A

abbreviations 25
absolute values 41, 59, 284, 422
accuracy
numerical 116, 230-231, 423-425
symbolic 421-422
admissible estimator 302
Ali-Mikhail-Haq 212, 249
ancillary statistic 337
animations
approximation error 286
bivariate Exponential pdf (Gumbel Model II) 11
bivariate Gamma pdf (McKay) 248
bivariate Normal pdf 217
bivariate Normal quantiles 219
bivariate Normal-Uniform pdf 214
bivariate Uniform pdf 213
conditional mean and variance 215
contours of bivariate Normal component-mix 249
contours of the trivariate Normal pdf 227
limit distribution of Binomial is Poisson 281
Lorenz curve for a Pareto distribution 44
non-parametric kernel density estimate 183
Pearson system 150
pmf of sum of two dice (fair vs shaved) 87
Robin Hood 223
arbitrary-precision numbers 423-424
Arc-Sine distribution 6
ARCH model 384-392
assumptions technology 8-9
asymptotic distribution 282-286
definition 282
of MLE (invariance property) 369-371
of MLE (maximum likelihood estimator) 367
of MLE (multiple parameters) 371-374
of MLE (with hypothesis testing) 393-394
of sample mean 287
of sample sum 287
asymptotic Fisher Information 375, 376
asymptotic theory 277-300
asymptotic unbiased estimator 366
asymptotic variance-covariance matrix 395-399, 404, 407, 410, 415, 418-419
augmented symmetric function 272-276
Azzalini's skew-Normal distribution 80, 225


B

bandwidth 181
Bates's distribution 139, 289-290
Bernoulli distribution 89-91
cumulant generating function 271
distribution of sample sum 141
likelihood 352
Logit model 90-91
method of moments estimator 184
pmf 89
sample mean vs sample median 309-310
sufficiency in Bernoulli trials 337
Berry-Esseen Theorem 453
best unbiased estimator (BUE) 325, 335-336, 362, 364
Beta distribution
as defining Pearson Type I(J) 185
as member of Pearson family 158
cumulants 64
fitted to censored marks 353-354
MLE 363
pdf 64
Beta-Binomial distribution 106
bias 306
Binomial distribution 91-95
as limiting distribution of Ehrenfest urn 95
as sum of n Bernoulli rv's 91, 141
cdf 92
kurtosis 93
limit distribution 280, 281
mgf 141, 281
Normal approximation 93, 281, 299
pmf 91
Poisson approximation 95, 280, 300
product cumulant 270
biology 107, 380
Birnbaum-Saunders distribution
cdf, pdf, quantiles 38-39
pseudo-random number generation 78
bivariate Cauchy distribution 237
bivariate Exponential distribution
Gumbel Model I, 204
Gumbel Model II, 11-13
bivariate Gamma (McKay) 248
bivariate Logistic distribution (Gumbel) 248, 249
bivariate Normal distribution 216-226
cdf 216, 217, 229-231
bivariate Normal distribution (cont.)
characteristic function 221
component-mixture 249
conditional distribution 220
contour plot 218
marginal distributions 220
mgf 220
orthant probability 231
pdf 216, 217
pseudo-random number generation 232-234
quantiles 218-219
truncated bivariate Normal 224-226
variance-covariance matrix 220
visualising random data 234
bivariate Normal-Uniform distribution 213-215
bivariate Poisson 243-248
mgf 246
moments 246-248
pgf 244
pmf 244-245
bivariate Student's t 237-238
bivariate Uniform (à la Morgenstern) 212-213
Black-Scholes option pricing 70-71, 447
Brownian motion 70


