random censoring, 260
    • Type I censoring, 260
    • Type II censoring, 259
• characteristic polynomials, 457–460
• Clopper–Pearson confidence interval, 321–322
• Cochran's theorem, 92
• Cochran, William, 92
• coefficient of correlation, 53–59, 116, 148
• coefficient of determination, 53–59, 116, 148, 166–167
• comparing two survivor functions, 329–332
• competing risks, 332–342
  • crude lifetimes, 335–337
  • general case, 338–342
  • net lifetimes, 333–334
• complete data sets, 266–273, 318–323
• conditional intensity function, 354
• conditional survivor function, 212–213
• confidence intervals
  • in simple linear regression, 69–90
  • in survival analysis, 269, 272, 275, 280–285
  • in time series analysis, 472–476, 515–517
• confidence regions
  • in simple linear regression, 90–91
  • in survival analysis, 288
• Cook's distances, 134–139
• Cook, R. Dennis, 134
• corrected Akaike's information criterion (AICC), 487
• counting function, 344
• counting process, 344
• covariates, see proportional hazards model
• Cox proportional hazardsmodel, see proportional hazards model
• Cox, David, 142, 241, 329
• coxph (Cox proportional hazards) function, 302, 305
• crude lifetimes, 335–337
• cumulative hazard function, 219
• cumulative intensity function, 350

D

• decomposition, 434–440
• dependent variable, 2, 6–9
• design matrix, 146, 158
• deterioration, 343
• deterministic models, 2–4
• detrending filters, 422–426
• DFR class, 216, 343
• diagnostics, 128–139, 480–486
• Dickey–Fuller test, 633
• difference–sign test, 491
• differencing, 425–426, 631–640
• duality, 495, 562–564
• Durbin–Watson test, 490

E

• Einstein, Albert, 378
• empirical survivor function, 319–323
• exponential distribution, 220–227, 231
  • hypothesis testing, 278
  • lifetime distribution representations, 220
  • memoryless property, 221–222
  • moments, 223–224
  • parameter estimation, 266–285
    • complete data sets, 266–273
    • randomly censored data sets, 281–285
    • Type I censored data sets, 279–281
    • Type II censored data sets, 274–279
  • self-reproducing property, 224–225
• exponential power distribution, 233
• exponentially-weighted moving average, 432–433

F

• filtering, 419–434
• Fisher information matrix, 197, 269, 274, 279, 286, 292
• Fisher, Ronald, 470
• fitted values, 32–38, 148
• Forbes, James, 44
• forecasting, 476–480
  • geometry, 477
  • prediction intervals, 478–480
• forward stepwise regression, 171–172

G

• Galton, Francis, 105
• gamma distribution, 232
• Gauss, Carl Friedrich, 9
• Gaussian white noise, 371
• general linear models, 446–456
• generalized Pareto distribution, 234
• glm (generalized linear model) function, 198
• Gompertz distribution, 234
• Greenwood's formula, 328–329

H

• Haenszel, William, 329
• hat matrix, 129, 150, 159
• hazard function, 214–219
  • bathtub-shaped, 217–218
• Hessian matrix, 11, 67–68
• hyperexponential distribution, 235
• hypergeometric distribution, 330
• hypoexponential distribution, 235

I

• IDB distribution, 234
• idempotent matrix, 159
• IFR class, 216, 343
• iid noise, 371
• improvement, 343
• independent increments, 345
• independent variables, 2, 6–9
• inference concerning β₀, 76–80, 116–117
• inference concerning β₀ and β₁, 90–91, 117–118, 150
• inference concerning β₁, 72–76, 101–102, 116–117
• inference concerning σ², 69–72, 116, 149
• inference concerning $E[Y_h]$, 80–84, 102–103, 149
• inference concerning $Y_h^{\star }$, 85–90, 103–105, 149
• inference in simple linear regression, 64–118
• influential points, 134–139
• information matrix, see Fisher information matrix, observed information matrix
• intensity function, 343, 350–355
• interaction terms, 163–165
• interval censoring, 260, 312
• inverse Gaussian distribution, 233, 254–255, 257
• invertibility, 451–456, 461–463

