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Preface | xi | |
Chapter 1. | Properties of Univariate Unimodal Distributions | |
1.0 | Summary | 1 |
1.1 | A General Definition of Unimodality on the Line | 1 |
1.2 | The Convex Structure of the Set of Unimodal Distributions | 4 |
1.3 | Convolution Properties of Unimodal Distributions | 11 |
1.4 | Strong Unimodality | 17 |
1.5 | The Gauss Inequality and the Three-Sigma Rule for Unimodal Distributions | 23 |
1.6 | The Mean-Median-Mode Inequality | 33 |
Chapter 2. | Concepts of Multivariate Unimodality | |
2.0 | Summary | 37 |
2.1 | Notation | 37 |
2.2 | Some Definitions of Unimodality for Distributions on R[superscript n] | 38 |
2.3 | Logconcave Distributions | 46 |
2.4 | Preservation of Unimodality Properties under Weak Limits | 53 |
2.5 | The Choquet Version of Central Convex Unimodality | 54 |
2.6 | Interrelationships among the Definitions | 57 |
2.7 | Marginal Distributions | 61 |
2.8 | Convolutions | 63 |
Chapter 3. | Some More Notions of Unimodality | |
3.0 | Summary | 67 |
3.1 | Schur Concavity and Related Concepts of Unimodality | 67 |
3.2 | Generalized Unimodality Indexed by a Positive Parameter | 72 |
3.3 | Classes of Concave Densities and Measures | 84 |
Chapter 4. | Unimodality for Discrete Distributions | |
4.0 | Summary | 101 |
4.1 | The Structure of Unimodal Discrete Distributions | 101 |
4.2 | Convolutions of Discrete Unimodal Distributions and Strong Unimodality | 108 |
4.3 | Unimodality of High Convolutions | 112 |
Chapter 5. | Unimodality of Infinitely Divisible Distributions | |
5.0 | Summary | 117 |
5.1 | Infinitely Divisible Distributions | 117 |
5.2 | Unimodality of Symmetric Infinitely Divisible Distributions | 122 |
5.3 | The Unimodality of all Distributions in L | 128 |
5.4 | The Unimodality of Multivariate Infinitely Divisible Distributions | 140 |
Chapter 6. | Unimodality and Notions of Dependence | |
6.0 | Summary | 147 |
6.1 | Preliminaries on Notions of Dependence | 147 |
6.2 | Monotonicity Properties of Probabilities Rectangular Regions | 151 |
Chapter 7. | Ordering of Distributions by Peakedness | |
7.0 | Summary | 159 |
7.1 | Basic Results on Peakedness Ordering | 159 |
7.2 | Peakedness Comparisons for the Multivariate Normal and Other Suitably Unimodal Distributions | 165 |
7.3 | The Use of Unimodality and Peakedness Comparisons in Determining the Recurrence of Symmetric Random Walks | 173 |
Chapter 8. | Applications of Unimodality in Statistical Inference | |
8.0 | Summary | 177 |
8.1 | Unimodality of the Likelihood Function | 177 |
8.2 | Estimation of a Mode | 187 |
8.3 | Monotonicity of the Power Functions of Certain Multivariate Tests | 200 |
8.4 | Shortest Confidence Intervals and Minimum Volume Confidence Regions | 208 |
Chapter 9. | Convexity in Reliability Theory | |
9.0 | Summary | 215 |
9.1 | Some Orderings for Distributions of Nonnegative Random Variables | 215 |
9.2 | Univariate IFR and IFRA Classes | 224 |
9.3 | Multivariate IFR and IFRA Classes | 233 |
9.4 | Unimodality and Other Convexity Properties | 243 |
Appendix | ||
A.1 | Convex Sets in R[superscript n] | 249 |
A.2 | Convergence of Probability Measures | 252 |
A.3 | Convex Sets of Probability Measures | 254 |
A.4 | The Brunn-Minkowski Inequality | 259 |
References | 261 | |
Author Index | 271 | |
Subject Index | 275 |
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Add Unimodality, Convexity, and Applications, In this book, the basic notions and tools of unimodality as they relate to probability and statistics are presented. In addition, many applications are covered; these include the use of unimodality to obtain monotonicity properties of power functions of m, Unimodality, Convexity, and Applications to the inventory that you are selling on WonderClubX
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Add Unimodality, Convexity, and Applications, In this book, the basic notions and tools of unimodality as they relate to probability and statistics are presented. In addition, many applications are covered; these include the use of unimodality to obtain monotonicity properties of power functions of m, Unimodality, Convexity, and Applications to your collection on WonderClub |