C

Cauchy distribution
as a stable distribution 58
as ratio of two Normals 134
as transformation of Uniform 119
characteristic function 143
compared to Sinc2 pdf 35-36
distribution of sample mean 143
mean 36
pdf 35, 143
product of two Cauchy rv's 148
cdf (cumulative distribution function)
definitions
   - continuous multivariate 191
   - continuous univariate 31
   - discrete multivariate 194
   - discrete univariate 81
limit distribution 279
numerical cdf 39
of Arc-Sine 7
of Binomial 92
of Birnbaum-Saunders 39
of bivariate Exponential
   - Gumbel Model I, 204
   - Gumbel Model II, 12
of bivariate Normal 216, 217, 229-231
of bivariate Normal-Uniform 214
of bivariate Uniform 213
of half-Halo 75
of Inverse Triangular 13
of Levy 74
of Maxwell-Boltzmann 32
of Pareto distribution 38
of Pascal 10
of Reflected Gamma 33
of stable distribution 59
of trivariate Normal 229-231
see also inverse cdf
censored data 354
censored distribution 68-69
and option pricing 70-71
and pseudo-random number generation 114
censored Lognormal 71
censored Normal 69
censored Poisson 327
Central Limit Theorem 286-292, 365
Generalised Central Limit Theorem 56
Lindeberg-Feller 453
Lindeberg-Lévy 287, 366, 368, 373
central moment 45, 200
characteristic function 50-60
definition
   - multivariate 203
   - univariate 50
inversion of cf
   - numerical 53, 55, 60
   - symbolic 53-60
Inversion Theorem 53
of bivariate Normal 221
of Cauchy 58, 143
of Levy 58
of Lindley 51
of Linnik 54
of Normal 50, 57
of Pareto 51
of stable distribution 56-57
relation to pgf 84
transformations 131
Uniqueness Theorem 52
Chebyshev's Inequality 295-296
Chi-squared distribution
as square of a Normal rv 129, 131, 299
asymptotic distribution of sample mean 283
distribution of sample sum 142
mean deviation 41, 421-422
method of moments estimator 283
mgf 131
mode 36
pdf 36, 41
ratio of two Chi-squared rv's 135
relation to Fisher F 135
van Beek's bound 284-285
see also noncentral Chi-squared
coefficient of variation 40
complete sufficient statistic 343, 346
component-mix distribution 102-104
bivariate Normal component-mixture 249
estimating a Poisson two-component-mix 405-411
conditional expectation E[Xa < X b] 66-67
odd-valued Poisson rv 97-98
truncated Normal 67
conditional expectation E[XY = y] 197-199
definitions: continuous 197, discrete 199
deriving conditional mean and variance
   - continuous 198, 215
   - discrete 199
Normal Linear Regression model 221-222
Rao-Blackwell Theorem 342
regression function 197, 221-222
conditional pdf f (Xa < X b) 65-67
conditional pdf f (XY = y) 197
of bivariate Exponential (Gumbel Model II) 12
of bivariate Normal 220
of bivariate Normal-Uniform 215
Normal Linear Regression model 221-222
conditional pmf f (XY = y) 199
conditional probability 65, 97
confidence interval 394-395
consistency 292-294, 367, 457
consistent estimator 294, 297
Continuous Mapping Theorem 366, 456
contour plot 188, 218, 227
convergence
in distribution 278-282, 293
in probability 292-298
to a constant 294
copulae 211-215
correlation 201
and independence 125, 211
and positive definite matrix 228
between k-statistics 268
between order statistics 314
definition 201
trivariate example 202
visualising correlation 212-213
see also covariance
covariance 201
between sample moments 266
definition 201
derived from central mgf 205
in terms of raw moments 206
of bivariate Exponential (Gumbel Model II) 12
trivariate example 202
see also correlation
Cramér-Rao lower bound 333-335
for Extreme Value 336
for Inverse Gaussian 334-335
for Poisson 334
cumulant generating function
definition 60, 203
of Bernoulli 271
of Beta 64
of Poisson 96
cumulants 60
in terms of moments 62, 206-207
of Bernoulli 271
of Beta 64
of k-statistics 267-271
of Poisson 96
product cumulant 209-210, 269
unbiased estimator of cumulants 256-260
cumulative distribution function (see cdf)


D

data
censored 354
population vs sample 151
raw vs grouped 151

American NFL matches 260
Australian age profile 239
Bank of Melbourne share price 384
censored student marks 354
death notices 405
grain 153
income and education 396
medical patients and dosage 90
NB1, NB2 418
nerve (biometric) 380, 418
psychiatric (suicide) 412
sickness 155
snowfall 181
student marks 151, 162, 170, 177, 354
Swiss bank notes 19, 185
US stock market returns 185
word count 418
degenerate distribution 103, 238, 280
delta method 456
density estimation
Gram-Charlier 175-180
Johnson 164-174
non-parametric kernel density 181-183
Pearson 149-163
dice 84-87
differentiation with respect to powers 326
Discrete Uniform distribution 115
distributions
asymptotic
censored
component-mix
degenerate
elliptical
empirical
limit distribution
mixing
parameter-mix
piecewise
spherical
stable family
stopped-sum
truncated
zero-inflated
distributions - Continuous
a-Laplace (see Linnik)
Arc-Sine
Azzalini's skew-Normal
Bates
Beta
Birnbaum-Saunders
Cauchy
Chi-squared
Double Exponential (see Laplace)
Exponential
Extreme Value
Fisher F
Gamma
Gaussian (see Normal)
half-Halo
half-Normal
Hyperbolic Secant
Inverse Gamma
Inverse Gaussian
Inverse Triangular
Irwin-Hall
Johnson family
Laplace
Levy
Lindley
Linnik
Logistic
Lognormal
Maxwell-Boltzmann
noncentral Chi-squared
noncentral F
Normal
Pareto
Pearson family
Power Function
Random Walk
Rayleigh
Rectangular (see Uniform)
Reflected Gamma
semi-Circular (see half-Halo)
Sinc2
stable
Student's t
Triangular
Uniform
Weibull
distributions - Discrete
Bernoulli
Beta-Binomial
Binomial
Discrete Uniform
Geometric
Holla
Hypergeometric
Logarithmic
Negative Binomial
Pascal
Poisson
Pólya-Aeppli
Riemann Zeta
Waiting-time Negative Binomial
Waring
Yule
Zero-Inflated Poisson
Zipf (see Riemann Zeta)
distributions - Multivariate
bivariate Cauchy
bivariate Exponential (Gumbel Model I and II)
bivariate Gamma (McKay)
bivariate Logistic (Gumbel)
bivariate Normal
bivariate Normal-Uniform (à la Morgenstern)
bivariate Poisson
bivariate Student's t
bivariate Uniform (à la Morgenstern)
Multinomial
multivariate Cauchy
multivariate Gamma (Cheriyan and Ramabhadran)
multivariate Normal
multivariate Student's t
Trinomial
trivariate Normal
truncated bivariate Normal
domain of support 31, 81-85
circular 191
non-rectangular 124, 125, 190-191, 314
rectangular 124, 190
triangular 191, 314, 317
dominant estimator 302
Dr Faustus 421