J

• Jeffreys Bayesian credible interval, 321–322
• Jenkins, Gwilym, 631

K

• Kaplan, Edward, 324
• Kaplan–Meier estimator, 324–329
• Kepler, Johannes, 207

L

• least absolute deviation criterion, 61
• least squares estimators, 9–17, 469, 508–511, 546–548, 576–577
  • geometry, 12, 16–17, 157
  • in matrix form, 147, 158
  • properties, 18–32, 158–159
    • Gauss–Markov theorem, 29–32, 148, 158
    • linear combinations, 22–25, 147, 158
    • unbiasedness, 18–22, 147, 158
    • variance–covariance matrix, 25–29, 147, 158
• left censoring, 260
• Legendre, Adrien–Marie, 9
• leverage, 128–134, 159
  • definition, 129
• lifetime distribution representations, 210–219
  • cumulative hazard function, 219
  • hazard function, 214–219
  • probability density function, 213–214
  • survivor function, 212–213
• likelihood ratio statistic, see log likelihood function
• likelihood theory, 253–255
• linear filters, 427–433
• Ljung–Box test, 483, 518–519, 553, 623–624
• lm (linear model) function, 15, 20, 33, 35, 44, 46, 49, 57, 72, 75, 76, 79, 83, 84, 88, 94, 97, 107, 113, 126, 131, 141, 161, 169, 179, 180
• log likelihood function, 282–285, 287–288
• log logistic distribution, 233
• log normal distribution, 233
• log-log model, 191–193
• log-rank test, 329–332
• logistic regression, 189–203
  • interpreting coefficients, 194–196, 200–203
  • log-log model, 191–193
  • logit model, 191–193
  • odds, 200–203
  • parameter estimation, 196–200
  • probit model, 191–193
• logit model, 191–193

M

• MA(q) models, 457, 618–621
  • comparison with AR(p) models, 621
  • definition, 618
  • invertibility, 620
  • population autocorrelation function, 620
  • shifted, 620
  • simulation, 620
  • stationarity, 620
• MA(1) models, 595–614
  • as AR(∞) models, 600
  • confidence intervals, 612–614
  • definition, 595
  • invertibility, 599–600
  • parameter estimation, 607–614
    • least squares estimators, 610–612
    • maximum likelihood estimators, 612–614
    • method of moments, 607–610
  • population autocorrelation function, 596–599
  • population partial autocorrelation function, 601–602
  • shifted, 602
  • simulation, 602–607
  • stationarity, 596–599
• MA(2) models, 615–618
  • definition, 615
  • invertibility, 615
  • model selection, 616–618
  • population autocorrelation function, 616
  • shifted, 615
  • simulation, 616
  • stationarity, 615
• Makeham distribution, 234
• Mantel, Nathan, 329
• Mantel–Haenszel test, see log-rank test
• matrix approach
  • in survival analysis, 290, 296
  • to multiple linear regression, 157–159
  • to simple linear regression, 146–155
  • to weighted least squares, 174–175
• maximum likelihood estimators, 65–69, 149, 159, 196–200, 254–255, 267, 274, 280, 285, 291, 469–476, 511–515, 548–552, 577–583
• mean square error, 40–41
• Meier, Paul, 324
• memoryless property, 221–222
• method of moments, 468-469, 507–508, 544–546, 573–576
• Mill's ratio, 214
• model selection
  • in multiple linear regression, 171–172
  • in time series analysis, 486–488
• moment ratio diagrams, 235–241
• Monte Carlo simulation, 19–22, 28–29, 48–50, 61, 118–120, 204, 206, 223, 247, 248, 308, 309, 360, 362, 363, 373, 443, 444, 475, 489, 490, 499–503, 537–541, 570–573, 616–618, 647
• moving average, 427–433
• moving average models, 594–621
• multicollinearity, 167–171
• multiple linear regression, 155–172
  • adjusted coefficient of determination, 166–167
  • ANOVA table, 166
  • categorical independent variables, 162–163
  • coefficient of determination, 166–167
  • examples, 155
  • fitted values, 158
  • geometry, 156
  • hat matrix, 159
  • interaction terms, 163–165
  • interpreting coefficients, 160
  • least squares estimators, 158
  • matrix approach, 157–159
  • maximum likelihood estimators, 159
  • model, 156, 158
  • model selection, 171–172
  • multicollinearity, 167–171
  • nonlinear terms, 161
  • normal equations, 158
  • normal error terms, 157, 159
  • residuals, 159
  • variance inflation factors, 168–170
• multivariate normal distribution, 381, 570–573
• Muth distribution, 231

