Identification of the Modes of Multinomial Probability Distributions with Application In Categorical Data Analysis
Keywords:
multinomial distribution, Dirichlet distribution, mode, modulo, sampling distributionAbstract
The mode of a probability distribution is defined as that point for which the probability density function is maximum. It is possible for a probability distribution to have more than one mode as in the case of multimodal distributions. This paper examines the location of the modes of multinomial distributions in two cases: when the probabilities pi of the multinomial distribution are equal and when they are not all equal. It is shown that the location of the modes depend on the congruence N modulo k in the case when pi = 1/k , i = 1,2,3,…k. If the probabilities can be expressed as pi = ai/bi for all i, then the location of the modes depend on the congruence Nai modulo bi. Two applications in statistical inference are provided which use the multivariate mode as basis rather than the univariate mean. Since the tests preserve the multivariate structure of the data, they are more meaningful than the traditional parametric tests employed in practice.
References
Adams, Fagot, & Robinson. (1965). A theory of appropriate statistics. Pm,B(2): 99-127.
A. Tversky & D. Kahneman: "The framing of decisions and the psychology of choice", Science, 211, 453-458 (1981).
Blackwell, David; MacQueen, James B. (1973). "Ferguson distributions via Polya urn schemes". Ann. Stat. 1 (2): 353–355.
Chapman, J.W. (1976). A comparison of the X2,-2 log R, and multinomial probability criteria for significance tests when expected frequencies are small. Journal of American Statistical Association, 71: 854-863.
Clark Glymour (1980), "Why I Am Not a Bayesian", Theory and Evidence, Princeton Univ Press, ISBN 0-691-07240-X (Chapter III)
Clayson, D. L. And Dormody, Thomas, J. (2007) ―Analyzing Data Measured by Individual Likert-Type Items‖( New Mexico State University)
Connor, Robert J.; Mosimann, James E (1969). "Concepts of Independence for Proportions with a Generalization of the Dirichlet Distribution". Journal of the American statistical association (American Statistical Association) 64
Cox, D. R. and Hinkley, D. V (1974). Theoretical Statistics. Chapman and Hall. ISBN 0041215370. page 432
Cox, D. R. and Hinkley, D. V (1974). Theoretical Statistics. Chapman and Hall. ISBN 0041215370. page 433)
Dawid, A.P. and Mortera, J. (1996) "Coherent analysis of forensic identification evidence". Journal of the Royal Statistical Society, Series B, 58,425–443.
Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles in Business" pg. 46, John Wiley & Sons, 2007
ET. Jaynes. Probability Theory: The Logic of Science Cambridge University Press, (2003). ISBN 0-521-59271-2
Evans, Merran; Hastings, Nicholas; Peacock, Brian (2000). Statistical Distributions. New York: Wiley. pp. 134–136. ISBN 0-471-37124-6. 3rd ed
Foreman, L.A; Smith, A.F.M. and Evett, I.W. (1997). "Bayesian analysis of deoxyribonucleic acid profiling data in forensic identification applications (with discussion)". Journal of the Royal Statistical Society, Series A, 160, 429-469.
Gardner-Medwin, A. (2005) "What probability should the jury address?". Significance, 2 (1), March 2005
H.M. Finucan [1964], The mode of a Multinomial Distribution. Biometrica 51, 513-517.
Gelman and J. B. Carlin and H. S. Stern and D. B. Rubin (2003). Bayesian Data Analysis (2nd ed.). pp. 582. ISBN 1-58488-388-X.
Hwang, J. T. and Casella, George (1982). "Minimax confidence sets for the mean of a multivariate normal distribution". Annals of Statistics 10: 868–881.
Jaynes, E.T. (2003) Probability Theory: The Logic of Science, Cambridge University Press. ISBN 0-521-59271-2 (Chapter 5)
JM. Bernardo (2005), "Reference analysis", Handbook of statistics, 25, 17–90
Jonathan J. Koehler. (1993). "The influence of prior beliefs on scientific judgments of evidence quality". Organizational Behavior and Human Decision Processes. 56, 28–55.
N.L. Johnson [1997], S. Kotz and N. Balkrishnan, Discrete Multivariate Distributions. J. Wiley 1997.
Kiefer, J. and Schwartz, R. (1965). "Admissible Bayes character of T2-, R2 -, and other fully invariant tests for multivariate normal problems". Annals of Mathematical Statistics 36: 747–770..
Kosko, Bart (1992). Neural Networks and Fuzzy Systems: a dynamical systems approach to machine intelligence (1st ed.). Englewood Cliffs, NJ, USA.: Prentice Hall. p. 265. ISBN 9780136114352.
Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. SpringerVerlag. (From "Chapter 12 Posterior Distributions and Bayes Solutions", page 324)
Lehmann, Erich (1986). Testing Statistical Hypotheses (Second ed.). (see page 309 of Chapter 6.7 "Admissibilty", and pages 17–18 of Chapter 1.8 "Complete Classes"
Robertson, B. and Vignaux, G.A. (1995) Interpreting Evidence: Evaluating Forensic Science in the Courtroom. John Wiley and Sons. Chichester.
Schwartz, R. (1969). "Invariant proper Bayes tests for exponential families". Annals of Mathematical Statistics 40: 270–283..
Stephen M. Stigler (1986) The history of statistics. Harvard University press. Chapter 3.
Stephen. E. Fienberg, (2006) "When did Bayesian Inference become "Bayesian"? Bayesian Analysis, 1 (1), 1–40. See page 5.
Wolpert, RL. (2004) A conversation with James O. Berger, Statistical science, 9, 205–218
W. Feller [1968], An Introduction to Probability Theory and its Applications. J. Wiley 1950, 1957 and 1968
Downloads
Published
Issue
Section
License
Copyright (c) 2013 The Treshhold
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
