Gamma univariate

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Suppose where xi are iid observations from the gamma distribution with density


In the parameterization above represents shape while is a rate parameter. An alternative parameterization is given by , and in that case becomes a scale parameter. Another possible parameterization is , whereso that and are orthogonal (the Fisher information is diagonal).

In what follows, is the trigamma function.

Overall Recommended (OR) prior
In other parameterization
Recommended prior if is of main interest

Formal endorsements

The OR prior is the reference prior for several ordering of the parameters including and .

It is predictive matching of order as shown in [1]. For the ordering of parameters (so is of main interest), the recommended prior is the one in the last row of the table as it is the reference prior for this ordering.


Posterior propriety

The marginal posterior for under the OR prior and the Jeffreys prior is proper .[2]


Other properties

Under the posterior credible interval for has a frequentist coverage rate of O(n-1) [3], because it is a Tibshirani prior [4] .

can be obtained from by a change of variable.

Further properties can be found in [5] and in [1].


Alternatives

Jeffreys prior is:


References

  1. 1.0 1.1 Sun, D. and Ye, K. (1996). Frequentist validity of posterior quantiles for a two-parameter exponential family. Biometrika 83, 55-65.
  2. Liseo, B. (1993). Elimination of nuisance parameters with reference priors. Biometrika, 80, 295-304
  3. Moala, F. A.  Ramos P. L., Achcar, J.A. (2013). Bayesian inference for two-parameter gamma distribution assuming different noninformative priors. Revista Colombiana de Estadística 36, 321-338. 
  4. Tibshirani, R. (1989), ‘Noninformative priors for one parameters of many’, Biometrika 76, 604–608.
  5. Yang, R. and Berger, J.O. (1996). A catalog of noninformative priors. http://www.stats.org.uk/priors/noninformative/YangBerger1998.pdf