Abstract
The problem of estimating the binomial sample size N from k observed numbers of successes is examined from a likelihood point of view. The direct use of the likelihood function for inference about N is illustrated when p is known, and the problem of inference is considered when p is unknown, and has to be eliminated in some way from the likelihood. Different methods (Bayesian, integrated likelihood, conditional likelihood, profile likelihood) for eliminating the nuisance parameter are found to lead to very different likelihoods in N in an example. This occurs because of a strong ridge in the two-parameter likelihood in N and p. Integrating out the parameter p is found to be unsatisfactory, but reparameterization of the model shows that the inference about N is almost unaffected by the new nuisance parameter. The resulting likelihood in N corresponds closely to the profile likelihood in the original parameterization.
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References
Aitkin, M. (1986). Statistical modelling: the likelihood approach. The Statistician 35, 103–113.
Carroll, R.J. and Lombard, F. (1985). Note on N estimators for the binomial distribution. J. Amer. Statist Assoc. 80, 423–426.
Casella, G. (1986). Stabilizing binomial n estimators. J. Amer. Statist. Assoc. 81, 172–175.
Cox, D.R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference (with Discussion). J. Roy. Statist Soc. B 49, 1–39.
Draper, N. and Guttman, I. (1971). Bayesian estimation of the binomial parameter. Technometrics 13, 667–673.
Edwards, A.W.F. (1972). Likelihood. Cambridge University Press.
Hinde, J.P. and Aitkin, M. (1987). Canonical likelihoods: a new likelihood treatment of nuisance parameters. Biometrika 74, 45–58.
Kahn, W.D. (1987). A cautionary note for Bayesian estimation of the binomial parameter n. Amer. Statist. 41, 38–39.
Kalbfleisch, J.D. and Sprott, D.A. (1970). Application of likelihood methods to models involving large numbers of parameters (with Discussion). J. Roy. Statist. Soc B 32, 175–208.
Olkin, I., Petkau, A.J. and Zidek, J.V. (1981). A comparison of n estimators for the binomial distribution. J. Amer. Statist. Assoc., 76, 637–642.
Raftery, A.E. (1988). Inference for the binomial N parameter: a hierarchical Bayes approach. Biometrika 75, 223–228.
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© 1989 Springer-Verlag New York, Inc.
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Aitkin, M., Stasinopoulos, M. (1989). Likelihood Analysis of a Binomial Sample Size Problem. In: Gleser, L.J., Perlman, M.D., Press, S.J., Sampson, A.R. (eds) Contributions to Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3678-8_28
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DOI: https://doi.org/10.1007/978-1-4612-3678-8_28
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