Index

A

Affluence line, 446
Age, 23, 32, 43, 175, 210, 218, 225, 240, 258, 345, 364, 371, 380, 395, 442, 458
at failure, 2, 43, 69, 282, 289, 315
Ageing, 11, 19, 54, 175, 178, 203, 210, 216, 224, 236, 275, 345, 360, 371, 380, 385, 416, 440
classes, 176, 178, 205, 223, 281, 362, 458
concepts, 62, 106, 126, 175, 178, 186, 203–205, 210, 236, 242, 359, 385
intensity function, 245
properties, 32, 235, 393, 413, 432, 452
Ageing factor, 
relative, 240, 243
specific, 239, 243
Almost lack of memory property (ALMP), 277
Arrow's theorem, 449
Association, 32, 40, 327, 333, 399, 404, 416, 419, 422, 456

B

Bathtub shaped (BT), 
distribution, 270
hazard rate, 249, 271
function, 173, 247, 257, 261
model, 247, 271, 435
shape, 249, 257, 435
Binomial distribution, 7, 108, 116, 143, 202, 278
negative, 17, 109
Bivariate decreasing hazard rate-2 (BDHR-2), 399, 421, 427
Bivariate decreasing hazard rate-3 (BDHR-3), 404
Bivariate decreasing hazard rate-4 (BDHR-4), 417
Bivariate decreasing hazard rate-5 (BDHR-5), 405
Bivariate decreasing hazard rate-6 (BDHR-6), 405
Bivariate decreasing hazard rate-7 (BDHR-7), 405
Bivariate decreasing hazard rate-9 (BDHR-9), 417
Bivariate decreasing hazard rate-10 (BDHR-10), 361, 368, 376, 390, 417
Bivariate decreasing hazard rate-11 (BDHR-11), 391
Bivariate decreasing hazard rate-12 (BDHR-12), 391
Bivariate decreasing mean residual life-2 (BDMRL-2), 376, 391
Bivariate decreasing mean residual life-5 (BDMRL-5), 376
Bivariate decreasing mean residual life-6 (BDMRL-6), 377
Bivariate decreasing mean residual life-7 (BDMRL-7), 377
Bivariate decreasing PMRL-1 (BDPMRL-1), 384
Bivariate decreasing PMRL-2 (BDPRML-2), 384
Bivariate distribution functions, 314
Bivariate distributions, 12, 26, 291, 296, 318, 321, 354, 381, 411, 416
Bivariate equilibrium distribution, 30, 336, 342
Bivariate geometric distribution-1 (BVGD-1), 389
Bivariate geometric distribution-2 (BVGD-2), 394
Bivariate geometric distribution-3 (BVGD-3), 401
Bivariate geometric distribution-4 (BVGD-4), 403
Bivariate geometric distribution-5 (BVGD-5), 405
Bivariate geometric distribution-6 (BVGD-6), 405
Bivariate geometric distribution-7 (BVGD-7), 406
Bivariate hypergeometric distribution, 359
Bivariate increasing hazard rate class, 361
Bivariate increasing hazard rate-1 (BIHR-1), 359, 375
Bivariate increasing hazard rate-2 (BIHR-2), 399, 405, 412, 417, 424
Bivariate increasing hazard rate-3 (BIHR-3), 404, 417
Bivariate increasing hazard rate-5 (BIHR-5), 390
Bivariate increasing hazard rate-6 (BIHR-6), 390
Bivariate increasing hazard rate-7 (BIHR-7), 390
Bivariate increasing hazard rate-8 (BIHR-8), 359, 390, 405
Bivariate increasing hazard rate-9 (BIHR-9), 366, 385, 390, 407, 411
Bivariate increasing hazard rate-10 (BIHR-10), 361, 368, 376
Bivariate increasing hazard rate-11 (BIHR-11), 364, 404
Bivariate increasing hazard rate-12 (BIHR-12), 365
Bivariate increasing mean residual life-2 (BIMRL-2), 376, 391, 417
Bivariate increasing mean residual life-5 (BIMRL-5), 376, 391
Bivariate increasing PMRL-1 (BIPMRL-1), 384
Bivariate increasing PMRL-2 (BIPMRL-2), 384
Bivariate model, 321
Bivariate Poisson distribution, 28
Bivariate probability, 30
Bivariate probability mass function, 336
Bivariate random vectors, 376
i.i.d., 20
Bivariate Schur-constant distribution, 415
Bivariate variance residual life, 306
Bivariate Waring distribution, 285, 335, 360, 422
Bivariate Waring distribution-1 (BWD-1), 416
Bivariate Waring distribution-2 (BWD-2), 420
Bivariate Weibull distribution, 425
Bivariate weighted distribution, 337
Bivariate Yule distribution, 423
Bivariate-Schur constant model (BSM), 409

