Scientists actively involved in theoretical and applied aspects of statistical science were invited to write articles for what has turned into a 38-chapter summary of the some of the recent developments in the field. They highlight new noteworthy results and illustrate their practical significance, and point out possible new directions to pursue. The general areas they cover are approximations, bounds, and inequalities; probability and stochastic processes; distributions, characterizations, and applications; time series, linear, and non- linear models; inference and applications; and applications to biology and medicine. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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" the book contains some chapters of interest for probabilists and theoretical and applied statisticians. Most of the articles end with a complete list of references of recent developments, which will make it easier for graduate students and researchers in finding articles of interest." -Short Book Reviews, Vol. 21, No. 2, August 2001
"deserves a place in the central reference section of any university mathematics library." - The Mathematical Gazette, March 1997
Preface v List of Contributors xix List of Tables xxv List of Figures xxvii Theophilos N. Cacoullos - A View of his Career xxxix Publications of Theophilos N. Cacoullos xxxii The Ten Commandments for a Statistician xxxviii Part I. Approximation, Bounds, and Inequalities Nonuniform Bounds in Probability Approximations Using Steins Method 3(12) Louis H. Y. Chen Introduction 3(1) Poisson Approximation 4(3) Binomial Approximation: Binary Expansion of a Random Integer 7(2) Normal Approximation 9(2) Conclusion 11(4) References 11(4) Probability Inequalities for Multivariate Distributions with Applications to Statistics 15(26) Joseph Glaz Introduction and Summary 15(2) Positive Dependence and Product-Type Inequalities 17(5) Negative Dependence and Product-Type Inequalities 22(1) Bonferroni-Type Inequalities 23(4) Applications 27(14) Sequential Analysis 27(1) A Discrete Scan Statistic 28(2) An Approximation for a Multinomial Distribution 30(1) A Conditional Discrete Scan Statistic 31(4) Simultaneous Prediction in Time Series Models 35(2) References 37(4) Applications of Compound Poisson Approximation 41(22) A. D. Barbour O. Chryssaphinou E. Vaggelatou Introduction 41(4) First Applications 45(5) Runs 45(3) Sequence Matching 48(2) Word Counts 50(8) Discussion and Numerical Examples 58(5) References 60(3) Compound Poisson Approximation for Sums of Dependent Random Variables 63(24) Michael V. Boutsikas Markos V. Koutras Introduction 63(3) Preliminaries and Notations 66(1) Main Results 67(9) Examples of Applications 76(11) A Compound Poisson Approximation for Truncated Moving Sum of i.i.d. r.v.s 76(4) The Number of Overlapping Success Runs in a Stationary Two-State Markov Chain 80(5) References 85(2) Unified Variance Bounds and a Stein-Type Identity 87(14) N. Papadatos V. Papathanasiou Introduction 87(2) Properties of the Transformation 89(5) Application to Variance Bounds 94(7) References 98(3) Probability Inequalities for U-Statistics 101(16) Tasos C. Christofides Introduction 101(2) Preliminaries 103(3) Probability Inequalities 106(11) References 112(5) Part II. Probability and Stochastic Processes Theory and Applications of Decoupling 117(30) Victor de la Pena T. L. Lai Complete Decoupling of Marginal Laws and One-Sided Bounds 118(5) Tangent Sequences and Conditionally Independent Variables 123(1) Basic Decoupling Inequalities for Tangent Sequences 124(4) Applications to Martingale Inequalities and Exponential Tail Probability Bounds 128(2) Decoupling of Multilinear Forms, U-Statistics and U-Processes 130(4) Total Decoupling of Stopping Times 134(5) Principle of Conditioning in Weak Convergence 139(2) Conclusion 141(6) References 142(5) A Note on the Probability of Rapid Extinction of Alleles in a Wright-Fisher Process 147(8) F. Papangelou Introduction 147(3) The Lower Bound for Boundary Sets 150(5) References 154(1) Stochastic Integral Functionals in an Asymptotic Split State Space 155(14) V. S. Korolyuk N. Limnios Introduction 155(1) Preliminaries 156(3) Phase Merging Scheme for Markov Jump Processes 159(1) Average of Stochastic Integral Functional 160(1) Diffusion Approximation of Stochastic Integral Functional 161(5) Single Splitting State Space 161(3) Double Split State Space 164(2) Integral Functional with Perturbed Kernel 166(3) References 167(2) Busy Periods for Some Queues with Deterministic Interarrival or Service Times 169(16) Claude Lefevre Philippe Picard Introduction 169(2) Preliminaries: A Basic Class of Polynomials 171(3) Construction of the Basic Polynomials 171(2) A Generalized Appell Structure 173(1) The Dg/M(Q)/1 Queue 174(5) Model and Notation 174(1) Exact Distribution of Nr 175(4) The M(Q)/Dg/1 Queue 179(6) Model and Notation 179(1) Exact Distribution of Nr 180(3) References 183(2) The Evolution of Population Structure of the Perturbed Non-Homogeneous Semi-Markov System 185(24) P.