PEkit (version 1.0.0.1000)

MLEp: Maximum Likelihood Estimate of \(\psi\)

Description

Numerically searches for the MLE of \(\psi\) given an abundance vector with a binary search algorithm.

Usage

MLEp(abund)

Arguments

abund

An abundance vector.

Value

The MLE of \(\psi\).

Details

Numerically searches for the MLE of \(\psi\) as the root of equation $$K=\sum_{i=1}^n\psi/(\psi+i-1),$$ where \(K\) is the observed number of different species in the sample. The right side of the equation is monotonically increasing when \(\psi>0\), so a binary search is used to find the root. An accepted \(\psi\) sets value of the right side of the equation within R's smallest possible value of the actual value of \(K\).

References

W.J. Ewens, The sampling theory of selectively neutral alleles, Theoretical Population Biology, Volume 3, Issue 1, 1972, Pages 87-112, ISSN 0040-5809, <10.1016/0040-5809(72)90035-4>.

Examples

# NOT RUN {
##Find the MLE of psi of the vector (1,2,2).
##The frequencies of the frequencies of the data vector are given as input:
MLEp(abundance(c(1,2,2)))

##Find the MLE of psi of a sample from the Poisson-Dirichlet distribution:
set.seed(1000)
x<-rPD(n=10000, psi=100)
MLEp(abundance(x))
# }