E

economics and finance 43-45, 56, 70-72, 108-109, 117, 121, 384
Ehrenfest urn 94-95
ellipse 218, 236
ellipsoid 227
elliptical distributions 234
empirical pdf 73, 77, 154, 381, 383
empirical pmf 16, 110, 111, 112
engineering 122
entropy 15
Epanechnikov kernel 182
estimator
admissible 302
asymptotic unbiased 366
BUE (best unbiased) 325, 335-336, 362, 364
consistent 294, 297
density (see density estimation)
dominant 302
estimator vs estimate 357
Fisher estimator 395-396, 397, 404
h-statistic 253-256
Hessian estimator 395-396, 398, 404
inadmissible 302, 321-322
k-statistic 256-261
maximum likelihood estimator (see MLE)
method of moments 183-184, 283
minimax 305
minimum variance unbiased 341-346, 364
non-parametric kernel density 181-183
ordinary least squares 385
Outer-product 395-396, 398
sample central moment 360
sample maximum 320-321
sample mean (see sample mean)
sample median 309-310, 318-320
sample range 320-321
sample sum 277, 287
unbiased estimator of parameters 325-347
unbiased estimator of population moments 251-261
expectation operator
basic properties 32
definitions
   - continuous 32
   - discrete 83
   - multivariate 200
when applied to sample moments 263
Exponential distribution
bivariate 11-13, 204
difference of two Exponentials 139-140
distribution of sample sum 141-142
likelihood 351
MLE (numerical) 381
MLE (symbolic) 358
order statistics 313-314
pdf 141, 313, 344, 358
relation to Extreme Value 121
relation to Pareto 121
relation to Rayleigh 122
relation to Uniform 121
sufficient statistic 344
sum of two Exponentials 136
Exponential regression 375-376, 396
Extreme Value distribution
Cramér-Rao lower bound 336
pdf 336, 377
relation to Exponential 121


F

factorial moment 60, 206-207, 247
factorial moment generating function 60, 203, 247
factorisation criterion 339-341
families of distributions
Gram-Charlier 175-180
Johnson 164-174
Pearson 149-163
stable family 56-61
fat tails 56, 108-109
see also kurtosis
first-order condition 21, 36, 357-361, 363
Fisher estimator 395-396, 397, 404
Fisher F distribution 135
Fisher Information 326-332
and MLE (regularity conditions) 367-368, 372-373
asymptotic Fisher Information 375, 376
first derivative form vs second derivative 329
for censored Poisson 327-328
for Gamma 331-332
for Inverse Gaussian 18
for Lindley 326
for Normal 330-331
for Riemann Zeta 329
for Uniform 330
Frank 212
frequency polygon 73, 77, 151, 154, 380
see also plotting techniques
Function Form 82
functions of random variables 117-148
fundamental expectation result 274


G

games
archery (Robin Hood) 222-224
cards, poker 101
craps 87-89, 115
dice (fair and unfair) 84-87
Gamma distribution
as member of Pearson family 157, 185
as sum of n Exponential rv's 141-142
bivariate Gamma (McKay) 248
Fisher Information 331-332
hypothesis testing 392-394
method of moments estimator 184
mgf 142, 456
MLE (numerical) 382-383
multivariate (Cheriyan & Ramabhadran) 208
pdf 73, 142
pseudo-random number generation 73
relation to Inverse Gamma 147
Gamma regression model 419
gas molecules 32
Gaussian kernel 19, 182
generating functions 46-56, 203-205
Geometric distribution
definition 98
distribution of difference of two rv's 148
pmf 98
Gini coefficient 40, 43-45
gradient 357-361
Gram-Charlier expansions 175-180
graphical techniques (see plotting techniques)
Greek alphabet 28


