Index

A

  • (a, b, 0) class of distributions, 88, 506
  • (a, b, 1) class of distributions, 92, 507
    • estimation, 264
  • actuarial exposure, 340
  • aggregate loss distribution, 150
    • approximating distribution, 171
    • compound negative binomial–exponential, 167
    • direct calculation, 172
    • distribution function, 152
    • exponential severity, 168
    • individual risk model, compound Poisson approximation, 193
    • moment generating function, 153
    • moments, 153
    • probability generating function, 153
    • recursive formula, 173, 515
      • compound frequency, 175
      • computational issues, 177
      • construction of arithmetic distributions, 179
      • continuous severity, 178
      • undiscretization, 518
    • recursive method, 172
    • simulation, 480
  • aggregate loss model, 148
    • advantages, 149
  • Akaike Information Criterion, 374
  • Anderson–Darling test, 363
  • anniversary-to-anniversary mortality study, 342
  • arithmetic distribution, 179
  • asymptotically unbiased, 207
    • maximum likelihood estimator, 243

B

  • bandwidth, 334
  • Bayesian central limit theorem, 282
  • Bayesian estimation, 275
    • Bayes estimate, 279
    • credibility interval, 280
    • highest posterior density (HPD) credibility set, 281
    • improper prior distribution, 276
    • joint distribution, 276
    • loss function, 279
    • marginal distribution, 276
    • model distribution, 276
    • posterior distribution, 277
    • predictive distribution, 277, 408
    • prior distribution, 275
  • Bayesian Information Criterion, 374
  • Bernoulli distribution, 88
  • beta–binomial distribution, 286
  • beta distribution, 503
  • beta function, 491
    • incomplete, 490
  • bias, 205
  • binomial-beta distribution, 112
  • binomial distribution, 87, 88, 506
    • estimation, 261
    • simulation, 474
  • bootstrap, 485
  • Brown, Hollander, and Korwar tail correction, 309
  • Bühlmann credibility model, 418
  • Bühlmann-Straub credibility model, 422
  • Burr distribution, 493

C

  • censoring
  • central limit theorem, 31
    • Bayesian, 282
  • central moment, 22
  • characteristic function, 114
  • chi-square goodness-of-fit test, 363
  • claim count random variable, 150
  • coefficient of variation, 22
  • coherent risk measure, 42
  • coinsurance, 136
  • collective risk model, 148
  • complete data, 296
  • complete expectation of life, 24
  • compound distribution, 152
  • compound frequency distributions, 99, 509
    • estimation, 268
  • compound Poisson distribution, 103
  • compound Poisson frequency distribution, 105
  • conditional distribution, 404
  • confidence interval, 216
  • conjugate prior distribution, 290
  • consistency, 211
    • maximum likelihood estimator, 243
  • construction of mortality tables, 338
  • continuous mixture, 112
  • continuous random variable, 13
  • convolution, 152
  • counting distributions, 81
  • Cramèr–Rao lower bound, 210
  • credibility
    • Bühlmann credibility factor, 419
    • expected hypothetical mean, 419
    • expected process variance, 419
    • fully parametric, 446
    • greatest accuracy, 388, 401
      • Bayesian, 408
      • Bühlmann, 418
      • Bühlmann–Straub, 422
      • exact credibility, 427
      • linear, 415
      • nonparametric, 448
      • semiparametric, 459
    • hypothetical mean, 418
    • interval, 280
    • limited fluctuation, 388, 389
      • full credibility, 390
      • partial credibility, 393
    • nonparametric, 446
    • partial, 393
    • process variance, 418
    • semiparametric, 446
    • variance of the hypothetical means, 419
  • credibility factor, 393
  • cumulative distribution function, 11
  • cumulative hazard rate function, 310

