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

by Iain M. Johnstone and Bernard W. Silverman

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The unpublished 1998 manuscript Empirical Bayes approaches to mixture problems and wavelet regression by Iain M. Johnstone and Bernard W. Silverman contains some early work of the authors on this topic but has been almost completely superseded by the current work.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.