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
- 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
- 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
- hypothetical mean, 418
- interval, 280
- limited fluctuation, 388, 389
- 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
- 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
- moments, 153
- 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
- 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
- zeta, 122, 382
- (a, b, 0) class, 88, 506
- 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
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
- 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
I
- incomplete beta function, 490
- incomplete gamma function, 63, 489
- individual loss distribution, 150
- individual risk model, 148, 189
- moments, 190
- infinitely divisible distribution, 114
- inflation
- 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
K
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
- partial, 393
- limited loss variable, 27
- limit
- 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
- maximum likelihood estimator
- 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
- model selection, 3
- modeling process, 3
- moment, 21
- mortality study
- mortality table construction, 338
- multiple decrement tables, 344
- multistate models, 350
N
O
P
- paralogistic distribution, 496
- parameter, 3
- parametric distribution, 52, 296
- family, 54
- Pareto distribution, 77, 494
- parsimony, 372
- partial credibility, 393
- percentile, 29
- percentile matching, 220
- plot
- point estimation, 203
- Poisson–binomial distribution, 510
- Poisson distribution, 82, 506
- 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
- 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
- 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
- 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
- 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
U
V
W
Y
Z
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