H

h-statistic 253-256
half-Halo distribution 75, 80
half-Normal distribution 225
Helmert transformation 145
HELP 5
Hermite polynomial 175, 179, 449
Hessian estimator 395-396, 398, 404
Hessian matrix 358, 360
histogram 18, 155 (see also plotting techniques)
Holla's distribution 105, 112
Hyperbolic Secant distribution 80
Hypergeometric distribution 100-101


I

inadmissible estimator 302, 321-322
income distribution 43-44, 121
independence
correlation and dependence 125, 211
mutually stochastically independent 210
independent product space 124, 190
Invariance Property 360, 369-371, 401, 410, 417
inverse cdf
numerical inversion 38-39, 75-77, 109
symbolic inversion 37-38, 74-75
of Birnbaum-Saunders 38-39
of half-Halo 75
of Levy 74
of Pareto 38, 43
Inverse Gamma distribution
as member of Pearson family 185
pdf 365
relation to Gamma 147, 365
relation to Levy 58
Inverse Gaussian distribution
Cramér-Rao lower bound 334-335
Fisher Information 18
pdf 18, 334
relation to Random Walk distribution 147
Inverse Triangular distribution 13-14
Inversion Theorem 53
Irwin-Hall distribution 55, 139
isobaric 272


J

Jacobian of the transformation 118, 123, 130, 223
Johnson family 164-174
as transformation of a Logistic rv 185
as transformation of a Normal rv 164
Types and chart 164
   - SL (Lognormal) 165-167
   - SU (Unbounded) 168-172
   - SB (Bounded) 173-174


K

k-statistic 20, 256-261
kernel density (see non-parametric kernel density)
Khinchine's Theorem 298
Khinchine's Weak Law of Large Numbers 278, 296-298, 366
Kronecker product 437
kurtosis
building your own function 446
definition 40-41
of Binomial 93
of Poisson 446
of Weibull 42
Pearson family 149-150


L

Laplace distribution
as Linnik 54
as Reflected Gamma 33
order statistics of 23, 315-317
relation to Exponential 139-140
latent variable 353, 412
Lehmann-Scheffé Theorem 346
Levy distribution
as a stable distribution 58
as an Inverse Gamma 58
cdf, pdf, pseudo-random number 74
likelihood
function 21, 350-357
observed 22, 351-357
see also log-likelihood
limit distribution
definition 279
of Binomial 280, 281
of sample mean (Normal) 279
limits in Mathematica 278
Lindley distribution
characteristic function 51
Fisher Information 326-327
pdf 51, 327
linear regression function 221
linex (linear-exponential) loss 322
linguistics 107
Linnik distribution 54
List Form 82, 111
log-likelihood
concentrated 361, 382-383, 418
function 21, 357-376, 381
observed log-likelihood
   - ARCH model (stock prices) 387
   - Exponential model (nerve data) 381
   - Exponential regression (income) 396
   - Gamma model (nerve data) 382-383
   - Logit model (dosage data) 90
   - Ordered Probit model (psychiatric data) 414-415
   - Poisson two-component-mix model 405-406
see also likelihood
Logarithmic distribution 115
Logistic distribution
as base for a Johnson-style family 185
bivariate 248, 249
pdf 23, 318
order statistics of 23
relation to Uniform 147
sample mean vs sample median 318-320
Logit model 90-91
Lognormal distribution
and stock prices 71
as member of Johnson family 165-167
as transformation of Normal 120, 165
censored below 71
moments of sample sum 276
pdf 71, 120
Lorenz curve 43-44
loss function 301-305
asymmetric 303-304
asymmetric quadratic 322, 323
linex (linear-exponential) 322
quadratic 306