D

  • data-dependent distribution, 57, 295
  • date-to-date mortality study, 342
  • deductible
    • effect of inflation, 132
    • effect on frequency, 140
    • franchise, 128
    • ordinary, 126, 327
  • delta method, 248
  • density function, 14
  • density function plot, 357
  • difference plot, 357
  • discrete distribution, 81
  • discrete failure rate, 306
  • discrete mixture, 112
    • simulation, 472
  • discrete random variable, 13
  • distribution
    • (a, b, 0) class, 88, 506
      • simulation, 474
    • (a, b, 1) class, 92, 507
      • estimation, 264
    • aggregate loss, 150
    • arithmetic, 179
    • Bernoulli, 88
    • beta, 503
    • beta–binomial, 286
    • binomial, 87, 88, 506
    • binomial-beta, 112
    • Burr, 493
    • claim count, 150
    • compound, 152
    • compound frequency, 99, 509
      • recursive formula, 102
    • compound Poisson, 103
    • conditional, 404
    • conjugate prior, 290
    • counting distributions, 81
    • data-dependent, 57, 295
    • discrete, 81
    • empirical, 296
    • equilibrium, 38
    • exponential, 46, 499
    • exponential dispersion family, 436
    • extended truncated negative binomial (ETNB), 95
    • extreme value, 76, 499
    • frailty, 68
    • Frechet, 500
    • frequency, 150
    • function, 11
      • empirical, 297
    • gamma, 33, 36, 59, 63, 497
    • generalized
    • geometric, 85, 506
    • geometric–ETNB, 510
    • geometric–Poisson, 510
    • Gumbel, 499
    • improper prior, 276
    • individual loss, 150
    • infinitely divisible, 114
    • integrated tail, 38
    • inverse
    • joint, 276, 404
    • kernel smoothed, 332
    • k-point mixture, 55
    • logarithmic, 96, 508
    • loglogistic, 71, 495
    • lognormal, 64, 76, 501
    • log-t, 501
    • marginal, 276, 404
    • mixed-frequency, 111
    • mixture/mixing, 54, 64, 112, 404
    • negative binomial, 85, 507
      • extended truncated, 95
      • as Poisson mixture, 86
    • negative hypergeometric, 112, 286
    • Neyman Type A, 100, 510
    • normal, 45
    • paralogistic, 496
    • parametric, 52, 296
    • parametric family, 54
    • Pareto, 33, 35, 36, 46, 77, 494
    • Poisson, 82, 506
    • Poisson–binomial, 510
    • Poisson–extended truncated negative binomial, 269, 510, 511
    • Poisson–inverse Gaussian, 269, 511
    • Poisson-logarithmic, 103
    • Poisson–Poisson, 100, 510
    • Polya–Aeppli, 511
    • Polya-Eggenberger, 112
    • posterior, 277
    • predictive, 277, 408
    • prior, 86, 275
    • scale, 53
    • Sibuya, 96
    • Sichel, 438
    • single parameter Pareto, 502
    • spliced, 69
    • tail weight of, 33
    • transformed, 63
    • transformed beta, 71, 493
    • transformed beta family, 74
    • transformed gamma, 496
    • transformed gamma family, 74
    • Tweedie, 159
    • variable-component mixture, 55
    • Waring, 113, 287
    • Weibull, 63, 498, 500
    • Yule, 113, 287
    • zero-modified, 93, 509
    • zero-truncated, 93
      • binomial, 508
      • geometric, 507
      • negative binomial, 508
      • Poisson, 507
    • zeta, 122, 382
  • distribution function plot, 355

E

  • Efron tail correction, 309
  • empirical Bayes estimation, 445
  • empirical distribution, 296
    • function, 297
  • empirical model, 57
  • equilibrium distribution, 38
  • estimation
    • (a, b, 1) class, 264
    • Bayesian, 275
    • binomial distribution, 261
    • compound frequency distributions, 268
    • credibility interval, 280
    • effect of exposure, 269
    • empirical Bayes, 445
    • maximum likelihood, 229
    • multiple decrement tables, 344
    • negative binomial, 259
    • Nelson–Aalen, 310
    • point, 203
    • Poisson distribution, 255
  • estimator
    • asymptotically unbiased, 207
    • Bayes estimate, 279
    • bias, 205
    • confidence interval, 216
    • consistency, 211
    • interval, 216
    • Kaplan–Meier, 306, 308
    • kernel density, 332
    • mean squared error, 212
    • method of moments, 219
    • percentile-matching, 220
    • product limit, 306
    • relative efficiency, 215
    • smoothed empirical percentile, 220
    • unbiased, 205
    • uniformly minimum variance unbiased, 212
  • exact credibility, 427
  • exact exposure, 340
  • excess loss variable, 24
  • expected information, 210
  • exponential dispersion family, 80
  • exponential distribution, 499
  • exposure base, 120
  • exposure, effect in estimation, 269
  • extreme value distributions, 76, 499

F

  • failure rate, 17
    • discrete, 306
  • Fisher information, 210, 243
  • force of mortality, 17
  • frailty model, 68
  • franchise deductible, 128
  • Frechet distribution, 500
  • frequency, 150
    • effect of deductible, 140
    • interaction with severity, 513
  • frequency/severity interaction, 186
  • full credibility, 390
  • function
    • characteristic, 114
    • density, 14
    • empirical distribution, 297
    • force of mortality, 17
    • gamma, 64, 492
    • hazard rate, 17
    • incomplete beta, 490
    • incomplete gamma, 63, 489
    • likelihood, 230
    • loglikelihood, 231
    • loss, 279
    • probability, 16, 81
    • probability density, 14
    • probability generating, 82
    • survival, 14

