HodgesLehmann: Hodges-Lehmann (1952) Modification of Bayes Priors for...

In REBayes: Empirical Bayes Estimation and Inference

Hodges-Lehmann (1952) Modification of Bayes Priors for Compound Decisions Given a prior G find a modified prior f that bounds minimax risk.

Description

There are two variants both minimize Fisher information for location via conic optimization:

\min \sum \frac{(f_{i+1}-f_i )^2}{( f_{i+1}+f_i )/2} = \sum \frac{u_i^2}{v_i/2} \approx I(F)

\Leftrightarrow \quad \min \sum w_i \; \mbox{s.t.} \; u_i^2 \leq 2 v_i w_i

Huber Variant as proposed in Efron and Morris (1971) imposing constraint

f(x) = \alpha \Phi * G + (1-\alpha) h(x)

Mallows Variant as proposed in Bickel (1983) imposing constraints

f(x) = \alpha \Phi * G + (1-\alpha) h(x), \; h(x) = \Phi * H

N.B. When the grid is not equispaced, one would have to include grid spacings.

Usage

HodgesLehmann(grid, G, alpha, type = "Huber", ...)

Arguments

grid	grid on which to interpolate
G	initial prior
alpha	contamination proportion
type	either "Huber" or "Mallows"
...	other arguments to be passed to Mosek.

Value

An object of class density with solution

Author(s)

R. Koenker and J. Gu

References

Bickel, P. (1983), Minimax estimation of the mean of a normal distribution subject to doing well at a point, in M. H. Rizvi, J. S. Rustagi & D. Siegmund, eds, ‘Recent Advances in Statistics: Papers in Honor of Herman Chernoff on his Sixtieth Birthday’, Academic Press, pp. 511–528

Efron, B. & Morris, C. (1971), ‘Limiting the risk of Bayes and empirical Bayes estimators part I: the Bayes case’, Journal of the American Statistical Association 66, 807–815.

Hodges, J. L. & Lehmann, E. L. (1952), ‘The use of previous experience in reaching statistical decisions’, The Annals of Mathematical Statistics pp. 396–407.

Huber, P. (1964), ‘Robust estimation of a location parameter’, The Annals of Mathematical Statistics pp. 73–101.

Huber, P. (1974) "Fisher Information and Spline Interpolation." Ann. Statist. 2 (5) 1029 - 1033,

Mallows, C. (1978), ‘Problem 78-4, minimizing an integral’, SIAM Review 20, 183– 183.

See Also

HuberSpline

REBayes documentation built on Dec. 3, 2025, 5:06 p.m.