M

machine-precision numbers 423-425
marginal distribution 195-196
and copulae 211
joint pdf as product of marginals 210, 211, 351, 355
more examples 12, 126, 133-137, 146, 204, 214, 220, 224-225, 237-238, 244
Markov chain 94, 447-448
Markov's inequality 295-296
Mathematica
assumptions technology 8-9
bracket types 27
changes to default behaviour 443-445
differentiation with respect to powers 326
Greek alphabet 28
how to enter 30
kernel (fresh and crispy) 5, 425
limits 278
lists 428-429
matrices 433-437, 445
notation (common) 27
notation entry 28-30
packages 425
replacements 27
subscripts 429-432
timings 30
upper and lower case conventions 24
using G in Input cells 443
vectors 438-443
see also plotting techniques
mathStatica
Basic vs Gold version 4
Continuous distribution palette 5
Discrete distribution palette 5
HELP 5
installation 3
loading 5
registration 3
working with parameters 8
maximum likelihood estimation (see MLE)
Maxwell-Boltzmann distribution 32
mean 35-36, 45
see also sample mean
mean deviation 40, 41, 299, 421-422
mean square error (see MSE)
median 37
of Pareto distribution 37-38
see also sample median
medical 90-91, 155, 380, 405, 412
method of moments estimator 183-184
for Bernoulli 184
for Chi-squared 283
for Gamma 184
mgf (moment generating function)
and cumulant generating function 60
and independence 210
central mgf 93, 203, 205, 247
definition 46, 203
Inversion Theorem 53
Uniqueness Theorem 52
of Binomial 93, 141, 281
of bivariate Exponential (Gumbel Model I) 204
of bivariate Exponential (Gumbel Model II) 12
of bivariate Normal 220
of bivariate Poisson 246
of Chi-squared 131
of Gamma 142, 456
of Multinomial 239, 241-242, 242-243
of multivariate Gamma 208
of multivariate Normal 249
of noncentral Chi-squared 144
of Normal 47
of Pareto 49
of sample mean 141
of sample sum 141
of sample sum of squares 141
of Uniform 48
MGF Method 52-56, 130-132, 141-147
MGF Theorem 52, 141
more examples 281, 364-365
minimax estimator 305
minimum variance unbiased estimation (see MVUE)
mixing distributions 102-109
component-mix 102-104, 249, 405-411
parameter-mix 105-109
MLE (maximum likelihood estimation) 357-376
asymptotic properties 365-366, 371-376
general properties 362
invariance property 369-371
more than one parameter 371-374
non-iid samples 374-376
numerical MLE (see Chapter 12)
   - ARCH model (stock prices) 387
   - Exponential model (nerve data) 381
   - Exponential regression model (income) 396
   - Gamma model (nerve data) 382-383
   - Logit model (dosage data) 90
   - Normal model (random data) 418
   - Ordered Probit model (psychiatric data) 414-415
   - Poisson two-component-mix model 405-406
regularity conditions
   - basic 367-369
   - more than one parameter 371-372
   - non-iid samples 374-375
small sample properties 363-365
symbolic MLE (see Chapter 11)
   - for Exponential 358
   - for Normal 359-360, 418
   - for Pareto 360-361
   - for Power Function 362-363
   - for Rayleigh 21
   - for Uniform 377
mode 36
moment conversion functions
univariate 62-64
multivariate 206-210
moment generating function (see mgf)
moments
central moment 45, 200
factorial moment 60, 206-207
fitting moments (see Pearson, Johnson, method of moments)
negative moment 80
population moments vs sample moments 251
product moment 200, 266
raw moment 45, 200
moments of moments 261-271
introduction 20
moments of sampling distributions 251-276
monomial symmetric function 273
Monte Carlo 290
see also pseudo-random number generation
see also simulation
Morgenstern 212
MSE (mean square error)
as risk 306-311
comparing h-statistics with polyaches 264-266
of sample median and sample mean (Logistic) 318-320
of sample range and sample maximum (Uniform) 320-321
weak law of large numbers 296-297
multinomial coefficient 451
Multinomial distribution 238-243
multiple local optima 400
multivariate Cauchy distribution 236
multivariate Gamma distribution (Cheriyan and Ramabhadran) 208
multivariate Normal distribution 216-235
multivariate Student's t 236
mutually stochastically independent 210
MVUE (minimum variance unbiased estimation) 341-346, 364