-C. G. Vassiliou H. Tsakiridou Introduction 185(2) The Perturbed Non-Homogeneous Semi-Markov System 187(3) The Expected Population Structure with Respect to the First Passage Time Probabilities 190(6) The Expected Population Structure with Respect to the Duration of a Membership in a State 196(3) The Expected Population Structure with Respect to the State Occupancy of a Membership 199(2) The Expected Population Structure with Respect to the Counting Transition Probabilities 201(8) References 203(6) Part III. Distributions, Characterizations, and Applications Characterizations of Some Exponential Families Based on Survival Distributions and Moments 209(16) M. Albassam C. R. Rao D. N. Shanbhag Introduction 209(2) An Auxiliary Lemma 211(1) Characterizations Based on Survival Distributions 212(6) Characterizations Based on Moments 218(7) References 222(3) Bivariate Distributions Compatible or Nearly Compatible with Given Conditional Information 225(14) B. C. Arnold E. Castillo J. M. Sarabia Introduction 225(1) Imprecise Specification 226(2) Precise Specification 228(5) An Example 233(6) References 237(2) A Characterization of a Distribution Arising from Absorption Sampling 239(8) Adrienne W. Kemp Introduction 239(3) The Characterization Theorem 242(2) An Application 244(3) References 245(2) Refinements of Inequalities for Symmetric Functions 247(6) Ingram Olkin References 250(3) General Occupancy Distributions 253(16) Ch. A. Charalambides Introduction 253(2) A General Random Occupancy Model 255(5) Special Occupancy Distributions 260(9) Geometric Probabilities 260(4) Bernoulli Probabilities 264(4) References 268(1) A Skew t Distribution 269(10) M. C. Jones Introduction 269(1) Derivation of Skew t Density 270(2) Properties of Skew t Distribution 272(2) A First Bivariate Skew t Distribution 274(1) A Second Bivariate Skew t Distribution 275(4) References 277(2) On the Posterior Moments for Truncation Parameter Distributions and Identifiability by Posterior Mean for Exponential Distribution with Location Parameters 279(14) Y. Ma N. Balakrishnan Introduction 279(2) Posterior Moments 281(5) Examples 286(1) Identifiability by Posterior Mean 287(2) An Illustrative Example 289(4) References 289(4) Distributions of Random Volumes without Using Integral Geometry Techniques 293(26) A. M. Mathai Introduction 294(5) Evaluation of Arbitrary Moments of the Random Volumes 299(20) Matrix-Variate Distributions for X 299(6) Type-1 Beta Distribution for X 305(2) The Case when the Rows of X are Independently Distributed 307(2) Type-1 Beta Distributed Independent Rows of X 309(1) Type-2 Beta Distributed Independent Rows of X 310(1) Independently Gaussian Distributed Points 311(1) Distributions of the r-Contents 312(4) References 316(3) Part IV. Time Series, Linear, and Non-Linear Models Cointegration of Economic Time Series 319(14) T. W. Anderson Introduction 319(1) Regression Models 320(1) Simultaneous Equation Models 321(2) Canonical Analysis and the Reduced Rank Regression Estimator 323(2) Autoregressive Processes 325(1) Nonstationary Models 326(1) Cointegrated Models 327(1) Asymptotic Distribution of Estimators and Test Criterion 328(5) References 331(2) On Some Power Properties of Goodness-of-Fit Tests in Time Series Analysis 333(16) Efstathios Paparoditis Testing Spectral Density Fits 333(4) Local Power Considerations 337(3) Comparison 340(9) References 348(1) Linear Constraints on a Linear Model 349(10) Somesh Das Gupta Introduction 349(2) Geometric Interpretation of the Role of the Linear Constraints 351(8) References 357(2) M-Methods in Generalized Nonlinear Models 359(20) Antonio I. Sanhueza Pranab K. Sen Introduction 359(2) Definitions and Assumptions 361(2) Asymptotic Results 363(8) Test of Significance and Computational Algorithm 371(8) Subhypothesis Testing 371(1) Nonlinear Hypothesis Testing 371(1) Computational Algorithm 372(1) References 373(6) Part V. Inference and Applications Extensions of a Variation of the Isoperimetric Problem 379(12) Herman Chernoff Introduction 379(1) Information Retrieval Problem 380(1) Information Retrieval without Measurement Error 381(1) Useful Information in a Variable 382(1) Allocation of Storage Space 383(1) The Isoperimetric Problem 383(1) Extensions 384(7) References 387(4) On Finding a Single Positive Unit in Group Testing 391(12) Milton Sobel Introduction 391(1) Description of Properties, Numerical Results 392(4) Some Formulas for Procedure RDH 396(2) The Greedy Procedure RG 398(1) Conclusions 398(1) Changing the Prior with Procedure RDH 399(1) Robustness of