Elements of Distribution Theory

Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.2 Basic Framework......Page 15
1.3 Random Variables......Page 19
1.4 Distribution Functions......Page 22
1.5 Quantile Functions......Page 29
1.6 Density and Frequency Functions......Page 34
1.7 Integration with Respect to a Distribution Function......Page 40
1.8 Expectation......Page 42
Expectation of a function of a random variable......Page 43
Inequalities......Page 47
1.9 Exercises......Page 48
1.10 Suggestions for Further Reading......Page 52
2.2 Marginal Distributions and Independence......Page 53
2.3 Conditional Distributions......Page 60
2.4 Conditional Expectation......Page 67
2.5 Exchangeability......Page 73
2.6 Martingales......Page 76
2.7 Exercises......Page 78
2.8 Suggestions for Further Reading......Page 81
3.1 Introduction......Page 83
3.2 Basic Properties......Page 86
Uniqueness and inversion of characteristic functions......Page 87
Characteristic function of a sum......Page 90
An expansion for characteristic functions......Page 91
Random vectors......Page 95
3.3 Further Properties of Characteristic Functions......Page 96
Symmetric distributions......Page 101
Lattice distributions......Page 102
3.4 Exercises......Page 104
3.5 Suggestions for Further Reading......Page 107
4.2 Moments and Central Moments......Page 108
Central moments......Page 109
Moments of random vectors......Page 110
Correlation......Page 111
Covariance matrices......Page 112
Laplace transforms......Page 113
Moment-generating functions......Page 117
Moment-generating functions for random vectors......Page 123
4.4 Cumulants......Page 124
Cumulants of a random vector......Page 132
4.5 Moments and Cumulants of the Sample Mean......Page 134
Central moments of X......Page 137
4.6 Conditional Moments and Cumulants......Page 138
4.7 Exercises......Page 141
4.8 Suggestions for Further Reading......Page 144
5.2 Parameters and Identifiability......Page 146
Identifiability......Page 148
Likelihood ratios......Page 150
5.3 Exponential Family Models......Page 151
Natural parameters......Page 152
Some distribution theory for exponential families......Page 156
5.4 Hierarchical Models......Page 161
Models for heterogeneity and dependence......Page 163
5.5 Regression Models......Page 164
5.6 Models with a Group Structure......Page 166
Transformation models......Page 169
Invariance......Page 171
Equivariance......Page 174
5.7 Exercises......Page 178
5.8 Suggestions for Further Reading......Page 183
6.1 Introduction......Page 184
6.2 Discrete Time Stationary Processes......Page 185
6.3 Moving Average Processes......Page 188
6.4 Markov Processes......Page 196
Markov chains......Page 197
6.5 Counting Processes......Page 201
Poisson processes......Page 203
Distribution of the interarrival times......Page 205
6.6 Wiener Processes......Page 206
Irregularity of the sample paths of a Wiener process......Page 207
TheWiener process as a martingale......Page 208
6.7 Exercises......Page 209
6.8 Suggestions for Further Reading......Page 212
7.2 Functions of a Real-Valued Random Variable......Page 213
7.3 Functions of a Random Vector......Page 216
Functions of lower dimension......Page 218
Functions that are not one-to-one......Page 222
Application of invariance and equivariance......Page 224
7.4 Sums of Random Variables......Page 226
7.5 Order Statistics......Page 230
Pairs of order statistics......Page 234
7.6 Ranks......Page 238
7.7 Monte Carlo Methods......Page 242
7.8 Exercises......Page 245
7.9 Suggestions for Further Reading......Page 248
8.2 Multivariate Normal Distribution......Page 249
Density of the multivariate normal distribution......Page 253
8.3 Conditional Distributions......Page 254
Conditioning on a degenerate random variable......Page 256
8.4 Quadratic Forms......Page 258
8.5 Sampling Distributions......Page 264
8.6 Exercises......Page 267
8.7 Suggestions for Further Reading......Page 270
Gamma function......Page 271
Incomplete gamma function......Page 274
9.3 Asymptotic Expansions......Page 278
Integration-by-parts......Page 280
9.4 Watson’s Lemma......Page 282
9.5 Laplace’s Method......Page 290
9.6 Uniform Asymptotic Approximations......Page 295
9.7 Approximation of Sums......Page 302
9.8 Exercises......Page 309
9.9 Suggestions for Further Reading......Page 311
10.2 General Systems of Orthogonal Polynomials......Page 313
Construction of orthogonal polynomials......Page 315
Zeros of orthogonal polynomials and integration......Page 318
Completeness and approximation......Page 321
Hermite polynomials......Page 326
Laguerre polynomials......Page 330
10.4 Gaussian Quadrature......Page 331
10.5 Exercises......Page 333
10.6 Suggestions for Further Reading......Page 335
11.1 Introduction......Page 336
11.2 Basic Properties of Convergence in Distribution......Page 339
Uniformity in convergence in distribution......Page 348
Convergence in distribution of random vectors......Page 349
11.3 Convergence in Probability......Page 352
Convergence in probability to a constant......Page 355
Convergence in probability of random vectors and random matrices......Page 359
11.4 Convergence in Distribution of Functions of Random Vectors......Page 360
11.5 Convergence of Expected Values......Page 363
11.6 Op and op Notation......Page 368
11.7 Exercises......Page 373
11.8 Suggestions for Further Reading......Page 378
12.2 Independent, Identically Distributed Random Variables......Page 379
12.3 Triangular Arrays......Page 381
12.4 Random Vectors......Page 390
12.5 Random Variables with a Parametric Distribution......Page 392
12.6 Dependent Random Variables......Page 400
12.7 Exercises......Page 409
12.8 Suggestions for Further Reading......Page 413
13.2 Nonlinear Functions of Sample Means......Page 414
Central order statistics......Page 418
Pairs of central order statistics......Page 422
Sample extremes......Page 426
13.4 U-Statistics......Page 429
13.5 Rank Statistics......Page 436
13.6 Exercises......Page 446
13.7 Suggestions for Further Reading......Page 448
14.2 Edgeworth Series Approximations......Page 449
Third- and higher-order approximations......Page 458
Expansions for quantiles......Page 459
14.3 Saddlepoint Approximations......Page 461
Integration of saddlepoint approximations......Page 465
14.4 Stochastic Asymptotic Expansions......Page 468
14.5 Approximation of Moments......Page 474
14.6 Exercises......Page 480
14.7 Suggestions for Further Reading......Page 483
A1.1 Introduction......Page 485
A1.2 A General Definition of Integration......Page 486
A1.3 Convergence Properties......Page 488
A1.4 Multiple Integrals......Page 489
A1.5 Calculation of the Integral......Page 490
A1.6 Fundamental Theorem of Calculus......Page 491
A1.7 Interchanging Integration and Differentiation......Page 492
A2.2 Complex Exponentials......Page 493
A2.3 Logarithms of Complex Numbers......Page 496
A3.1 Sets......Page 497
A3.2 Sequences and Series......Page 500
A3.3 Functions......Page 503
A3.4 Differentiation and Integration......Page 507
A3.5 Vector Spaces......Page 511
References......Page 517
Name Index......Page 521
Subject Index......Page 523