N

• net lifetimes, 333–334
• nls (nonlinear least squares) function, 185
• noise terms in time series analysis
  • Gaussian white noise, 371
  • iid noise, 371
  • white noise, 371
• nonhomogeneous Poisson processes, 350–355
• noninformative censoring, see randomly censored data sets
• nonlinear least squares, 184–185
• nonparametric methods, 318–332
• nonstationary time series models, 627–640
  • ARIMA($p,d,q$) models, 631–635
  • detrending, 627–631
  • SARIMA models, 635–640
• normal equations, 10, 147, 158
• normal error terms, 64–65, 149, 157
• normalized spectral density function, 644–645
• Nyquist frequency, 642

O

• observed information matrix, 197, 269, 274, 279, 282, 286, 292
• odds, 200–203
• overdispersed renewal processes, 347

P

• Pareto distribution, 233
• partial autocorrelation function, see population partial autocorrelation function, sample partial autocorrelation function
• partitioning the total sum of squares, 51–59, 148
  • geometry, 152
• periodogram, 645–649
• Peto, Julian, 332
• point processes, 342–355
  • nonhomogeneous Poisson processes, 350–355
  • Poisson processes, 345–346
  • renewal processes, 346–350
• Poisson processes, 345–346
• population autocorrelation function, 380–388, 497, 529–533, 564–567
• population autocovariance function, 380–388
• population mean function, 382
• population partial autocorrelation function, 411–417, 497–499, 533–535, 567–569
• prediction intervals
  • in simple linear regression, 87, 104
  • in survival analysis, 308
  • in time series analysis, 478–480, 522–525
• probability density function, 213–214
• probit model, 191–193
• product–limit estimator, 324–329
• proportional hazards model, 241–244, 290–306
  • known baseline function, 291–297
  • unknown baseline function, 297–306

Q

• QQ plot, 100, 109–111, 114, 143
• quadratic regression, 180–182

R

• random variate generation, see Monte Carlo simulation
• random walk, 372–376, 385–386, 388
• randomly censored data sets, 281–289
• rank vector, 299
• Rayleigh distribution, 263–265, 315
• redistribute-to-the-right algorithm, 326–327
• regression, see simple linear regression, multiple linear regression
• regression function, 6
• regression models with nonlinear terms, 180–188
  • nonlinear least squares, 184–185
  • quadratic regression, 180–182
  • transformation technique, 182–184
• regression through the origin, 122–128
  • sum of squares for error, 128
• regression to the mean, 105
• remedial procedures, 140–145
• renewal processes, 346–350
  • overdispersed, 347
  • underdispersed, 347
• residuals, 33–38, 47, 98–101, 108–111, 148, 480–486
  • standardized, 108–110, 113–114, 143
• ridge regression, 170–171
• right censoring, 259–265
  • estimating $S(t)$, 323–329
  • likelihood function, 263
  • random censoring, 260, 281–285
  • Type I censoring, 260, 279–281
  • Type II censoring, 259, 274–279

S

• sample autocorrelation function, 399–411, 482–483
• sample autocovariance function, 399–411
• sample partial autocorrelation function, 417–418
• SARIMA models, 635–640
• scatterplot, 13–14, 97, 107, 111–113
• seasonal moving average, 430–431
• Shapiro–Wilk test, 101, 110, 114, 143, 485
• shifted AR(p) models, 569–570
• shifted AR(1) models, 499
• shifted AR(2) models, 536–537
• shifted ARMA($p,q$) models, 463–465, 622
• simple linear regression, 2–59
  • ANOVA table, 56–59, 92–93, 114–116
  • coefficient of correlation, 53–59, 116, 148
  • coefficient of determination, 53–59, 116, 148
  • diagnostics, 128–139
    • influential points, 134–139
    • leverage, 128–134
  • fitted values, 32–38, 148
  • geometry, 12, 16–17, 51, 65, 81–82, 152
  • hat matrix, 129
  • inference in, 64–118
    • concerning β₀, 76–80, 116–117
    • concerning β₀ and β₁, 90–91, 117–118, 150
    • concerning β₁, 72–76, 101–102, 116–117
    • concerning σ², 67–72, 116, 149
    • concerning $E[Y_h]$, 80–84, 102–03, 149
    • concerning $Y_h^{\star }$, 85–90, 103–105, 149
  • least squares estimators, 9–17, 147
  • matrix approach, 146–155
  • maximum likelihood estimators, 65–69, 149
  • mean square error, 40–41, 148
  • model, 6–9, 147
  • normal equations, 10, 147
  • normal error terms, 64–65, 149
  • partitioning the total sum of squares, 51–59, 148, 152
  • procedure, 7–8
  • regression through the origin, 122–128
  • remedial procedures, 140–145
  • residuals, 33–38, 47, 98–101, 108–111, 113–114, 148
  • sum of squares for error, 40–41, 52–59
  • sum of squares for regression, 52–59
  • total sum of squares, 52–59
  • variance of error terms, 39–51
  • with nonlinear terms, 180–188
• spectral analysis, 641–649
  • periodogram, 645–649
  • spectral density function, 642–645
• spectral density function, 642–645
• stationarity, 388–399, 461–463
  • covariance, 391
  • second-order, 391
  • strict, 389
  • weak, 391
• statistical models, 5–6
• stepwise regression, 171–172
• sum of squares for error, 40–41, 52–59, 152
• sum of squares for regression, 52–59, 152
• superpositioning, 354–355
• survfit (fit survival curve) function, 326, 329
• survival analysis
  • probability models, 210–244
  • statistical methods, 253–306
  • topics in, 318–355
• survivor function, 212–213
• survivor function estimation
  • complete data sets, 318–323
  • right-censored data sets, 323–329

