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Four Bootstrap Confidence Intervals for the Binomial-Error Model

Published online by Cambridge University Press:  01 January 2025

Miao-Hsiang Lin  and
Chao A. Hsiung
Miao-Hsiang Lin*
Affiliation:
Institute of Statistical Science, Academia Sinica
Chao A. Hsiung
Affiliation:
Institute of Statistical Science, Academia Sinica
*
Requests for reprints should be sent to Miao-Hsiang Lin, Institute of Statistical Science, Academia Sinica, Taipei, 11529, Taiwan, R.O.C.
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Abstract

Confidence intervals for the mean function of the true proportion score (μζ|x), where ζ and x respectively denote the true proportion and observed test scores, can be approximated by the Efron, Bayesian, and parametric empirical Bayes (PEB) bootstrap procedures. The similarity of results yielded by all the bootstrap methods suggests the following: the unidentifiability problem of the prior distribution g(ζ) can be bypassed with respect to the construction of confidence intervals for the mean function, and a beta distribution for g(ζ) is a reasonable assumption for the test scores in compliance with a negative hypergeometric distribution. The PEB bootstrap, which reflects the construction of Morris intervals, is introduced for computing predictive confidence bands for ζ|x. It is noted that the effect of test reliability on the precision of interval estimates varies with the two types of confidence statements concerned.

Keywords

binomial-error modelEfron's bootstrapsBayesian bootstrapparametric empirical Bayes bootstrap
Type
Original Paper
Information
Psychometrika , Volume 57 , Issue 4 , December 1992 , pp. 499 - 520
Copyright
Copyright © 1992 The Psychometric Society

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Footnotes

The Authors are indebted to the Editor and anonymous reviewers for constructive suggestions and comments. The authors wish to thank Min-Te Chao and Cheng-Der Fuh for some useful suggestions at earlier stages of writing this paper.

References

Bickel, P. J., Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. The Annals of Statistics, 9, 1196–1217.CrossRefGoogle Scholar
Deely, J. J., Lindley, D. V. (1981). Bayes empirical Bayes. Journal of the American Statistical Association, 76, 833–841.CrossRefGoogle Scholar
Efron, B. (1982). The jacknife, bootstrap, and other resampling plans, Philadelphia: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. The Annals of Statistics, 1, 209–230.CrossRefGoogle Scholar
Keats, J. A., Lord, F. M. (1962). A theoretical distribution for mental test scores. Psychometrika, 27, 59–72.CrossRefGoogle Scholar
Laird, N. M., Louis, T. A. (1987). Empirical Bayes confidence intervals based on bootstrap samples. Journal of the American Statistical Association, 82, 739–754.CrossRefGoogle Scholar
Lo, A. Y. (1987). A large sample study of the Bayesian bootstrap. The Annals of Statistics, 15, 360–375.CrossRefGoogle Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Lord, F. M. (1976). An interval estimate for making statistical inferences about true scores. Psychometrika, 41, 79–87.CrossRefGoogle Scholar
Morris, C. N. (1983). Parametric empirical Bayes inference: Theory and application. Journal of the American Statistical Association, 78, 47–59.CrossRefGoogle Scholar
Morris, C. N. (1983). Parametric empirical Bayes confidence intervals in scientific inference, data analysis, and robustness, New York: Academic Press.Google Scholar
Rao, C. R. (1973). Linear statistical inference and applications, New York: John Wiley.CrossRefGoogle Scholar
Rubin, D. B. (1981). The Bayesian bootstrap. The Annals of Statistics, 9, 130–134.CrossRefGoogle Scholar
Rubin, D. B. (1982). Estimation in parallel randomized experiments. Journal of Educational Statistics, 6, 377–400.CrossRefGoogle Scholar
Wilcox, R. R. (1981). A review of the beta-binomial model and its extensions. Journal of Educational Statistics, 6, 3–32.CrossRefGoogle Scholar