EbayesThresh (version 1.4-12)

wfromx: Find Empirical Bayes weight from data

Description

Suppose the vector $(x_1,\dots ,x_n)$ is such that $x_i$ is drawn independently from a normal distribution with mean ${\theta }_i$ and standard deviation $s_i$ (s_i equals 1 for Cauchy prior). The prior distribution of the ${\theta }_i$ is a mixture with probability $1-w$ of zero and probability $w$ of a given symmetric heavy-tailed distribution. This routine finds the marginal maximum likelihood estimate of the parameter $w$.

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.

Value

The numerical value of the estimated weight.

Details

The weight is found by marginal maximum likelihood.

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

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

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.

References

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

See Also

wandafromx, tfromx, tfromw, wfromt

Examples

# NOT RUN {
wfromx(x = rnorm(100, s = c(rep(0,90),rep(5,10))), prior = "cauchy")
# }