N

Negative Binomial distribution 99, 105, 418
noncentral Chi-squared distribution
as Chi-squared-Poisson mixture 105
derivation 144
exercises 299
noncentral F distribution 135
non-parametric kernel density 181-183
with bi-weight, tri-weight kernel 182
with Epanechnikov kernel 182
with Gaussian kernel 19, 182
non-rectangular domain 124, 125, 190-191, 320-321
Normal distribution
and Gram-Charlier expansions 175
as a stable distribution 57
as limit distribution of a Binomial 93, 281, 299
as member of Johnson family 164-165, 167
as member of Pearson family 150, 158
asymptotic distribution of MLE of (m, s2) 372-374
basics 8
bivariate Normal 216-226
censored below 69
central moments 265
characteristic function 50, 57
characteristic function of X1X2 132
conditional expectation of sample median, given sample mean 342-343
distribution:
   - of product of two Normals 132, 133
   - of ratio of two Normals 134
   - of X2 129, 131
   - of sample mean 143, 294-295
   - of sample sum of squares 144
   - of sample sum of squares about the mean 145
estimators for the Normal variance 307-308
finance 56, 108-109
Fisher Information 330-331
limit distribution of sample mean 279
limit Normal distribution 362, 367
   - examples 369, 392-395
mgf 47
mgf of X2 131
MLE of (m, s2) 359-360, 418
MVUE of (m, s2) 346
Normal approximation to Binomial 93, 281, 299
pseudo-random number generation
   - approximate 291-292
   - exact 72-73, 418
QQ plot 291
raw moments 46
relation to Cauchy 134
relation to Chi-squared 129, 131
relation to Lognormal 120
risk of a Normally distributed estimator 303-304
sample mean as consistent estimator of population mean 294-295
standardising a Normal rv 120
sufficient statistics for (m, s2) 340-341
trivariate Normal 226-228
truncated above 65-66, 67
working with s vs s2 326, 377, 455
   see also Invariance Property
Normal linear regression model 221-222, 385, 457
notation
Mathematica notation
   - bracket types 27
   - Greek alphabet 28
   - how to enter 30
   - notation (common) 27
   - notation entry 28-30
   - replacements 27
   - subscripts 429-432
   - upper and lower case conventions 24
   - using G in Input cells 443
statistics notation
   - abbreviations 25
   - sets and operators 25
   - statistics notation 26
   - upper and lower case conventions 24


O

one-to-one transformation 118
optimisation
differentiation with respect to powers 326
first-order condition 21, 36, 357-361, 363
gradient 357-361
Hessian matrix 358, 360
multiple local optima 400
score 357-361
second-order condition 22, 36-37, 357-360
unconstrained vs constrained numerical optimisation 369, 379, 388-389, 401, 414
optimisation algorithms 399-405
Armijo 408
BFGS (Broyden-Fletcher-Goldfarb-Shanno) 399-400, 403, 405-411, 459
DFP (Davidon-Fletcher-Powell) 403
direct search 400
genetic 400
Golden Search 401
Goldstein 408
gradient method 400, 401-405
line search 401
Method -> Newton 390-391, 397, 403, 415, 459
Method -> QuasiNewton 403, 406-407, 419, 459
NR (Newton Raphson) 390-391, 397, 399-400, 403, 412-417, 458-459
numerical convergence 404-405
Score 403-404
simulated annealing 400
taboo search 400
option pricing 70-72
order statistics 311-322
distribution of:
   - sample maximum 312, 321
   - sample minimum 312
   - sample median 318-320
   - sample range 320-321
for Exponential 313-314
for Laplace 23, 315-317
for Logistic 23
for Uniform 312
joint order statistics 23, 314, 316, 320
Ordered Probit model 412-417
ordinary least squares 385
orthant probability 231
Outer-product estimator 395-396, 398


