R: Estimation of a parameter in the prior weight sequence in the...

zetafromx {EbayesThresh}

R Documentation

Estimation of a parameter in the prior weight sequence in the EbayesThresh paradigm.

Description

Suppose a sequence of data has underlying mean vector with elements \theta_i. Given the sequence of data, and a vector of scale factors cs and a lower limit pilo, this routine finds the marginal maximum likelihood estimate of the parameter zeta such that the prior probability of \theta_i being nonzero is of the form median(pilo, zeta*cs, 1).

Usage

zetafromx(xd, cs, pilo = NA, prior = "laplace", a = 0.5)

Arguments

`xd`	A vector of data.
`cs`	A vector of scale factors, of the same length as `xd`.
`pilo`	The lower limit for the estimated weights. If `pilo=NA` it is calculated according to the sample size to be the weight corresponding to the universal threshold `\sqrt{2 \log n}`.
`prior`	Specification of prior to be used conditional on the mean being nonzero; can be `cauchy` or `laplace`.
`a`	Scale factor if Laplace prior is used. Ignored if Cauchy prior is used. If, on entry, `a=NA` and `prior="laplace"`, then the scale parameter will also be estimated by marginal maximum likelihood. If `a` is not specified then the default value 0.5 will be used.

Details

An exact algorithm is used, based on splitting the range up for zeta into subintervals over which no element of zeta*cs crosses either pilo or 1.

Within each of these subintervals, the log likelihood is concave and its maximum can be found to arbitrary accuracy; first the derivatives at each end of the interval are checked to see if there is an internal maximum at all, and if there is this can be found by a binary search for a zero of the derivative.

Finally, the maximum of all the local maxima over these subintervals is found.

Value

A list with the following elements:

`zeta`	The value of `zeta` that yields the marginal maximum likelihood.
`w`	The weights (prior probabilities of nonzero) yielded by this value of `zeta`.
`cs`	The factors as supplied to the program.
`pilo`	The lower bound on the weight, either as supplied or as calculated internally.

Note

Once the maximizing zeta and corresponding weights have been found, the thresholds can be found using the program tfromw, and these can be used to process the original data using the routine threshld.

Author(s)

Bernard Silverman

References

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