Lagrangian Probablility Distributions

By: Prem C. Consul, Felix Famoye, Samuel Kotz (Foreword by)

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Hardcover | 1 December 2005

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Lagrangian expansions can be used to obtain numerous useful probability models, which have been applied to real life situations including, but not limited to: branching processes, queuing processes, stochastic processes, environmental toxicology, diffusion of information, ecology, strikes in industries, sales of new products, and production targets for optimum profits. This book presents a comprehensive, systematic treatment of the class of Lagrangian probability distributions, along with some of its families, their properties, and important applications.

Key features:

* Fills a gap in book literature

* Examines many new Lagrangian probability distributions, their numerous families, general and specific properties, and applications to a variety of different fields

* Presents background mathematical and statistical formulas for easy reference

* Detailed bibliography and index

* Exercises in many chapters

Graduate students and researchers with a good knowledge of standard statistical techniques and an interest in Lagrangian probability distributions will find this work valuable. It may be used as a reference text or in courses and seminars on Distribution Theory and Lagrangian Distributions. Applied scientists and researchers in environmental statistics, reliability, sales management, epidemiology, operations research, optimization in manufacturing and marketing, and infectious disease control will benefit immensely from the various applications in the book.

Industry Reviews

From the reviews:

"The book is the first comprehensive presentation of the theory of distributions which are derived and/or closely connected with Lagrange expansions. It gives an overview of the research on this topic published during the last thirty-five years. The text is illustrated with many surprising examples. The book can be recommended for both applied and theoretical statistics." -Mathematical Reviews

"The purpose of this book is to collect most of the research materials in the various journals during the last thirtyfive years and to give a reasonable and systematic account of the class of Lagrangian distributions and some of their properties and applications. ... it is meant for graduate students and researchers who have knowledge of standard statistical techniques. Every chapter ... are supplied with exercises. The list of references contains 252 positions. The volume of the book is over 350 pages." (Nijole Kalinauskaite, Zentralblatt MATH, Vol. 1101 (3), 2007)

"The book acquaints researchers and graduate students with good mathematical, probabilistic and statistical knowledge, with the present-day stage of the theory of distributions derived from and/or closely connected with Lagrange expansions. ... The authors are leading persons in the field. They collected most of the results published in various journals during the last forty years and their work can be considered a textbook for advanced courses and seminars in theoretical and/or applied statistics." (Gejza Wimmer, Applications of Mathematics, Vol. 56 (6), 2011)

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You Can Find This Book In

Non-Fiction Mathematics Probability & Statistics Optimisation Applied Mathematics Economics Econometrics Economic Statistics Medicine Medicine in General Public Health & Preventive Medicine Epidemiology & Medical Statistics