C

Change points, 248, 266, 435
unique, 249
Clayton measure, 329, 393, 413
Coherent structures, 449
COM-Poisson distribution, 29
bivariate, 30
Compatibility conditions, 31, 293, 304
Competing risk models, 254, 443
Concave, 21, 89, 177, 181, 227, 235, 243, 440
Conditional distributions, 24, 292, 321, 349, 388, 396, 401, 411
bivariate geometric, 293, 343
univariate, 290
Conditional probability, 12, 44, 69, 226, 282, 313, 360
Conditional probability mass functions, 24, 31, 79, 293, 304, 338
Conditionally increasing sequence (CIS), 38
Constancy, 277, 291, 355
Construction, 2, 48, 248, 257, 269, 276, 442
Continuous distributions, 3, 70, 157, 168, 173, 257, 392, 444
Convergent, 4, 24
Convex, 21, 86, 181, 197, 215, 235, 243, 270, 440
Convex functions, 10, 270
Convolution, 15, 168, 178, 185, 191, 202, 209, 220, 272, 277
n-fold, 16, 104, 459
Correlation, 25, 32, 389, 407
Pearson, 396
Correlation coefficient, 25, 32, 40, 389, 410, 417, 422
Covariance, 25, 123, 305, 327, 332, 404, 418

D

Data sets, 50, 107, 247, 256, 275, 387
Deaths, 247, 429
number of, 93
Decreasing function, 74, 92, 139, 197, 229, 246, 355, 409, 425, 443
Decreasing hazard rate average class (DHRA), 190, 238, 440
Decreasing hazard rate average-1 class (DHRA-1), 191, 435
Decreasing hazard rate (DHR), 181, 196, 225, 238, 245, 252, 265, 357, 411, 435, 448
Decreasing hazard rate function, 251
Decreasing initially and then decreasing mean residual life (DIMRL), 248
Decreasing likelihood ratio (DLR), 186
Decreasing mean inactivity time (DMIT), 229
Decreasing mean residual life (DMRL), 195–209, 216, 221, 235–240, 261, 377, 412, 449, 458
Decreasing reversed hazard rate average (DRHRA), 232
Decreasing reversed hazard rate average-1 (DRHRA-1), 233
Decreasing reversed hazard rate (DRHR), 224, 437
Decreasing uncertainty of residual life (DURL), 453
Decreasing variance residual life (DVRL), 206, 221, 235
Dependence, 32, 34, 286, 306, 327, 334, 363, 394
negative, 34, 331, 389, 415
positive, 329, 415
Device, 
multi-component, 380, 387, 416
p-component, 282, 313, 372
DFR, 235, 250, 435, 453
Discrete distributions, 22, 50, 77, 107, 157, 179, 254, 373, 387, 444
gamma, 163
inverse Weibull, 161, 189, 251
life, 3, 206, 240
Lindley, 165, 180
Pareto, 48, 131
Discrete lifetimes, 3, 23, 90, 180, 192, 203, 218, 227, 269, 363, 437, 452
Discrete models, 3, 66, 157, 167, 185, 202, 258, 387, 452
Discrete random variables, 3, 8, 22, 43, 52, 71, 84, 90, 98, 121, 129, 140, 155, 262, 278, 454
i.i.d., 11
Discrete uniform distribution, 12, 75, 89, 130, 454
Discretized quadratic hazard model (DQHM), 258
Distribution function, 3, 23, 27, 54, 70, 74, 84, 92, 105, 118, 124, 130, 144, 154, 173, 185, 193, 217, 226, 251, 254, 313, 335, 347, 437, 448, 458
joint, 322
Distributions, 
Abel series, 141, 151
additive Weibull (AW), 255
arithmetic, 14, 70, 79, 230
baseline, 105, 194, 343, 442
beta, 7, 186
bivariate, 
Bernoulli, 30
exponential, 403
component, 99
compound, 17, 452
discrete half-logistic, 159
equilibrium compound, 452
family of, 108, 267
finite range, 84, 279
hypergeometric, 108
negative, 13, 109, 131, 423
identical, 356, 363
income, 2, 444
k-geometric, 278
linear hazard rate, 261, 267
log-concave, 224
logarithmic series, 115, 138, 149
mixture, 7, 10, 93, 193, 270
multivariate Zipf, 426
new, 173, 442
normal, 112
power, 168, 447
probability, 11, 43, 107, 147, 177, 275
quasi-binomial, 145
specified, 107
standard, 7, 65, 71
“S”, 170
zeta, 133, 444