T

• time series analysis
  • basics, 366–442
    • classification, 377–378
    • goals, 376–377
  • computing, 378–380, 418, 440–442, 592–594
  • nonstationary models, 627–640
    • ARIMA($p,d,q$) models, 631–635
    • detrending, 627–631
    • SARIMA models, 635–640
  • operations, 419–442
    • backshift operator, 425, 448–451
    • decomposition, 434–440
    • detrending filters, 422–426
    • differencing, 425–426
    • exponentially-weighted moving average, 432–433
    • filtering, 419–434
    • linear filters, 427–433
    • moving average, 427–433
    • seasonal moving average, 430–431
    • transformations, 420–422
  • probability models, 446–465
    • AR(p) models, 457, 561–592
    • AR(1) models, 493–526
    • AR(2) models, 526–561
    • ARMA($p,q$) models, 456–465, 621–627
    • general linear models, 446–456
    • MA(q) models, 457, 618–621
    • MA(1) models, 595–614
    • MA(2) models, 615–618
    • shifted ARMA($p,q$) models, 463–465
  • properties, 380–418
    • causality, 451–456
    • invertibility, 451–456, 461–463, 599–600, 620
    • population autocorrelation function, 380–388, 497, 529–533, 564–567, 596–599, 620
    • population autocovariance function, 380–388
    • population partial autocorrelation function, 411–417, 497–499, 533–535, 567–569, 601–602
    • sample autocorrelation function, 399–411
    • sample autocovariance function, 399–411
    • sample partial autocorrelation function, 417–418
    • stationarity, 388–399, 461–463, 494–497, 527–529, 562, 596–599, 620
  • spectral analysis, 641–649
    • periodogram, 645–649
    • spectral density function, 642–645
  • statistical methods, 466–488
    • forecasting, 476–480, 521–525, 557–560, 588–591, 626–627
    • model assessment, 480–486, 517–520, 552–555, 583–588, 623–626
    • model selection, 486–488, 520–521, 555–557
    • parameter estimation, 466–476, 507–516, 544–552, 573–583, 607–614
• total sum of squares, 52–59, 152
• transformations, 420–422
• Tucker, Justin, 189
• turning point test, 483–484, 519, 554–555, 624–625
• Type I censored data sets, 279–281
• Type II censored data sets, 274–279

U

• underdispersed renewal processes, 347
• uniform distribution, 233
• unit roots analysis, 461–463, 496, 562

V

• variance inflation factors, 168–170
• variance of error terms, 39–51

W

• Wald confidence interval, 321–322
• Walker, Gilbert, 529
• Weibull distribution, 215–216, 223, 227–231, 231
  • lifetime distribution representations, 227–228
  • moments, 229
  • parameter estimation, 285–289
  • self-reproducing property, 231
• Weibull, Waloddi, 227
• weighted least squares, 172–180, 185–188
  • estimators, 174
  • matrix approach, 174–175
  • normal equations, 173–174
• white noise, 371
• Wilson–score confidence interval, 321–322
• Working–Hotelling confidence band, 121

Y

• Yule, George, 529
• Yule–Walker equations, 529–530, 567, 574
• Yule–Walker estimators, 567, 574