Algebra

Foreword	p. vii
Preface	p. ix
List of Tables	p. xviii
Abbreviations	p. xix
Preliminary Information	p. 1
Introduction	p. 1
Mathematical Symbols and Results	p. 1
Combinatorial and Factorial Symbols	p. 1
Gamma and Beta Functions	p. 4
Difference and Differential Calculus	p. 5
Stirling Numbers	p. 7
Hypergeometric Functions	p. 9
Lagrange Expansions	p. 10
Abel and Gould Series	p. 13
Faa di Bruno's Formula	p. 13
Probabilistic and Statistical Results	p. 14
Probabilities and Random Variables	p. 14
Expected Values	p. 15
Moments and Moment Generating Functions	p. 16
Cumulants and Cumulant Generating Functions	p. 18
Probability Generating Functions	p. 18
Inference	p. 19
Lagrangian Probability Distributions	p. 21
Introduction	p. 21
Lagrangian Probability Distributions	p. 22
Equivalence of the Two Classes of Lagrangian Distributions	p. 30
Moments of Lagrangian Distributions	p. 33
Applications of the Results on Mean and Variance	p. 36
Convolution Property for Lagrangian Distributions	p. 36
Probabilistic Structure of Lagrangian Distributions L (f; g; x)	p. 39
Modified Power Series Distributions	p. 41
Modified Power Series Based on Lagrange Expansions	p. 42
MPSD as a Subclass of Lagrangian Distributions L (f; g; x)	p. 42
Mean and Variance of a MPSD	p. 45
Maximum Entropy Characterization of some MPSDs	p. 46
Exercises	p. 48
Properties of General Lagrangian Distributions	p. 51
Introduction	p. 51
Central Moments of Lagrangian Distribution L (f; g; x)	p. 51
Central Moments of Lagrangian Distribution L[subscript 1] (f[subscript 1]; g; y)	p. 56
Cumulants of Lagrangian Distribution L[subscript 1] (f[subscript 1]; g; y)	p. 60
Applications	p. 61
Relations between the Two Classes L (f; g; x) and L[subscript 1] (f[subscript 1]; g; y)	p. 62
Some Limit Theorems for Lagrangian Distributions	p. 66
Exercises	p. 67
Quasi-Probability Models	p. 69
Introduction	p. 69
Quasi-Binomial Distribution I (QBD-I)	p. 70
QBD-I as a True Probability Distribution	p. 70
Mean and Variance of QBD-I	p. 71
Negative Moments of QBD-I	p. 73
QBD-I Model Based on Difference-Differential Equations	p. 75
Maximum Likelihood Estimation	p. 77
Quasi-Hypergeometric Distribution I	p. 80
Quasi-Polya Distribution I	p. 81
Quasi-Binomial Distribution II	p. 82
QBD-II as a True Probability Model	p. 83
Mean and Variance of QBD-II	p. 83
Some Other Properties of QBD-II	p. 85
Quasi-Hypergeometric Distribution II	p. 85
Quasi-Polya Distribution II (QPD-II)	p. 86
Special and Limiting Cases	p. 87
Mean and Variance of QPD-II	p. 88
Estimation of Parameters of QPD-II	p. 88
Gould Series Distributions	p. 89
Abel Series Distributions	p. 90
Exercises	p. 90
Some Urn Models	p. 93
Introduction	p. 93
A Generalized Stochastic Urn Model	p. 94
Some Interrelations among Probabilities	p. 100
Recurrence Relation for Moments	p. 101
Some Applications of Prem Model	p. 103
Urn Model with Predetermined Strategy for Quasi-Binomial Distribution I	p. 104
Sampling without Replacement from Urn B	p. 104
Polya-type Sampling from Urn B	p. 105
Urn Model with Predetermined Strategy for Quasi-Polya Distribution II	p. 105
Sampling with Replacement from Urn D	p. 106
Sampling without Replacement from Urn D	p. 107
Urn Model with Inverse Sampling	p. 107
Exercises	p. 108
Development of Models and Applications	p. 109
Introduction	p. 109
Branching Process	p. 109
Queuing Process	p. 111
G[vertical bar]D[vertical bar]1 Queue	p. 112
M[vertical bar]G[vertical bar]1 Queue	p. 113
Stochastic Model of Epidemics	p. 115
Enumeration of Trees	p. 116
Cascade Process	p. 117
Exercises	p. 118
Modified Power Series Distributions	p. 121
Introduction	p. 121
Generating Functions	p. 122
Moments, Cumulants, and Recurrence Relations	p. 122
Other Interesting Properties	p. 125
Estimation	p. 128
Maximum Likelihood Estimation of [theta]	p. 128
Minimum Variance Unbiased Estimation	p. 130
Interval Estimation	p. 132
Some Characterizations	p. 132
Related Distributions	p. 136
Inflated MPSD	p. 136
Left Truncated MPSD	p. 137
Exercises	p. 140
Some Basic Lagrangian Distributions	p. 143
Introduction	p. 143
Geeta Distribution	p. 143
Definition	p. 143
Generating Functions	p. 144
Moments and Recurrence Relations	p. 145
Other Interesting Properties	p. 145
Physical Models Leading to Geeta Distribution	p. 146
Estimation	p. 148
Some Applications	p. 150
Consul Distribution	p. 151
Definition	p. 151
Generating Functions	p. 152
Moments and Recurrence Relations	p. 152
Other Interesting Properties	p. 153
Estimation	p. 154
Some Applications	p. 157
Borel Distribution	p. 158
Definition	p. 158
Generating Functions	p. 158
Moments and Recurrence Relations	p. 159
Other Interesting Properties	p. 159
Estimation	p. 160
Weighted Basic Lagrangian Distributions	p. 161
Exercises	p. 162
Generalized Poisson Distribution	p. 165
Introduction and Definition	p. 165
Generating Functions	p. 166
Moments, Cumulants, and Recurrence Relations	p. 167
Physical Models Leading to GPD	p. 167
Other Interesting Properties	p. 170
Estimation	p. 170
Point Estimation	p. 170
Interval Estimation	p. 173
Confidence Regions	p. 175
Statistical Testing	p. 175