E

Entropy, 453
Equilibrium distributions, 11, 31, 104, 177, 186, 203, 235, 274, 336, 343, 376, 385, 440
Equivalence classes, 238
Estoup distribution, 134
Estoup model, 132
Events, 58, 69, 85, 145, 275, 362, 395, 429, 448, 450
Expectation, 7, 24, 180
Expected frequencies, 59, 259, 271
Exponential distribution, 46, 127, 160

F

Factorial moment generating function, 5, 14
Factorial moments, 14, 63, 104, 129, 138, 191, 274
Factorial partial moments, 66
Failure, 1, 18, 43, 58, 85, 97, 107, 145, 175, 225, 247, 275, 291, 360, 383, 458
Form-invariant, 10
Formation of coherent systems, 178, 190, 191
Formation of mixtures, 185, 223, 225
and convolutions, 209
Formation of roller-coaster shape, 279
Functional forms, 6, 43, 58, 284, 291, 296, 315, 320, 327, 387
Functions, 
characteristic, 5, 10, 12, 407
concave, 22
convex, 22, 270
covariance, 326, 380
decreasing, 21
doubly truncated mean, 54
expected value, 4, 24
increasing, 21
log-concave, 22
log-convex, 22
log-odds, 85, 89
non-decreasing, 35, 54, 65, 86, 180, 226, 234, 259, 450, 457
non-increasing, 35
odds, 85, 167, 181, 212
probability density (mass), 456
reliability, 175
survival, 175
TP2 (totally positive of order 2), 22, 181, 331
vitality, 52

G

Geeta distribution, 116, 155
Generalized power series distribution (GPSD), 115
Geometric distribution, 13, 20, 45, 54, 55, 57, 71, 104, 125, 139, 148, 156, 168, 177, 194, 223, 237, 267, 276, 388
bivariate, 30, 288, 307, 329, 349, 382, 389, 403, 425
multivariate, 126, 293, 351, 388, 407
negative hyper, 8
reversed, 71, 233
univariate, 355
Geometric distributions, independent univariate reversed, 318
Geometric marginals, 291, 398, 403
Geometric mixture, 100, 433
Good distribution, 138, 168
Gumbel's bivariate exponential distribution, 392

