# AM_find_gamma_NegBin: Given that the prior on M is a Negative Binomial, find the... In AntMAN: Anthology of Mixture Analysis Tools

## Description

Once the prior on the number of mixture components M is assumed to be a Negative Binomial with parameter `r>0` and `0<p<1`, with mean is 1+ r*p/(1-p), this function adopts a bisection method to find the value of `gamma` such that the induced distribution on the number of clusters is centered around a user specifed value K^{*}, i.e. the function uses a bisection method to solve for γ \insertCiteargiento2019infinityAntMAN. The user can provide a lower γ_{l} and an upper γ_{u} bound for the possible values of γ. The default values are γ_l= 10^{-3} and γ_{u}=10. A defaault value for the tolerance is ε=0.1. Moreover, after a maximum number of iteration (default is 31), the function stops warning that convergence has not bee reached.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```AM_find_gamma_NegBin( n, r, p, Kstar = 6, gam_min = 0.001, gam_max = 10000, tolerance = 0.1 ) ```

## Arguments

 `n` The sample size. `r` The dispersion parameter `r` of the Negative Binomial. `p` The probability of failure parameter `p` of the Negative Binomial. `Kstar` The mean number of clusters the user wants to specify. `gam_min` The lower bound of the interval in which `gamma` should lie. `gam_max` The upper bound of the interval in which `gamma` should lie. `tolerance` Level of tolerance of the method.

## Value

A value of `gamma` such that E(K)=K^{*}

## Examples

 ```1 2 3 4 5 6``` ```n <- 82 r <- 1 p <- 0.8571 gam_nb= AM_find_gamma_NegBin(n,r,p,Kstar=6, gam_min=0.001,gam_max=10000, tolerance=0.1) prior_K_nb= AM_prior_K_NegBin(n,gam_nb, r, p) prior_K_nb%*%1:n ```

AntMAN documentation built on July 23, 2021, 5:08 p.m.