# [Rd] Beta binomial and Beta negative binomial

Joan Maspons j.maspons at creaf.uab.cat
Sat Mar 17 19:38:41 CET 2012

Hello,

El 16 de març de 2012 20:34, Christophe Dutang <dutangc at gmail.com> ha escrit:
> Hi,
>
> Please look at the distribution task view (http://cran.r-project.org/web/views/Distributions.html) and the package gamlss.dist.

Thanks for the tip. There are Beta binomial functions but they don't
have the number of trials parameter so I supose it's a Beta Bernoulli
distribution.

>
> Regards
>
> Christophe
>
> --
> Christophe Dutang
> Ph.D. student at ISFA, Lyon, France
> website: http://dutangc.free.fr
>
> Le 16 mars 2012 à 18:41, Joan Maspons a écrit :
>
>> Hi,
>> I need Beta binomial and Beta negative binomial functions ...
>>
>> Can I implement these new functions inside stats
>> package following the
>> same patterns as other probability distributions?
>>
>> Yours,
>> --
>> Joan Maspons

I have implemented a prototype of the beta negative binomial:

# mu<- a/(a+b)          [mean]
# sigma<- ab/((a+b)^2 (a+b+1))  [variance]
# Maxima: solve([mu= a/(a+b) , sigma= a*b/((a+b)^2 * (a+b+1))], [a,b]);
a<-  -(mu * sigma + mu^3 - mu^2) / sigma
b<- ((mu-1) * sigma + mu^3 - 2 * mu^2 + mu) / sigma
if (a <= 0 | b <= 0) return (NA)
return (data.frame(a,b))
}

#Rmpfr::pochMpfr()?
pochhammer<- function (x, n){
return (gamma(x+n)/gamma(x))
}

# PMF:
# P (X = x) = ((alpha)_n (n)_x (beta)_x)/(x! (alpha+beta)_n
(n+alpha+beta)_x) |  for  | x>=0
# (a)_b Pochhammer symbol
dbetanbinom<- function(x, size, mu, sigma){
if (is.na(sum(param))) return (NA) #invalid Beta parameters
if (length(which(x<0))) res<- 0
else
res<- (pochhammer(param\$a, size) * pochhammer(size, x) *
pochhammer(param\$b, x)
/ (factorial(x) * pochhammer(param\$a + param\$b, size)
* pochhammer(size + param\$a + param\$b, x)))
return (res)
}

curve(dbetanbinom(x, size=12, mu=0.75, sigma=.1), from=0, to=24, n=25, type="p")

# CDF:
# P (X<=x) = 1-(Gamma(n+floor(x)+1) beta(n+alpha, beta+floor(x)+1)
#            genhypergeo(1, n+floor(x)+1, beta+floor(x)+1;floor(x)+2,
n+alpha+beta+floor(x)+1;1))
#            /(Gamma(n) beta(alpha, beta) Gamma(floor(x)+2)) |  for  | x>=0
pbetanbinom<- function(q, size, mu, sigma){
require(hypergeo)
if (is.na(sum(param))) return (NA) #invalid Beta parameters
res<- numeric(length(q))
for (i in 1:length(q)){
if (q[i]<0) res[i]<- 0
else res[i]<- (1-(gamma(size+floor(q[i])+1) *
beta(size+param\$a, param\$b+floor(q[i])+1)
* genhypergeo(c(1, 1+size+floor(q[i]), 1+param\$b+floor(q[i])),
c(2+floor(q[i]),1+size+param\$a+param\$b+floor(q[i])), 1))
/ (beta(param\$a, param\$b) * gamma(size) * gamma(2+floor(q[i]))))
}
return (res)
}

## genhypergeo not converge. Increase iterations or tolerance?
pbetanbinom(0:10x, size=20, mu=0.75, sigma=0.03)

I have to investigate
http://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html
Any tip on how to solve the problem?

--
Joan Maspons
CREAF (Centre de Recerca Ecològica i Aplicacions Forestals)
Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalonia
Tel +34 93 581 2915            j.maspons at creaf.uab.cat
http://www.creaf.uab.cat