H

Haight distribution, 113, 153
Harmonically new better than used in expectation (HNBUE), 222, 240, 383
Harmonically new worse than used in expectation (HNWUE), 222
Hazard functions, 44, 61, 99, 149, 169, 247, 255, 266, 279, 430, 431, 443
Hazard of death, 430
Hazard positive dependence (HPD), 331
Hazard rate, 11, 44, 57, 62, 69, 75, 90, 94, 103, 110, 125, 151, 159, 168, 175, 178, 184, 192, 203, 226, 237, 245, 250, 254, 261, 269, 270, 273, 282, 287, 291, 306, 328, 337, 355, 360, 368, 399, 402, 411, 430, 432, 439, 452
alternative, 47, 56, 73, 90, 96, 161, 185, 194, 214, 255, 272, 431
conditional, 292, 296, 303, 361, 390, 399, 400
cumulative, 214, 240, 244
reversed, 77, 92, 319, 320, 437
average, 190
bathtub, 257
bivariate, 288, 302, 358
conditional, 96, 287, 322, 361, 390, 402
constant, 47, 57, 293
cumulative, 47, 91, 194, 214
decreasing, 186, 247, 270, 279
reversed, 224
functions, 43, 119, 139, 144, 161, 234, 248, 251, 270, 406, 429, 438, 449
increasing, 186, 188, 197, 198, 200, 247, 270, 272, 273, 365, 444
reversed, 226
marginal, 282
multivariate, 282, 287, 298, 345, 366
reversed, 313
non-monotonic, 8, 275
order, 176, 269, 341, 364
periodic, 275
predictive, 355
reversed, 69, 86, 92, 102, 125, 146, 157, 224, 252
scalar, 284, 337, 390, 403, 411, 426
reversed, 313
total, 291, 399
univariate, 282, 290, 322, 329, 349
reversed, 314
vector-valued, 295
bivariate, 289
Hurwitz-Lerch zeta distributions (HLZD), 117
Hurwitz-Zeta distribution, 134

I

IFR, 201, 212, 229, 235, 453
Inactivity time, 74, 224
mean, 74, 224, 323
reversed mean, 177
Increasing convex functions, 181, 196
Increasing convex order, 177, 216
Increasing function, 35, 74, 95, 170, 200, 217, 226, 242, 269, 325, 346, 366, 396, 411, 419, 457
linearly, 51
strictly, 75, 160
Increasing hazard rate average class (IHRA), 190, 238, 365, 435, 440, 443, 458
Increasing hazard rate average (IHRA-1), 191, 435
Increasing hazard rate (IHR), 181, 190, 196, 237, 243, 252, 261, 357, 364, 411, 435, 437, 449, 458
Increasing initially and then increasing mean residual life (IDMRL), 248
Increasing likelihood ratio (ILR), 186, 235
Increasing mean inactivity time (IMIT), 227, 236
Increasing mean residual life (IMRL), 195–209, 239, 415, 448
Increasing mean residual life in harmonic average (IMRLHA), 202
Increasing proportional failure rate (IPFR), 448
Increasing reversed hazard rate (IRHR), 226, 437, 449
Increasing sequence, 21, 227
Increasing set, 191
Increasing variance inactivity time (IVIT), 230
Increasing variance residual life (IVRL), 206
Induction, 212, 285, 294, 348
Inequality of incomes, 445
Infant mortality, 247

J

Joint distribution, 20, 31, 281, 287, 338, 363, 369, 398, 409, 459
Joint probability, 36, 340, 395
generating function, 28
mass function, 23, 36, 282, 363, 388, 396, 420

