**Risk bounds for Empirical Bayes estimates of
sparse sequences, with applications to wavelet smoothing**

Software to implement the
methods of the papers is written in the S-PLUS language. Mostly the software will also work in R but
it hasn’t been fully tested. The
routines for single sequences use the standard parts of the language. Those for
wavelet transforms make use of the S+Wavelets module.

The main work of the
simulations is carried out using the authors' EbayesThresh
package but some supplementary routines are used.

- Description of S-PLUS
routines (.pdf)
- Routines for
carrying out simulations reported in “Needles and straw in
haystacks: Empirical Bayes
estimates of possibly sparse sequences”. The main routine is simulation1full, and
the routines ebayesstable1 and ebayesstable2 produce the summaries
presented in the paper. The main
routine returns an array giving the mean square error of every realization
considered. (Other error norms can
also be calculated.)
- The results
obtained by running simulation1full.
Save this file and use source to load it into SPlus or R. There are two objects. The mean square errors are in the array
ebayess.res and the mean absolute errors in the array ebayess1.res.

- S-PLUS
functions for
calculation of estimates, both for single sequences and for wavelet
transforms
- S-PLUS
functions carrying out the main simulations described in the paper
- additional functions
to carry out simulations for the FDR method
- additional functions
to carry out simulations for the method of Efromovich (JASA 1998)
- the full simulation
results themselves, to 6 significant figures. Use the routines
`table2()`and`table3()`to provide the summaries given in the paper. The S-PLUS objects are as follows:

sim1.r results of simulation1()

sim1a.r results of simulation1a()

sim2.r results of simulation2()

sim2dec.r results of simulation2(nondecimated=F)

sim3.r results of simulation3()

sim4.r results of fdrsimulation()

sim4dec.r results of fdrsimulation(nondecimated=F)

· The NMR data and
results as a table, some plots of the data, and
some S-PLUS commands used to carry out their
analysis.

The unpublished 1998 manuscript Empirical Bayes
approaches to mixture problems and wavelet regression by Iain M. Johnstone
and Bernard W. Silverman contains some early work of the authors on this topic
but has been almost completely superseded by the current work. An Splus implementation
of the methods described is available here. The programs
use the WaveThresh
package.

This page constructed by
Bernard Silverman.