P

p-value 393-394
parameter identification problem 414
parameter-mix distribution 105-109
Pareto distribution
characteristic function 51
median 37-38
mgf 49
MLE 360-361
pdf 37, 49, 51, 360
quantiles 38
relation to Exponential 121
relation to Power Function 147
relation to Riemann Zeta 107
Pascal distribution 10, 99
pdf (probability density function)
definition 31, 187
see also Distributions
see also pmf (for discrete rv's)
peakedness 40-41, 108-109
Pearson family 149-163
animated tour 150
Pearson coefficients in terms of moments 159-160
Types and chart 150
   - Type I, 17, 156, 158, 185
   - Type II, 158
   - Type III, 154, 157, 185
   - Type IV, 151-153, 157
   - Type V, 158, 185
   - Type VI, 158
   - Type VII, 157
unimodal 179
using a cubic polynomial 161-163
penalty function 400, 407, 415
pgf (probability generating function)
definitions 60, 84, 203
deriving probabilities from pgf 85, 85-86, 86, 104, 245
of bivariate Poisson 244-245
of Hypergeometric 100
of Negative Binomial 99
of Pascal 11
of Zero-Inflated Poisson 104
physics 32, 94-95
piecewise distributions
Bates's distribution 289-290
Inverse Triangular 13
Laplace 23, 315-317
order statistics of 23
Reflected Gamma 33
plotting techniques (some examples)
arrows 37, 81, 280
contour plots 188, 218, 227
data
   - bivariate / trivariate 233-235
   - grouped data 18, 155
   - raw 151
   see also frequency polygon
   - scatter plot 397
   - time-series 384
   see also empirical pdf / pmf
domain of support (bivariate) 125, 138, 140
empirical pdf 73, 77, 154, 381, 383
empirical pmf 16, 110, 111, 112
filled plot 44, 68
frequency polygon 73, 77, 151, 154, 380
graphics array 32, 38, 68, 109, 118, 124, 168, 174, 218
histogram 18, 155
Johnson system 170
non-parametric kernel density 19, 182-183
parametric plot 167
pdf plots 6, 139, etc.
   - as parameters change 8, 14, 32, 145, 165, 225, 313, 315
   - 3D 11, 188, 198, 213, 214, 217, 316
Pearson system 17, 152
pmf plots 10, 83, 98, 101, 103
   - as parameters change 87, 92, 96
   - 3D 190
QQ plots 291
scatter plot 397
superimposing plots 34, 35, 37, 42, 54, 55, 69, 91, 133, 219, 302, 306
text labels 32, 37, 54, 145, 302, 306, 313
wireframe 228
see also animations
pmf (probability mass function)
definitions 82, 189
see also Distributions - Discrete
see also pdf (for continuous rv's)
Poisson distribution 95-98
as limit distribution of Binomial 95, 280, 300
bivariate Poisson 243-248
censoring 327-328
Cramér-Rao lower bound 334
cumulant generating function 96
distribution of sample sum 137
kurtosis 446
odd-valued Poisson 97-98
pmf 16, 95, 110, 334
Poisson two-component-mix 102-103, 406
pseudo-random number generation 16, 110
sufficient statistic for l 340
zero-inflated Poisson 104
poker 101
Pólya-Aeppli distribution 105
polyache 255-256
polykay 257-259
Power Function distribution
as a Beta rv 185, 363
as defining Pearson Type I(J) 185
MLE 362-363
relation to Pareto 147
sufficient statistic 363-364
power sum 252, 272-276
probability
conditional 65, 97
multivariate 191-194
orthant probability 231
probability content of a region 192-193, 230-231
throwing a die 84-87
see also cdf
probability density function (see pdf)
probability generating function (see pgf)
probability mass function (see pmf)
probit model 412-413
product moment 200, 266
products / ratios of random variables 133-136
see also:
   - deriving the pdf of the bivariate t 237-238
   - product of two Uniforms 126-127
Proportional-hazards model 412
Proportional-odds model 412
pseudo-random number generation
methods
   - inverse method (numerical) 75-77, 109-115
   - inverse method (symbolic) 74-75
   - Mathematica's Statistics package 72-73
   - rejection method 77-79
and censoring 114
computational efficiency 113, 115
List Form 111
of Birnbaum-Saunders 78
of Gamma 73
of half-Halo 75-77
of Holla 112
of Levy 74
of multivariate Normal 232-234
of Normal 291-292, 418
of Poisson 16, 110
of Riemann Zeta 113
visualising random data in 2D, 3D 233-235


Q

QQ plot 291
quantiles 37
of Birnbaum-Saunders 38-39
of bivariate Normal 218-219
of bivariate Student's t 237
of Pareto 38
of trivariate Normal 227-228


R

random number (see pseudo-random number)
random variable
continuous 31, 81, 187
discrete 81-82, 189
see also Distributions
Random Walk distribution 147
random walk with drift 355, 384-386
Rao-Blackwell Theorem 342
raw moment 45, 200
Rayleigh distribution
MLE 21
relation to Exponential 122
rectangular domain 124, 190
reference computer 30
Reflected Gamma distribution 33-34
registration 3
regression 384-392
regression function 197, 221-222
regularity conditions
for Fisher Information 329-330
for MLE
   - basic 367-369
   - more than one parameter 371-372
   - non-iid samples 374-375
relative mean deviation 299
re-parameterisation 369, 388-389, 401, 406, 410, 414
Riemann Zeta distribution
area of application 107
Fisher Information 329
pmf 113, 329
pseudo-random number generation 113
risk 301-305
Robin Hood 222-224