Procedure RDH for q Known 400(3) References 401(2) Testing Hypotheses on Variances in the Presence of Correlations 403(16) A. M. Mathai P. G. Moschopoulos Bivariate Normal Population 403(3) Modifying the Hypothesis 406(2) Nonnull Moments 408(5) Null Case 413(2) The Conditional Hypothesis 415(4) References 418(1) Estimating the Smallest Scale Parameter: Universal Domination Results 419(10) Stavros Kourouklis Introduction 419(1) Main Results 420(9) References 426(3) On Sensitivity of Exponential Rate of Convergence for the Maximum Likelihood Estimator 429(18) James C. Fu Introduction 429(2) Main Results 431(5) Some Applications 436(6) Exponential Model 437(1) Families of t-distributions with Location Parameter 438(4) Discussion 442(5) References 443(4) A Closer Look at Weighted Likelihood in the Context of Mixtures 447(22) Marianthi Markatou Introduction 447(2) Background 449(3) Simulation Experiments and Results 452(12) Normal Mixture with Equal Component Variance 454(4) Normal Mixtures with Unequal Component Variance 458(4) Other Models 462(2) Model Selection 464(1) Conclusions 465(4) References 465(4) On Nonparametric Function Estimation with Infinite-Order Flat-Top Kernels 469(16) Dimitris N. Politis Introduction: A General Family of Flat-Top Kernels of Infinite Order 469(3) Multivariate Density Estimation: A Review 472(3) Further Issues on Density Estimation 475(10) Case of Smooth Density over a Finite Domain 476(2) Case of Infinite Domain with Some Discontinuities 478(3) References 481(4) Multipolishing Large Two-Way Tables 485(18) Kaye Basford Stephan Morgenthaler John W. Tukey Introduction 485(1) Bilinear Multipolishers 486(3) Choosing the Auxiliary Variables Equal to the Effects of the Additive Fit 488(1) Robust Alternatives 489(1) Matrix Approximations 489(3) Approximations of Two-Way Tables 491(1) Displays 492(1) Example 493(2) Concluding Remarks 495(8) References 496(7) On Distances and Measures of Information: A Case of Diversity 503(14) Takis Papaioannou Introduction 503(2) Measuring Information - Measures of Information 505(3) Properties of Measures of Information 508(1) Measures of Information and Inference 509(2) Applications 511(1) Conclusions 512(5) References 512(5) Representation Formulae for Probabilities of Correct Classification 517(20) Wolf-Dieter Richter Introduction 517(2) Vector Algebraic Preliminaries 519(6) Distributional Results 525(12) Representation Formulae Based upon the Two-Dimensional Gaussian Law 525(6) Representation Formulae Based upon the Doubly Noncentral F-Distribution 531(3) References 534(3) Estimation of Cycling Effect on Reliability 537(12) Vilijandas Bagdonavicius Mikhail Nikulin Models 537(2) Semiparametric Estimation 539(10) The First Model 539(3) The Second Model 542(2) References 544(5) Part VI. Applications to Biology and Medicine A New Test for Treatment vs. Control in an Ordered 2 X 3 Contingency Table 549(16) Arthur Cohen H. B. Sackrowitz Introduction 549(2) New Test, Implementation and Example 551(2) Simulation Study 553(1) Theoretical Properties 554(11) Appendix 559(4) References 563(2) An Experimental Study of the Occurrence Times of Rare Species 565(8) Marcel F. Neuts Statement of the Problem 565(1) The Design of the Experiment 566(2) Stage 2 of the Experiment 568(1) Findings 569(4) References 571(2) A Distribution Functional Arising in Epidemic Control 573(10) Niels G. Becker Sergey Utev Introduction 573(1) Properties of the Functional 574(3) Proof of the Theorem 577(2) Application to Epidemic Control 579(4) References 581(2) Birth and Death Urn for Ternary Outcomes: Stochastic Processes Applied to Urn Models 583(18) A. Ivanova N. Flournoy Introduction 583(1) A Birth and Death Urn with Immigration for Ternary Outcomes 584(2) Embedding the Urn Scheme in a Continuous-Time Birth and Death Process 586(1) The Probability Generating Function for the Number of Success on Treatment i in the Continuous-Time Birth and Death Process 587(3) The Probability Generating Function for the Number of Trials on Treatment i in the Continuous-Time Birth and Death Process 590(1) The Number of Trials on Treatment i in the Continuous-Time Birth and Death Process 591(2) The Joint Probability Generating Function for the Number of Successes and the Number of Trials in the Continuous-Time Birth and Death Process 593(1) Adopting a Stopping Rule to Convert Continuous-Time Statistics to the Urn Design 594(2) Limiting Results for the Proportion of Trials on Treatment i 596(1) Limiting Results for the Proportion of Successes on Treatment i in the Urn 597(1) Asymptotic Inference Pertaining to the Success Probabilities 598(1) Concluding Remarks 599(2) References 600(1) Author Index 601(10) Subject Index 611
CH. A. Charalambides, M. V. Koutras, N. Balakrishnan
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