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Correlation in a Bayesian framework. (English) Zbl 0966.62012

Summary: The authors consider the correlation between two arbitrary functions of the data and a parameter when the parameter is regarded as a random variable with given prior distribution. They show how to compute such a correlation and use closed form expressions to assess the dependence between parameters and various classical or robust estimators thereof, as well as between \(p\)-values and posterior probabilities of the null hypothesis in the one-sided testing problem. Other applications involve the Dirichlet process and stationary Gaussian processes. Using this approach, the authors also derive a general nonparametric upper bound on Bayes risk.

MSC:

62F15 Bayesian inference
62M99 Inference from stochastic processes
62H20 Measures of association (correlation, canonical correlation, etc.)
62C10 Bayesian problems; characterization of Bayes procedures
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References:

[1] Anderson, An Introduction to Multivariate Statistical Analysis (1984) · Zbl 0651.62041
[2] Basu, On statistics independent of a complete sufficient statistic, Sankhyā 15 pp 377– (1988)
[3] Beran, Statistics for Long Memory Processes (1994) · Zbl 0869.60045
[4] Berger, Testing a point null hypothesis: the irreconcilability of P values and evidence (with discussion), Journal of the American Statistical Association 82 pp 112– (1987) · Zbl 0612.62022
[5] Bickel, Minimax estimation of the mean of a normal distribution when the parameter space is restricted, The Annals of Statistics 9 pp 1301– (1981) · Zbl 0484.62013
[6] Borovkov, Estimates for averaged quadratic risk, Probability and Mathematical Statistics 1 pp 185– (1980) · Zbl 0507.62024
[7] Brown, Admissible estimators, recurrent diffusions, and insoluble boundary value problems, The Annals of Mathematical Statistics 42 pp 855– (1971) · Zbl 0246.62016
[8] Brown, Information inequalities for the Bayes risk, The Annals of Statistics 18 pp 1578– (1990)
[9] Casella, Reconciling Bayesian and frequentist evidence in the one-sided testing problem, Journal of the American Statistical Association 82 pp 106– (1987) · Zbl 0612.62021
[10] David, Order Statistics (1981)
[11] M. Delampady, A. DasGupta, G. Casella, H. Rubin & W. E. Strawderman (2000). A new approach to default priors and robust Bayes methodology. Submitted for publication. · Zbl 1005.62021
[12] Ferguson, Prior distributions on spaces of probability measures, The Annals of Statistics 2 pp 615– (1974) · Zbl 0286.62008
[13] Grenander, Statistical Analysis of Stationary Time Series (1957)
[14] Huber, Robust Statistics (1981)
[15] Karlin, The theory of decision procedures for distributions with monotone likelihood ratio, The Annals of Mathematical Statistics 27 pp 272– (1956) · Zbl 0070.37203
[16] Lehmann, Theory of Point Estimation (1983) · Zbl 0522.62020 · doi:10.1007/978-1-4757-2769-2
[17] Levit, Second-order asymptotic minimaxity, Teoriya Veroyatnosteñ ee Primeneniya 25 pp 561– (1980)
[18] Oh, Comparison of the P- value and posterior probability, Journal of Statistical Planning and Inference 76 pp 93– (1999) · Zbl 0930.62025
[19] Rao, Linear Statistical Inference and its Applications (1973)
[20] C. Zang & A. DasGupta (1998). Bayesian inference under long range dependence. Submitted for publication.
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