具体描述
Lattice path combinatorics has developed greatly as a branch of probability studies recently, and the need for new books on the subject is obvious. The present monograph, by one who has made significant contributions to combinatorics and its applications to probability and statistics, will be useful to research workers, teachers, professional statisticians, and advanced students alike. It treats several recent results and it offers a powerful new tool for studying many problems in mathematical statistics.
The emphasis in the five chapters is on ‘dominance.’ From a consideration of exceedances in the lattice path problem, the text goes on to provide solutions to tests of hypotheses and simple sampling plans, displaying the usefulness of Young chains in the enumeration of the latter. The fourth chapter, on knock-out tournaments, represents one approach to paired comparisons quite close in spirit to dominance and lattice path combinatorics, and the final chapter considers the advantages of using combinatorial methods in statistical problems (including the
Frame-Robinson—Thrall theorem to derive properties of non--parametric tests) and mentions current trends of research. Numerous examples, exercises and references round out the text.
作者简介
T. V. NARAYANA is a member of the Department of Mathematics at the University of Alberta.
目录信息
l
LATTICE PATH PROBLEMS AND VECTORS 0F INTEGERS
1. Representation of subsets of {1, ..., N}
2. A refinement of the Chung-Feller Theorem
3. Lattice paths and the ballot theorem
4. Repeated reflections and applications
Exercises
References
II
THE DOMINANCE THEOREM AND SMIRNOV TEST-STATISTICS
l. A theorem on domination and its geometrical interpretation
2. Discussion of the dominance theorem and some special cases
3. Duality and application to Smirnov statistics
Exercises
References
III
SOME APPLICATIONS OF DOMINANCE T0 STATISTICAL PROBLEMS
1. The role of dominance in combinatorial problems
2. Dominance tests for Lehmann alternatives
Exercises
References
IV
THE COMBINATORICS OF KNOCK-OUT TOURNAMENTS
1. The classical case
2. Random knock-out tournaments
3. A comparison of tournaments
Exercises
References
V
A MISCELLANY OF FURTHER RESEARCH PROBLEMS
1. A comparison of selection procedures
2. The numbers {n choose r}{n choose r-1}/n
3. Weak inadmissibility of tests
Exercise
References
APPENDIX
On some convolution identities from lattice path combinatorics
References
NOTES AND SOLUTIONS
SUPPLEMENTARY BIBLIOGRAPHY
INDEX
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