K

Katz family, 112

L

Lack of memory (LMP), 71, 126, 159, 179, 240, 252, 275, 348
Left-tail decreasing in sequence (LTDS), 38
Left-tail increasing (LTI) concept, 35
Lagrangian expansion, 143, 152
Lerch distribution, 122
Lerch family, 120, 130
Life distributions, 7, 43, 48, 65, 97, 175, 185, 194, 215, 230, 240, 248, 266, 283, 314, 345, 355, 371, 376, 381, 452, 458
ordering, 244
residual, 99, 181, 194, 236, 274, 305, 376, 452
weaker class of, 194, 206
Lifetime data, 69, 107
Lifetime distributions, 47, 52, 95, 107, 179, 250, 452
multivariate continuous, 428
Lifetimes, 
component, 281, 363, 432
exponential, 129
geometric, 126
of the device, 1, 247, 259, 416
Likelihood ratio order, 177, 269, 347
Log-concave, 10, 21, 181, 186, 197, 225, 236
Log-concavity, 21, 186
Log-convex, 10, 21, 181, 236
Log-convexity, 21
Lomax distribution, 244

M

Marginal distributions, 12, 20, 26, 284, 295, 300, 321, 337, 340, 388, 396, 400, 416, 426
Marginals, 296, 322, 337, 365, 405, 420, 427
Maximum likelihood estimates, 62, 155, 166
Mean residual life, 49, 55, 58, 80, 98, 110, 120, 126, 133, 156, 175, 195, 200, 218, 230, 247, 256, 277, 306, 330, 372, 380, 391, 402, 414, 424, 438, 448, 452
conditional, 302, 343, 376, 391
dynamic, 304
reversed, 55, 74, 88, 101, 133, 145, 157, 167, 224, 323, 447
univariate, 301, 341
Mean residual life function, 50, 58, 75, 88, 99, 129, 141, 144, 153, 167, 180, 195, 235, 248, 261, 274, 298, 308, 349, 407, 420, 438
bivariate, 300, 417
conditional, 303, 376, 391
hazard and, 236
monotonicity of the, 371
multivariate, 299, 372
multivariate reversed, 323
univariate, 301
univariate reversed, 325
Mean residual life order, 177, 180, 347
harmonic, 177
multivariate, 347
Median, 34, 260
Memory, 179, 199
Method of maximum likelihood, 123, 155, 255
Mixing constant, 6, 101
Mixtures, 6, 93, 127, 178, 185, 193, 202, 213, 270, 283
Model hazard rate, 355
Models, 
additive hazards, 432
discrete lifetime, 53, 175
discretized quadratic hazard rate, 259, 272
linear regression, 255
multivariate hazard rate, 360
non-monotonic hazard rate, 247
proportional hazard rate, 238, 440
proportional hazards, 48, 430
quadratic hazard rate, 267
simple linear regression, 161, 251
Modified power series distributions (MPSD), 13, 115, 143
Modified Weibull distribution (MWD), 254, 259, 268
Modified Weibull law, 254
Moments, 4, 66, 109, 120, 132, 148, 153, 162, 251, 259, 274
central, 4, 121, 155
Moments of residual life, 66, 99
Monotonicity, 71, 80, 195, 224, 262, 345, 356, 371, 376, 384, 399
Multivariate decreasing hazard rate 1 (MDHR-1), 356
Multivariate decreasing hazard rate 2 (MDHR-2), 356, 367
Multivariate decreasing hazard rate 3 (MDHR-3), 357
Multivariate decreasing hazard rate 4 (MDHR-4), 357
Multivariate decreasing hazard rate 5 (MDHR-5), 359
Multivariate decreasing hazard rate 6 (MDHR-6), 359
Multivariate decreasing hazard rate 7 (MDHR-7), 359
Multivariate decreasing hazard rate 8 (MDHR-8), 359
Multivariate decreasing hazard rate 9 (MDHR-9), 360, 366, 369
Multivariate decreasing hazard rate average 1 (MDHRA-1), 366
Multivariate decreasing hazard rate average 2 (MDHRA-2), 367, 368
