Paper presentation at 28th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, July 2008

1. THE EMPIRICAL BAYES
ESTIMATOR AND MIXED
DISTRIBUTIONS
Nestor Ruben Barraza
Facultad de Ingenier´a
ı
Universidad de Buenos Aires
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.1/12

2. Introduction
n samples, r successful
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.2/12

3. Introduction
n samples, r successful
Maximum likelihood
ˆ= r
θ
n
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.2/12

4. Introduction
n samples, r successful
Maximum likelihood
ˆ= r
θ
n
Smoothing
ˆ= r+a
θ
n+b
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.2/12

5. Estimators
Laplace Succession Law
Lidstone Law
Good-Turing
Discount
Katz
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6. Urn Model
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7. The Empirical Bayes Estimator
General formulation using Mixed Distributions
s
ˆ i=1 θP (r/θ, i, n)p(i)dU (θ/i)
θ = E[θ/r, n] = s
i=1 P (r/θ, i, n)p(i)dU (θ/i)
ˆ θP (r/θ, n)dS(θ)
θ=
P (r/θ, n)dS(θ)
where:
s
dS(θ) = p(i)dU (θ/i)
i=1
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8. Mixed Binomial Models
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9. Mixed Poisson Models
λr
ˆr,n = 1 λ r! exp−λ dS(λ)
θ λr
n r! exp−λ dS(λ)
ˆr,n = r + 1 P (r + 1)
θ
n P (r)
Smoothing
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10. Mixed Poisson
Known estimators
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11. Mixed Poisson
Inverse Gaussian Mod. Extr. Value Weibull
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12. Cross-Entropy
1 p(r + 1)
Hp (r) = − r log2 (r + 1) + r log2 + log2 n
n p(r)
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13. Conclusions
A new general formulation of the Empirical
Bayes Estimator has been presented
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.11/12

14. Conclusions
A new general formulation of the Empirical
Bayes Estimator has been presented
It allows adding some information about the
events behavior through the mixing
distribution. The general formulation allows
working with any mixed discrete probability
function
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.11/12

15. Conclusions
A new general formulation of the Empirical
Bayes Estimator has been presented
It allows adding some information about the
events behavior through the mixing
distribution. The general formulation allows
working with any mixed discrete probability
function
A new increase or discount correction factor
has also been introduced. This factor
depends on the mixing distribution
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.11/12

16. Conclusions
Examples with some well known mixing
distributions used in Queuing Theory and
Reliability were displayed
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17. Conclusions
Examples with some well known mixing
distributions used in Queuing Theory and
Reliability were displayed
An interesting change in concavity can be
seen for the Inverse Gaussian Mixing
distribution
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.12/12

18. Conclusions
Examples with some well known mixing
distributions used in Queuing Theory and
Reliability were displayed
An interesting change in concavity can be
seen for the Inverse Gaussian Mixing
distribution
Applications for real data will be shown in a
future work.
THE EMPIRICAL BAYES ESTIMATOR AND MIXED DISTRIBUTIONS– p.12/12