Test about Parameters	p. 175
Chi-Square Test	p. 176
Empirical Distribution Function Test	p. 177
Characterizations	p. 177
Applications	p. 179
Truncated Generalized Poisson Distribution	p. 180
Restricted Generalized Poisson Distribution	p. 182
Introduction and Definition	p. 182
Estimation	p. 182
Hypothesis Testing	p. 185
Other Related Distributions	p. 186
Compound and Weighted GPD	p. 186
Differences of Two GP Variates	p. 187
Absolute Difference of Two GP Variates	p. 188
Distribution of Order Statistics when Sample Size Is a GP Variate	p. 188
The Normal and Inverse Gaussian Distributions	p. 189
Exercises	p. 189
Generalized Negative Binomial Distribution	p. 191
Introduction and Definition	p. 191
Generating Functions	p. 192
Moments, Cumulants, and Recurrence Relations	p. 192
Physical Models Leading to GNBD	p. 194
Other Interesting Properties	p. 197
Estimation	p. 200
Point Estimation	p. 201
Interval Estimation	p. 204
Statistical Testing	p. 205
Characterizations	p. 207
Applications	p. 215
Truncated Generalized Negative Binomial Distribution	p. 217
Other Related Distributions	p. 218
Poisson-Type Approximation	p. 218
Generalized Logarithmic Series Distribution-Type Limit	p. 218
Differences of Two GNB Variates	p. 219
Weighted Generalized Negative Binomial Distribution	p. 220
Exercises	p. 220
Generalized Logarithmic Series Distribution	p. 223
Introduction and Definition	p. 223
Generating Functions	p. 224
Moments, Cumulants, and Recurrence Relations	p. 225
Other Interesting Properties	p. 226
Estimation	p. 229
Point Estimation	p. 229
Interval Estimation	p. 233
Statistical Testing	p. 233
Characterizations	p. 234
Applications	p. 236
Related Distributions	p. 236
Exercises	p. 238
Lagrangian Katz Distribution	p. 241
Introduction and Definition	p. 241
Generating Functions	p. 241
Moments, Cumulants, and Recurrence Relations	p. 242
Other Important Properties	p. 244
Estimation	p. 245
Applications	p. 248
Related Distributions	p. 248
Basic LKD of Type I	p. 248
Basic LKD of Type II	p. 249
Basic GLKD of Type II	p. 250
Exercises	p. 251
Random Walks and Jump Models	p. 253
Introduction	p. 253
Simplest Random Walk with Absorbing Barrier at the Origin	p. 254
Gambler's Ruin Random Walk	p. 254
Generating Function of Ruin Probabilities in a Polynomial Random Walk	p. 255
Trinomial Random Walks	p. 259
Quadrinomial Random Walks	p. 260
Binomial Random Walk (Jumps) Model	p. 262
Polynomial Random Jumps Model	p. 263
General Random Jumps Model	p. 264
Applications	p. 265
Exercises	p. 266
Bivariate Lagrangian Distributions	p. 269
Definitions and Generating Functions	p. 269
Cumulants of Bivariate Lagrangian Distributions	p. 271
Bivariate Modified Power Series Distributions	p. 272
Introduction	p. 272
Moments of BMPSD	p. 273
Properties of BMPSD	p. 275
Estimation of BMPSD	p. 276
Some Bivariate Lagrangian Delta Distributions	p. 280
Bivariate Lagrangian Poisson Distribution	p. 281
Introduction	p. 281
Moments and Properties	p. 282
Special BLPD	p. 283
Other Bivariate Lagrangian Distributions	p. 283
Bivariate Lagrangian Binomial Distribution	p. 283
Bivariate Lagrangian Negative Binomial Distribution	p. 284
Bivariate Lagrangian Logarithmic Series Distribution	p. 286
Bivariate Lagrangian Borel-Tanner Distribution	p. 287
Bivariate Inverse Trinomial Distribution	p. 288
Bivariate Quasi-Binomial Distribution	p. 289
Exercises	p. 290
Multivariate Lagrangian Distributions	p. 293
Introduction	p. 293
Notation and Multivariate Lagrangian Distributions	p. 293
Means and Variance-Covariance	p. 297
Multivariate Lagrangian Distributions (Special Form)	p. 299
Multivariate Lagrangian Poisson Distribution	p. 299
Multivariate Lagrangian Negative Binomial Distribution	p. 300
Multivariate Lagrangian Logarithmic Series Distribution	p. 300
Multivariate Lagrangian Delta Distributions	p. 301
Multivariate Modified Power Series Distributions	p. 302
Multivariate Lagrangian Poisson Distribution	p. 303
Multivariate Lagrangian Negative Binomial Distribution	p. 304
Multivariate Lagrangian Logarithmic Series Distribution	p. 304
Moments of the General Multivariate MPSD	p. 304
Moments of Multivariate Lagrangian Poisson Distribution	p. 305
Moments of Multivariate Lagrangian Negative Binomial Distribution	p. 307
Multivariate MPSDs in Another Form	p. 308
Multivariate Lagrangian Quasi-Polya Distribution	p. 312
Applications of Multivariate Lagrangian Distributions	p. 312
Queuing Processes	p. 313
Random Branching Processes with k Types of Females	p. 314
Exercises	p. 315
Computer Generation of Lagrangian Variables	p. 317
Introduction and Generation Procedures	p. 317
Inversion Method	p. 317
Alias Method	p. 318
Basic Lagrangian Random Variables	p. 319
Simple Delta Lagrangian Random Variables	p. 321
Generalized Poisson Random Variables	p. 324
Generalized Negative Binomial Random Variables	p. 326
Generalized Logarithmic Series Random Variables	p. 330
Some Quasi-Type Random Variables	p. 334
References	p. 337
Index	p. 347
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Lagrangian Probablility Distributions

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