wfromx {EbayesThresh}R Documentation

Find Empirical Bayes weight from data

Description

Suppose the vector (x1,,xn)(x_1, \ldots, x_n) is such that xix_i is drawn independently from a normal distribution with mean θi\theta_i and standard deviation sis_i (s_i equals 1 for Cauchy prior). The prior distribution of the θi\theta_i is a mixture with probability 1w1-w of zero and probability ww of a given symmetric heavy-tailed distribution. This routine finds the marginal maximum likelihood estimate of the parameter ww.

Usage

wfromx(x, s = 1, prior = "laplace", a = 0.5, universalthresh = TRUE)

Arguments

x

Vector of data.

s

A single value or a vector of standard deviations if the Laplace prior is used. If a vector, must have the same length as x. Ignored if Cauchy prior is used.

prior

Specification of prior to be used; can be "cauchy" or "laplace".

a

Scale factor if Laplace prior is used. Ignored if Cauchy prior is used.

universalthresh

If universalthresh = TRUE, the thresholds will be upper bounded by universal threshold; otherwise, the thresholds can take any non-negative values.

Details

The weight is found by marginal maximum likelihood.

The search is over weights corresponding to threshold tit_i in the range [0,si2logn][0, s_i \sqrt{2 \log n}] if universalthresh=TRUE, where nn is the length of the data vector and (s1,...,sn)(s_1, ... , s_n) (s_i is 1 for Cauchy prior) is the vector of sampling standard deviation of data (x1,...,xn)(x_1, ... , x_n); otherwise, the search is over [0,1][0, 1].

The search is by binary search for a solution to the equation S(w)=0S(w)=0, where SS is the derivative of the log likelihood. The binary search is on a logarithmic scale in ww.

If the Laplace prior is used, the scale parameter is fixed at the value given for a, and defaults to 0.5 if no value is provided. To estimate a as well as w by marginal maximum likelihood, use the routine wandafromx.

Value

The numerical value of the estimated weight.

Author(s)

Bernard Silverman

References

See ebayesthresh and http://www.bernardsilverman.com

See Also

wandafromx, tfromx, tfromw, wfromt

Examples

wfromx(x = rnorm(100, s = c(rep(0,90),rep(5,10))), prior = "cauchy")

[Package EbayesThresh version 1.4-12 Index]