S

sample information 332, 338, 376
sample maximum 311, 312, 320-321, 377
sample mean
as consistent estimator (Khinchine) 298
as consistent estimator (Normal) 294-295
as MLE (for Exponential parameter) 358
as MLE (for Normal parameter) 359-360
asymptotic distribution of sample mean 287
definition 277
distribution of sample mean
   - for Cauchy 143
   - for Normal 143
   - for Uniform 139, 288-292
Khinchine's Theorem 298
limit distribution of sample mean (Normal) 279
mgf of 141
variance of the sample mean 264
vs sample median, for Bernoulli trials 309-310
vs sample median, for Logistic trials 318-320
sample median
conditional expectation of sample median, given sample mean 342-343
vs sample mean, for Bernoulli trials 309-310
vs sample mean, for Logistic trials 318-320
sample minimum 311, 312
sample moment 251
sample central moment 251, 360
   - covariance between sample central moments 266
   - in terms of power sums 252
   - variance of 264
sample raw moment 251
   - as unbiased estimators of population raw moments 253
   - in terms of power sums 252
sample range 320-321
sample sum
asymptotic distribution of sample sum 287
definition 277
distribution of sample sum
   - for Bernoulli 141
   - for Chi-squared 142
sample sum (cont.)
distribution of sample sum (cont.)
   - for Exponential 141-142
   - for Poisson 137
   - for Uniform 55, 139
mgf of sample sum 141
moments of sample sum 261-271, 276
sample sum of squares
distribution of (Normal) 144
mgf of 141
sampling with or without replacement 100
scedastic function 197
score 357-361
second-order condition 22, 36-37, 357-360
security (stock) price 70-72, 108-109, 384
Sheather-Jones optimal bandwidth 19, 182
signal-to-noise ratio 299
Silverman optimal bandwidth 182
simulation 87-89, 126-127, 298-299
see also Monte Carlo
see also pseudo-random number
Sinc2 distribution 35-36
skewness
definition 40
of Weibull 42
Pearson family 149-150
Skorohod's Theorem 456
small sample accuracy 289-292
smoothing methods 181-183
spherical distributions 234, 451
stable distributions 56-61
standard deviation 40, 45
standard error 395, 399
standardised random variable 40, 120, 281, 287
statistic 251
stopped-sum distribution 108
Student's t distribution
as member of Pearson family 157
as Normal-InverseGamma mixture 105
bivariate Student's t 237-238
derivation, pdf 134
sufficient statistic 337-341, 344, 362, 363-364
sums of random variables 136-147
deriving pmf of bivariate Poisson 244-245
sum of Bernoulli rv's 141
sum of Chi-squared rv's 142
sum of Exponentials 136, 141-142
sum of Poisson rv's 137
sum of Uniform rv's 54-55, 138-139
see also sample sum
Swiss bank notes 19, 185
symmetric function 253, 272-276
systems of distributions (see families) 149-180


T

t distribution (see Student's t)
t-statistic 395, 399
theorems
Berry-Esseen 453
Central Limit Theorem 286-292
Continuous Mapping Theorem 366, 456
Inversion Theorem 53
Khinchine 298
Lehmann-Scheffé 346
Lindeberg-Feller 453
Lindeberg-Lévy 287
MGF Theorem 52, 141
Rao-Blackwell Theorem 342
Skorohod's Theorem 456
transformation theorems
   - univariate 118
   - multivariate 123
   - not one-to-one 127
Uniqueness Theorem 52
timings 30
transformations 117-148
MGF Method 52-56, 130-132, 141-147
transformation method 118-130
   - univariate 118
   - multivariate 123
   - manual 130
   - Jacobian 118, 123, 130, 223
   - one-to-one transformation 118
   - not one-to-one 127
Helmert transformation 145
non-rectangular domain 124, 125
transformation to polar co-ordinates 222-223
see also:
   - products / ratios of random variables
   - sums of random variables
Triangular distribution
as sum of two Uniform rv's 55, 138-139
Trinomial distribution 239
trivariate Normal 226-228
cdf 229-231
orthant probability 231
pseudo-random number generation 232-234
visualising random data 235
truncated distribution 65-67
truncated (above) standard Normal 65-66, 67
truncated bivariate Normal 224-226


U

unbiased estimators of parameters 325-347
asymptotic unbiasedness 366
unbiased estimators of population moments 251-261
introduction 20
multivariate 259-261
of central moments 253-254, 259-261
of cumulants 256-258, 260
of Normal population variance 307-308
of population variance 253, 254
of raw moments 253
Uniform distribution
bivariate Uniform (à la Morgenstern) 212-213
Fisher Information 330
mgf 48
MLE 377
order statistics 312
other transformations of a Uniform rv 122
pdf 48, 122, 312, 320, 330
product of two Uniform rv's 126-127
relation to Bates 139, 289-290
relation to Cauchy 119
relation to Exponential 121
relation to Irwin-Hall 55, 139
relation to Logistic 147
sample mean and Central Limit Theorem 288-292
sample range vs sample maximum 320-321
sum of Uniform rv's 54-55, 138-139
unimodal 36, 179, 182-183
Uniqueness Theorem 52


V

van Beek bound 283-285, 453
variance
definition 40, 45
of sample mean 264
of 2nd sample central moment 264
variance-covariance matrix
asymptotic variance-covariance matrix 395-399, 404, 407, 410, 415, 418-419
definition 201
of bivariate Exponential
   - Gumbel Model I, 205
   - Gumbel Model II, 12
of bivariate Normal 220
of bivariate Normal-Uniform 215
of bivariate Uniform 213
of trivariate models 202, 211
of truncated bivariate Normal 226
of unbiased estimators 333-335


W

Waiting-time Negative Binomial distribution 99
Waring distribution 418
weak law of large numbers 296-298
Weibull distribution 42


X

xenium (see book cover)


Y

Yule distribution 107


Z

zero-inflated distributions 103-104
Zipf distribution (see Riemann Zeta) 107