Abstract
We consider asymptotic coverage properties of one-sided posterior confidence intervals for discrete distributions, with a unidimensional parameter of interest and a nuisance parameter of arbitrary dimension. In this case, no higher order asymptotic expansion of the frequentist coverage for these intervals is established, unless some randomization is added. We study here the existence of such frequentist expansions and propose simple continuity corrections based on a uniform random vector. This helps in determining a family of matching priors for one sided intervals in the discrete case.
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References
Babu, J. G. and Singh, K. (1989). Note on Edgeworth expansion for lattice distributions, J. Multivariate Anal., 30, 27–33.
Bickel, P. and Ghosh, J. K. (1990). A decomposition for the likelihood ratio statistic and the Bartlett correction—A Bayesian argument, Ann. Statist., 18, 1070–1090.
Bhattacharya, R. and Rao, R. (1986). Normal Approximations and Asymptotic Expansions, 2nd ed., Wiley, New York.
Blyth, C. R. (1986). Approximate binomial confidence limits, J. Amer. Statist. Assoc., 81, 843–855.
Cox, D. R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference, J. Roy. Statist. Soc. Ser. B, 49, 1–39.
Davison, A. C. (1988). Approximate conditional inference in generalized linear models, J. Roy. Statist. Soc. Ser. B, 50, 445–461.
Ghosh, J. K. and Mukerjee, R. (1993). Frequentist validity of highest posterior density regions in the multiparameter case, Ann. Inst. Statist. Math., 45, 293–302.
Hall, P. (1982). Improving the normal approximation when constructing one-sided confidence intervals for binomial or Poisson parameters, Biometrika, 69(3), 647–652.
Kolassa, J. E. and McCullagh, P. (1990). Edgeworth series for lattice distributions, Ann. Statist., 18, 981–985.
Lehmann, E. L. (1986). Testing Statistical Hypotheses, Wiley, New York.
Mukerjee, R. and Dey, D. K. (1993). Frequentist validity of posterior quantiles in the presence of a nuisance parameter: Higher order asymptotics, Biometrika, 80, 499–505.
Mukerjee, R. and Ghosh, M. (1997). Second order probability matching priors, Biometrika (to appear).
Pólya, G. and Szegö, G. (1972). Problems and Theorems in Analysis, Vol. 1, Springer, New York.
Peers, W. H. (1965). On confidence points and Bayesian probability points in the case of several parameters, J. Roy. Statist. Soc. Ser. B, 27, 9–16.
Severini, T. A. (1993). Bayesian estimates which are also confidence intervals, J. Roy. Statist. Soc. Ser. B, 55, 533–540.
Tibshirani, R. (1989). Noninformative priors for one parameters of many, Biometrika, 76, 604–608.
Welch, B. L. and Peers, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihoods, J. Roy. Statist. Soc. Ser. B, 25, 318–329.
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Rousseau, J. Coverage Properties of One-Sided Intervals in the Discrete Case and Application to Matching Priors. Annals of the Institute of Statistical Mathematics 52, 28–42 (2000). https://doi.org/10.1023/A:1004128814284
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DOI: https://doi.org/10.1023/A:1004128814284