G

  • gamma distribution, 63, 497
  • gamma function, 64, 492
    • incomplete, 489
  • gamma kernel, 334
  • generalized beta distribution, 502
  • generalized inverse Gaussian distribution, 438
  • generalized Pareto distribution, 493, 500
  • generalized Poisson–Pascal distribution, 511
  • generalized Waring distribution, 113, 287
  • generating function
    • moment, 31
    • probability, 31
  • geometric distribution, 85, 506
  • geometric–ETNB distribution, 510
  • geometric–Poisson distribution, 510
  • greatest accuracy credibility, 388, 401
  • Greenwood's approximation, 313
  • Gumbel distribution, 499

H

  • hazard rate, 17
    • cumulative, 310
    • tail weight, 35
  • histogram, 300
  • hyperparameter, 86
  • hypothesis tests, 224, 360
    • Anderson–Darling, 363
    • chi-square goodness-of-fit, 363
    • Kolmogorov–Smirnov, 360
    • likelihood ratio test, 367, 373
    • p-value, 227
    • significance level, 225
    • uniformly most powerful, 226
  • hypothetical mean, 418

I

  • incomplete beta function, 490
  • incomplete gamma function, 63, 489
  • individual loss distribution, 150
  • individual risk model, 148, 189
  • infinitely divisible distribution, 114
  • inflation
    • effect of, 132
    • effect of limit, 135
  • information, 210, 243
  • insuring ages, 341
  • integrated tail distribution, 38
  • interval estimator, 216
  • inverse Burr distribution, 494
  • inverse exponential distribution, 499
  • inverse gamma distribution, 497
  • inverse Gaussian distribution, 59, 501
  • inverse paralogistic distribution, 496
  • inverse Pareto distribution, 495
  • inverse transformed distribution, 63
  • inverse transformed gamma distribution, 76, 497
  • inverse Weibull distribution, 63, 77, 498
  • inversion method, 469

J

  • joint distribution, 276, 404

K

  • Kaplan–Meier estimator, 306, 308
    • large data sets, 338
    • variance, 311
  • kernel density estimator, 332
    • bandwidth, 334
    • gamma kernel, 334
    • triangular kernel, 334
    • uniform kernel, 333
  • kernel smoothed distribution, 332
  • Klein and Moeschberger tail correction, 309
  • Klein's estimate, 313
  • Kolmogorov–Smirnov test, 360
  • k-point mixture distribution, 55
  • kurtosis, 23

L

  • large data sets, 338
  • left censored and shifted variable, 25
  • left truncated and shifted variable, 24
  • left truncation, 237, 327
  • life table, simulation, 473
  • likelihood function, 230
  • likelihood ratio test, 367, 373
  • limited expected value, 27
  • limited fluctuation credibility, 388, 389
  • limited loss variable, 27
  • limit
    • effect of inflation, 135
    • policy, 327
  • linear exponential family, 78, 160
  • logarithmic distribution, 96, 508
  • loglikelihood function, 231
  • loglogistic distribution, 71, 495
  • lognormal distribution, 64, 76, 501
    • simulation, 476
  • log-t distribution, 501
  • loss elimination ratio, 132
  • loss function, 279

M

  • marginal distribution, 276, 404
  • maximum covered loss, 137
  • maximum likelihood estimation, 229
    • binomial, 262
    • inverse Gaussian, 235
    • negative binomial, 259
    • Poisson, 256
      • variance, 257
    • truncation and censoring, 236
  • maximum likelihood estimator
    • consistency, 243
    • unbiased, 243
  • mean, 21
  • mean excess loss, 24
  • mean residual life, 24
    • tail weight, 35
  • mean squared error, 212
  • median, 29
  • method of moments, 219
  • mixed-frequency distributions, 111
  • mixed random variable, 13
  • mixing distribution, 112
  • mixture, continuous, 112
  • mixture, discrete, 112
  • mixture distribution, 54, 64, 404
  • mode, 18
  • model
    • advantages, 5
    • collective risk, 148
    • empirical, 57
    • individual risk, 148
    • multistate, 350
  • model selection, 3
    • Akaike, 374
    • graphical comparison, 355
    • Schwarz Bayesian, 374
  • modeling process, 3
  • moment, 21
    • of aggregate loss distribution, 153
    • factorial, 505
    • generating function, 31
    • generating function, for aggregate loss, 153
    • individual risk model, 189
    • limited expected value, 27
  • mortality study
    • actuarial exposure, 340
    • anniversary-to-anniversary, 342
    • date-to-date, 342
    • exact exposure, 340
    • insuring ages, 341
    • seriatim, 339
  • mortality table construction, 338
  • multiple decrement tables, 344
  • multistate models, 350