Multivariate decreasing hazard rate average 3 (MDHRA-3), 368
Multivariate decreasing hazard rate (MDHR), 356
Multivariate decreasing mean residual life 1 (MDMRL-1), 372, 381
Multivariate decreasing mean residual life 2 (MDMRL-2), 372
Multivariate decreasing mean residual life 3 (MDMRL-3), 372
Multivariate decreasing mean residual life 4 (MDMRL-4), 372, 381
Multivariate decreasing mean residual life (MDMRL), 371
Multivariate distributions, 22, 291, 387
Multivariate equilibrium distribution, 31
Multivariate geometric distribution-2, 394
Multivariate geometric distribution-3, 401
Multivariate geometric distribution-4, 403
Multivariate increasing hazard rate average 1 (MIHRA-1), 366
Multivariate increasing hazard rate average 2 (MIHRA-2), 367
Multivariate increasing hazard rate average 3 (MIHRA-3), 368
Multivariate increasing hazard rate average (MIHRA), 365
Multivariate increasing hazard rate (MIHR), 356
Multivariate increasing hazard rate-1 (MIHR-1), 356
Multivariate increasing hazard rate-2 (MIHR-2), 356, 367
Multivariate increasing hazard rate-3 (MIHR-3), 357
Multivariate increasing hazard rate-4 (MIHR-4), 357
Multivariate increasing hazard rate-5 (MIHR-5), 358, 379
Multivariate increasing hazard rate-6 (MIHR-6), 358
Multivariate increasing hazard rate-7 (MIHR-7), 358, 359
Multivariate increasing hazard rate-8 (MIHR-8), 358, 379
Multivariate increasing hazard rate-9 (MIHR-9), 360, 366
Multivariate increasing hazard rate-11 (MIHR-11), 363
Multivariate increasing mean residual life 1 (MIMRL-1), 372
Multivariate increasing mean residual life 2 (MIMRL-2), 372
Multivariate increasing mean residual life 3 (MIMRL-3), 372
Multivariate increasing mean residual life 4 (MIMRL-4), 372
Multivariate increasing mean residual life (MIMRL), 371
Multivariate left-tail decreasing (MLTD), 38
Multivariate length-biased distribution, 28
Multivariate life distributions, 370
Multivariate models, 26, 369, 416
Multivariate new better than used in expectation-1 (MNBUE-1), 379
Multivariate new better than used in expectation-2 (MNBUE-2), 380
Multivariate new better than used in expectation-3 (MNBUE-3), 380
Multivariate new better than used in expectation-4 (MNBUE-4), 380
Multivariate new better than used in expectation-5 (MNBUE-5), 381
Multivariate new better than used in expectation-6 (MNBUE-6), 381
Multivariate new better than used in expectation-7 (MNBUE-7), 381
Multivariate new better than used in expectation-8 (MNBUE-8), 381
Multivariate new better than used in hazard rate 1 (MNBUHR-1), 369
Multivariate new better than used in hazard rate 2 (MNBUHR-2), 369
Multivariate new better than used-1 (MNBU-1), 378
Multivariate new better than used-2 (MNBU-2), 378
Multivariate new better than used-3 (MNBU-3), 378, 379
Multivariate new better than used-4 (MNBU-4), 378
Multivariate new worse than used in hazard rate (MNWUHR-1), 369
Multivariate new worse than used in hazard rate 2 (MNWUHR-2), 369
Multivariate Schur-constant family of distributions, 407
Multivariate totally positive of order 2 (MTP2), 39
Multivariate weighted distributions, 28, 343

