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 θ_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 θ_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 √{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

### See Also

`tfromw`, `threshld`, `wmonfromx`, `wfromx`

[Package EbayesThresh version 1.4-12 Index]