# etienne: Etienne's sampling formula In untb: Ecological Drift under the UNTB

## Description

Function `etienne()` returns the probability of a given dataset given `theta` and `m` according to the Etienne's sampling formula. Function `optimal.params()` returns the maximum likelihood estimates for `theta` and `m` using numerical optimization

## Usage

 ```1 2``` ```etienne(theta, m, D, log.kda = NULL, give.log = TRUE, give.like = TRUE) optimal.params(D, log.kda = NULL, start = NULL, give = FALSE, ...) ```

## Arguments

 `theta` Fundamental biodiversity parameter `m` Immigration probability `D` Dataset; a count object `log.kda` The KDA as defined in equation A11 of Etienne 2005. See details section `give.log` Boolean, with default `TRUE` meaning to return the logarithm of the value `give.like` Boolean, with default `TRUE` meaning to return the likelihood and `FALSE` meaning to return the probability `start` In function `optimal.params()`, the start point for the optimization routine (theta,m). `give` In function `optimal.params()`, Boolean, with `TRUE` meaning to return all output of the optimization routine, and default `FALSE` meaning to return just the point estimate `...` In function `optimal.params()`, further arguments passed to `optim()`

## Details

Function `etienne()` is just Etienne's formula 6:

omitted...see PDF

where \log K(D,A) is given by function `logkda()` (qv). It might be useful to know the (trivial) identity for the Pochhammer symbol [written (z)_n] documented in `theta.prob.Rd`. For convenience, Etienne's Function `optimal.params()` uses `optim()` to return the maximum likelihood estimate for theta and m.

Compare function `optimal.theta()`, which is restricted to no dispersal limitation, ie m=1.

Argument `log.kda` is optional: this is the K(D,A) as defined in equation A11 of Etienne 2005; it is computationally expensive to calculate. If it is supplied, the functions documented here will not have to calculate it from scratch: this can save a considerable amount of time

## Author(s)

Robin K. S. Hankin

## References

R. S. Etienne 2005. “A new sampling formula for biodiversity”. Ecology letters 8:253-260

`logkda`,`optimal.theta`
 ```1 2 3 4 5 6 7 8 9``` ```data(butterflies) ## Not run: optimal.params(butterflies) #takes too long without PARI/GP #Now the one from Etienne 2005, supplementary online info: zoo <- count(c(pigs=1, dogs=1, cats=2, frogs=3, bats=5, slugs=8)) l <- logkda.R(zoo, use.brob=TRUE) # Use logkda() if pari/gp is available optimal.params(zoo, log.kda=l) #compare his answer of 7.047958 and 0.22635923. ```