N

Negative binomial distribution, 112, 115, 126, 147
Negative hypergeometric law, 237, 360, 425
Negatively quadrant dependence (NQD), 35, 331, 394, 400, 404, 425
New better than used in convex order (NBUC), 215
New better than used in expectation (NBUE), 217, 236, 377, 446, 458
New better than used in hazard rate average (NBUHRA), 194, 436
New better than used in hazard rate average-1 (NBUHRA-1), 194, 435
New better than used in hazard rate average-2 (NBUHRA-2), 194
New better than used in hazard rate (NBUHR), 192, 211, 239, 436
New better than used in hazard rate-1 (NBUHR-1), 369, 435
New better than used in pgf order (NBUpg), 217, 218
New better than used (NBU), 210, 236, 379, 436, 451, 458
New better than used-1 (NBU-1), 214
New better than used-3 (NBU-(3)), 222
New worse than used in convex order (NWUC), 215
New worse than used in expectation (NWUE), 218, 448, 456
New worse than used in hazard rate average-1 (NWUHRA-1), 194
New worse than used in hazard rate average-2 (NWUHRA-2), 194
New worse than used in hazard rate (NWUHR), 192
New worse than used in pgf order (NWUpg), 217
New worse than used (NWU), 210, 240, 436, 443, 451
New worse than used-1 (NWU-1), 214
No-ageing, 126, 179, 277, 345, 356, 360, 361, 395, 403, 407
conditionally, 354
locally, 349
univariate, 348, 394
No-ageing property, 126, 175, 179, 210, 242, 351, 363, 388, 400, 404, 416
Bayesian, 356
bivariate, 399
conditional, 352
locally, 349
multivariate, 347, 360
strong, 348, 361, 404
univariate, 348, 394

O

Odds ratio, 85, 213, 441
Ord family, 108, 167

P

Partial moments, 49, 67
Partial order, 176, 193, 238
Periodic hazard rate function, 275
Permutations, 38, 290, 351, 363
PF2 (Polya frequency function of order 2), 22, 181
pgf order, 217
Poisson distribution, 10, 16, 30, 62, 109, 145, 179, 387, 434
compound weighted, 11
exponentiated, 254
generalized, 143, 153
mixed, 206
weighted, 10
univariate, 29
Pólya-Aeppli distribution, 27
bivariate, 28
Type II bivariate, 28
Population, 97, 110, 132, 151, 251, 445
Positively lower orthant dependent (PLOD), 38
Positively quadrant dependence (PQD), 35, 331, 404, 416, 422, 427
Positively upper orthant dependent (PUOD), 38
Probability density function, 7, 43, 116, 128, 168, 173, 355
Probability generating function, 5, 15, 18, 26, 113, 124, 129, 138, 142, 152, 165, 205, 211, 395
order, 177, 217
Product moment of residual life (PMRL), 383
Proportional hazard rates model, 96
Proportional hazards model (PHM), 48, 430, 438
Proportional mean residual life model (PMRLM), 438
Proportional odds model (POM), 440
Proportional reversed hazard model (PRHM), 436

Q

Quasi-binomial distribution II (QBD II), 149
Quasi-geometric distribution (QGD), 147
Quasi-negative binomial distribution (QNBD), 147

R

Random lifetime, 23, 43, 238, 368
univariate, 281
Random variables, 
absolutely continuous, 70, 331
continuous, 105, 116, 127, 157, 168, 173, 335
geometric, 127, 181, 210, 219, 238, 273, 355, 409
independent, 127, 130, 277, 314, 337, 438
independent geometric, 126, 309
mutually independent, 16
non-negative discrete, 51
Random vector, 
discrete, 334, 341, 357, 366, 378, 392, 419
exchangeable, 364, 377
independent, 33, 397
non-negative, 392
Real-valued functions, 22, 41, 52, 65, 77, 113
Recurrence relation, 17, 55, 88, 115, 136, 145, 155, 253, 261, 266, 283, 301, 309
Reliability, 1, 18, 26, 32, 43, 49, 69, 117, 175, 185, 195, 220, 237, 275, 399, 411, 420, 444, 450, 456
Reliability analysis, 1, 7, 17, 31, 67, 85, 128, 203, 236, 247, 335, 429, 453
Reliability concepts, 2, 21, 121, 429, 443, 452
Reliability functions, 3, 48, 69, 107, 118, 128, 154, 164, 177, 224, 257, 269, 281, 287, 306, 327, 341, 387, 412, 426, 432, 450, 452
Reliability properties, 3, 130, 155, 365, 395, 411, 419, 458
Reliability theory, 1, 21, 43, 47, 62, 85, 175, 407, 429, 449, 458
Renewal process, 19, 217, 458
Residual life, 
reversed, 74, 87, 327
variance of, 62, 250, 426
Residual lifetime, 49, 175, 179, 226, 356, 453
Reverse regular of order 2 (RR-2), 36
Reversed geometric law, 71
Reversed hazard rate functions, 139, 159
Reversed proportional hazard rates model, 97
Right-corner set increasing (RCSI), 36, 329, 334, 416
Right-tail decreasing (RTD) concept, 35
Right-tail increasing in sequence (RTIS), 38

