Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-08T09:56:17.557Z Has data issue: false hasContentIssue false
  • English
  • Français

An Asymptotic Likelihood Ratio Test for a Markov Chain with Bernoullian or Contingent Input

Published online by Cambridge University Press:  01 January 2025

Richard S. Bogartz
Richard S. Bogartz*
Affiliation:
University of Illinois
Get access Rights & Permissions [Opens in a new window]

Abstract

A Markov chain with transition probabilities pij(θ) that are functions of a parameter vector θ is defined. One of t input values is delivered to the chain on each trial. Under the hypothesis H0 the parameter vector is independent of the input; under the hypothesis H1 the vector is in general different for each different input. A likelihood ratio test for a single observation on a chain of great length is given for testing H0 against H1, given that the distribution of the inputs depends at most on the previous input and the present state of the chain. The test is therefore one of stationarity of the transition probabilities against a specific alternative form of nonstationarity. Application of the approach to a statistical test of lumpability of the states of a chain is indicated. Tests for other related hypotheses are suggested. Application of the test to “within-subject effects” for individual subjects is considered. Finally, some applications of the results in psychophysical contexts are suggested.

Keywords

Markov ChainLikelihood RatioPublic PolicyLikelihood Ratio TestPresent State
Type
Original Paper
Information
Psychometrika , Volume 33 , Issue 4 , December 1968 , pp. 405 - 422
Copyright
Copyright © 1968 Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Part of the final version of this paper was written at Stanford University during work sessions of a 1967 Summer Conference on Mathematical Models for Perception and Learning. Support was provided by the National Science Foundation through the Committee on Mathematics in the Social Sciences of the Institute for Advanced Study in the Behavioral Sciences.

References

Anderson, T. W., and Goodman, L. A. Statistical inference about Markov chains. Annals of Mathematical Statistics, 1957, 28, 89–110.CrossRefGoogle Scholar
Bartlett, M. S. The frequency goodness of fit test for probability chains. Proceedings of the Cambridge Philosophical Society, 1951, 47, 86–95.CrossRefGoogle Scholar
Billingsley, P. Statistical methods in Markov chains. Annals of Mathematical Statistics, 1961, 32, 12–40.CrossRefGoogle Scholar
Bogartz, R. S. Test of a theory of predictive behavior in young children. Psychonomic Science, 1966, 4, 433–434.CrossRefGoogle Scholar
Bogartz, R. S. Extension of a theory of predictive behavior to immediate recall by preschool children. In Riegel, Klaus F. (Ed.) The development of language functions. Technical Report No. 8, 1966, Language Development Program, University of Michigan, 1–14. (b).Google Scholar
Bogartz, R. S. Extension of a theory of predictive behavior in young children to the effects of intertrial interval duration. Psychonomic Science, 1967, 8, 521–522.CrossRefGoogle Scholar
Dixon, W. J., and Mood, A. M. A method for obtaining and analyzing sensitivity data. Journal of the American Statistical Association, 1948, 43, 109–126.CrossRefGoogle Scholar
Feller, W. An introduction to probability theory and its applications. Vol. I, New York: Wiley, 1957.Google Scholar
Fiacco, A. V., and McCormick, G. P. The sequential unconstrained minimization technique for nonlinear programming, a primal-dual method. Management Science, 1964, 10(2), 360–366.CrossRefGoogle Scholar
Gold, R. Z. Tests auxiliary to chi square tests in a Markov chain. Annals of Mathematical Statistics, 1963, 34, 56–74.CrossRefGoogle Scholar
Kemeny, J. G., and Snell, J. L. Finite Markov chains, Princeton: Van Nostrand, 1960.Google Scholar
Smith, J. E. K. Stimulus programming in psychophysics. Psychometrika, 1961, 26, 27–33.CrossRefGoogle Scholar
Suppes, P., and Atkinson, R. C. Markov learning models for multiperson interactions, Stanford: Stanford Univ. Press, 1960.Google Scholar
Wolfe, P. Foundations of nonlinear programming: Note on linear programming and extensions—Part 65. Memorandum RM-4669-PR, August, 1965, The Rand Corporation.Google Scholar