# wfromx: Find Empirical Bayes weight from data In EbayesThresh: Empirical Bayes Thresholding and Related Methods

## Description

Suppose the vector (x_1, …, x_n) is such that x_i is drawn independently from a normal distribution with mean θ_i and variance 1. 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

 `1` ```wfromx(x, prior = "laplace", a = 0.5) ```

## Arguments

 `x` vector of data `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.

## Details

The weight is found by marginal maximum likelihood. The search is over weights corresponding to thresholds in the range [0, √{2 \log n}], where n is the length of the data vector.

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`.

## Value

The numerical value of the estimated weight.

## Author(s)

Bernard Silverman

## References

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

`wandafromx`, `tfromx`, `tfromw`, `wfromt`
 `1` ```wfromx(x=rnorm(100, c( rep(0,90), rep(5,10))), prior="cauchy") ```