S

Schur-concave, 41, 363
Schur-constant distributions, 363, 412
Schur-constant models, 356, 415
Schur-convex, 41
Shock model, 19, 193, 395
Shocks, 18, 159, 170, 193, 225, 395
consecutive, 18–20
Stochastic decreasing (SD), 36
Stochastic dominance of order 1 (SD1), 188
Stochastic dominance of order 2 (SD2), 188, 224
Stochastic dominance of order 3 (SD3), 188
Stochastic monotonicity, 35
Stochastic orders, 176, 215, 227, 238, 244, 269, 345, 364, 436, 451, 458
Survival, 32, 85, 175, 210, 409, 430, 441
probability of, 210, 240, 371, 395
Survival functions, 3, 14, 18, 23, 30, 43, 49, 60, 63, 87, 125, 133, 148, 157, 169, 183, 188, 210, 222, 234, 239, 250, 265, 282, 295, 306, 329, 335, 348, 356, 363, 372, 382, 388, 398, 405, 416, 423, 434, 441, 452, 458
baseline, 340, 410, 432
bivariate, 362, 378
conditional, 31, 363, 389
discrete, 254
empirical, 158, 258, 419
joint, 339, 394, 409
marginal, 27, 284, 330, 409, 433
univariate, 407, 411
System failure, 19
System lifetime, 20, 225, 277, 432

T

Tail value at risk (TVaR), 450
Time, 
between failures, 458
between shocks, 18
of death, 58
Time-dependent measures, 32, 39, 176, 393, 413, 422
Total positivity, 36, 456
Total positivity of order 2 (TP-2), 36

U

Uncertainty, 1, 107, 355, 450
Shannon's measure, 452
Uniform distribution, 14, 84, 131, 227, 230, 238, 267, 316, 323, 454
bivariate, 316, 322
Unimodal, 109, 145, 155, 161, 251
Units, 
new, 215, 222, 240
old, 215, 240
Unity, 24, 40, 70, 128, 139, 245, 312, 330, 373, 393, 407, 422, 440
Univariate case, 24, 30, 281, 293, 345, 358, 364, 365, 371, 380, 385
Univariate equilibrium distributions, 337, 459
Upside-down bathtub shape (UBT), 
distribution, 264, 271
hazard rate, 249, 261
Used better than aged in expectation (UBAE), 206, 236
Used better than aged (UBA), 205, 236
Used worse than aged in expectation (UWAE), 206
Used worse than aged (UWA), 205
Useful life, 247

V

Values, 
chi-square, 59, 259
expected, 4, 24, 25, 32, 79
Variance, 4, 9, 20, 25, 40, 49, 64, 71, 78, 79, 113, 117, 129, 133, 139, 145, 156, 305, 327, 401, 410, 457
Variance residual life function, 62, 80, 305, 306, 393, 418
bivariate, 308
reversed, 326
Variance-covariance matrix, 305, 327
Vector hazard rate, 290, 308, 328, 346, 357, 367, 374, 404, 411, 417, 426

W

Waring distribution, 7, 13, 45, 65, 98, 104, 179, 186, 237, 266, 310, 439
Wear out, 247, 279
Weibull distribution, 169, 237, 425
discrete, 158, 255
Weight function, 9, 234, 337
Weighted distributions, 8, 28, 103, 233
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