N

  • negative binomial distribution, 85, 507
    • as compound Poisson-logarithmic, 103
    • estimation, 259
    • as Poisson mixture, 86
    • simulation, 474
  • negative hypergeometric distribution, 112, 286
  • Nelson–Aalen estimator, 310
    • variance, 313
  • Neyman Type A distribution, 100, 510
  • noninformative prior distribution, 276
  • normal distribution, 391
    • bivariate, 432
    • simulation, 476

O

  • observed information, 245
  • ogive, 300
  • ordinary deductible, 126, 327

P

  • paralogistic distribution, 496
  • parameter, 3
    • scale, 53
    • uncertainty, 86
  • parametric distribution, 52, 296
    • family, 54
  • Pareto distribution, 77, 494
  • parsimony, 372
  • partial credibility, 393
  • percentile, 29
  • percentile matching, 220
  • plot
    • density function, 357
    • difference, 357
    • distribution function, 355
  • point estimation, 203
  • Poisson–binomial distribution, 510
  • Poisson distribution, 82, 506
    • estimation, 255
    • simulation, 474
  • Poisson–ETNB distribution, 269, 511
  • Poisson–inverse Gaussian distribution, 269, 511
  • Poisson-logarithmic distribution, 103
  • policy limit, 134, 327
  • Polya–Aeppli distribution, 511
  • Polya-Eggenberger distribution, 112
  • posterior distribution, 277
  • predictive distribution, 277, 408
  • prior distribution, 86
    • improper, 276
    • noninformative or vague, 276
  • probability density function, 14
  • probability function, 16, 81
  • probability generating function, 31, 82
    • for aggregate loss, 153
  • probability mass function, 16
  • process variance, 418
  • product–limit estimator, 306, 308
    • large data sets, 338
    • variance, 311
  • pseudorandom variables, 468
  • pure premium, 387
  • p-value, 227

R

  • random variable
    • central moment, 22
    • coefficient of variation, 22
    • continuous, 13
    • discrete, 13
    • excess loss, 24
    • kurtosis, 23
    • left censored and shifted, 25
    • left truncated and shifted, 24
    • limited expected value, 27
    • limited loss, 27
    • mean, 21
    • mean excess loss, 24
    • mean residual life, 24
    • median, 29
    • mixed, 13
    • mode, 18
    • moment, 21
    • percentile, 29
    • right censored, 27
    • skewness, 22
    • standard deviation, 22
    • support, 13
    • variance, 22
  • recursive formula, 515
    • aggregate loss distribution, 173
    • for compound frequency, 102
    • continuous severity distribution, 178, 517
  • reinsurance, 157
  • relative efficiency, 215
  • reputational risk, 42
  • right censored variable, 27
  • right censoring, 236, 304
  • right truncation, 327
  • risk measure, 41
    • coherent, 42
  • risk model
  • risk set, 305

S

  • scale distribution, 53
  • scale parameter, 53
  • Schwarz Bayesian Criterion, 374
  • score function, 207
  • severity/frequency interaction, 186, 513
  • Sibuya distribution, 96
  • Sichel distribution, 438
  • significance level, 225
  • simulation, 467
    • aggregate loss calculations, 480
    • binomial distribution, 474
    • discrete mixtures, 472
    • life table, 473
    • lognormal distribution, 476
    • negative binomial distribution, 474
    • normal distribution, 476
    • Poisson distribution, 474
  • single parameter Pareto distribution, 502
  • skewness, 22
  • smoothed empirical percentile estimate, 220
  • span, 179
  • spliced distribution, 69
  • standard deviation, 22
  • stop-loss insurance, 157
  • support, 13
  • survival function, 14

T

  • tail correction, 309
  • tail value at risk, 41, 44
  • tail weight, 33
  • transformed beta distribution, 71, 493
  • transformed beta family, 74
  • transformed distribution, 63
  • transformed gamma distribution, 74, 496
  • transformed gamma family, 74
  • triangular kernel, 334
  • truncation
  • Tweedie distribution, 159

U

  • unbiased, 4, 205
    • maximum likelihood estimator, 243
  • uniform kernel, 333
  • uniformly minimum variance unbiased estimator (UMVUE), 212
  • uniformly most powerful test, 226

V

  • vague prior distribution, 276
  • Value at Risk, 41, 43, 44
  • variable-component mixture, 55
  • variance, 22, 407
    • conditional, 406
    • delta method, 248
    • Greenwood's approximation, 313
    • Kaplan–Meier estimator, 311
    • Nelson–Aalen estimator, 313
    • product–limit estimator, 311

W

Y

Z

  • zero-modified distribution, 93, 509
  • zero-truncated binomial distribution, 508
  • zero-truncated distribution, 93
  • zero-truncated geometric distribution, 507
  • zero-truncated negative binomial distribution, 508
  • zero-truncated Poisson distribution, 507
  • zeta distribution, 382
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