# Unconditional NPML estimator for the SPECIES number

### Description

This function calculate the unconditional NPML estimator of the species number by Norris and Pollock 1996, 1998. This estimator was obtained from the full likelihood based on a Poisson mixture model. The confidence interval is calculated based on a bootstrap procedure.

### Usage

1 |

### Arguments

`n` |
a matrix or a numerical data frame of two columns. It is also called the “frequency of frequencies” data in literature. The first column is the frequency |

`t` |
a positive integer. |

`C` |
integer either 0 or 1. It specifies whether bootstrap confidence interval should be calculated. “ |

`method` |
string either “N-P” or “W-L”(default). If |

`b` |
integer. |

`conf` |
a positive number |

`seed` |
a single value, interpreted as an integer. Seed for random number generation |

`dis` |
0 or 1. 1 for on-screen display of the mixture output, and 0 for none. |

### Details

The computing is intensive if `method`

=“N-P” is used particularly when extrapolation is large.
It may takes hours to compute the bootstrap confidence interval. If `method`

=“W-L” is used, computing usually
is much much faster. Estimates from both methods are often identical.

### Value

The function `unpmle`

returns a list of: `Nhat`

, `CI`

(if “`C`

=1”)

`Nhat` |
point estimate of N |

`CI` |
bootstrap confidence interval. |

### Note

The unconditional NPML estimator is unstable from either `method='N-P'`

or `method='W-L'`

. Extremely large estimates may occur.
This is also reflected in that the upper confidence bound often greatly vary from different runs of bootstrap procedure. In contrast the penalized NPMLE by `pnpmle`

function is much more stable.

### Author(s)

Ji-Ping Wang, Department of Statistics, Northwestern University

### References

Norris, J. L. I., and Pollock, K. H. (1996), Nonparametric MLE Under Two Closed Capture-Recapture Models With Heterogeneity, Biometrics, 52,639-649.

Norris, J. L. I., and Pollock, K. H.(1998), Non-Parametric MLE for Poisson Species Abundance Models Allowing for Heterogeneity Between Species, Environmental and Ecological Statistics, 5, 391-402.

Bonhing, D. and Schon, D., (2005), Nonparametric maximum likelihood estimation of population size based on the counting distribution, Journal of the Royal Statistical Society, Series C: Applied Statistics, 54, 721-737.

Wang, J.-P. Z. and Lindsay, B. G. ,(2005), A penalized nonparametric maximum likelihood approach to species richness estimation. Journal of American Statistical Association, 2005,100(471):942-959

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
library(SPECIES)
##load data from the package,
## "butterfly" is the famous butterfly data by Fisher 1943.
data(butterfly)
##output estimate without confidence interval using cutoff t=15
#unpmle(butterfly,t=15,C=0)
##output estimate with confidence interval using cutoff t=15
#unpmle(butterfly,t=15,C=1,b=200)
``` |