## Negative binomial Mixtures, construction of negative binomial mixtures and their properties

##### Abstract

The objective of this project was to construct Negative Binomial mixtures. We consider a class
of mixture distributions generated by randomizing the success parameter p and fixing
parameter r of a Negative Binomial distribution where we obtained a number of mixtures. We
parametized p to e-..1. and p to 1 - e-..1..
The mixing distributions used are Exponential, Gamma, Exponeniated Exponential, Beta
Exponential, Variate Gamma, Variate Exponential, Inverse Gaussian, and Lindely.
Some of the results were expressed in the explicit, expectations and recursive form. The explicit
involves using [(x) = fC+;-l) pr(1- PYg(p)dp where x = 0,1,2, .... and g(p) is the mixing
distribution. By using this method the Negative Binomial- Exponential mixture was obtained as
[( x ) = - itrr(rx+r)r(r+it-l) 0 her rni Id b b . d usi I···· . r+l1l+x-l )r'r t er mixtures cou not e 0 tame usmg exp iclt since integration
was not possible.
The Expectations method involved using the Laplace or method of moments where [(x) =
(r+;-l) L~=oG) (-l)k Lt(r + k) for x > 0; r > 0; andk = 0,1,2 ..... x. The mixtures obtained
using this method are NB- Exponential (x) = C+xX-l) L~-O- (Xk) (-l)k _r+..k1+..._1,. NB-Lindely
()
(}Z (r + x - 1) (X) k ((}+r+k+l) .
prob x = ((}+1) x L~=o k (-1) ((}+r+k)Z' NB- Inverse Gaussian
p(X = x) = (r + ~- 1)L~=o (Z) (-l)k exp (; [1-Jl- 2(r:;)ll
z
])
NB-Exponentiated exponential [(x) = (r+;-l) ex L~=oG) (-l)k B Cfl;+k, ex}
Gamma [(x) = (r+Xx-l) L~--O(Xk) (r+(-kl)+kl)a Beta Exponential
(b r+k: )
p(X = x) = (r + x - 1)LX _ (X) (-1)k B +-c ,a
X k-O k B(a,b)
, Variate Gamma [(x) = C+Xx-l) Inl(-b) [In (Tr++kk++ab) + _r+a_k+a - _r+bk_+]b L~=O(Xk)(_1)k, Variate
a
l (b+r+k)
Exponential [(x) = C+~-l) nl:m L~=oG) (_1)k.
NB - Inverse Gaussian distribution was also obtained using recursive relations as P; (x) =
-r-+x-l [Pr(x - 1) - --Prr+1 (x - 1)] . x r+x-l
iv
Geometric mixtures have been obtained by putting r = 1 in the Negative Binomial mixtures;
we came up with Geometric-exponential, Geometric- Gamma, Geometric-Beta Exponential,
Geometric-inverse Gaussian, and Geometric-Lindley mixtures.
Cases in which the parameters p is fixed and r is a random variable where it has a continuous
mixing distribution is considered, the probability generating function used is G (s) = L~=o PkSk
where Pk is a Negative Binomial mixture. The results obtained were: NB- Exponential
A [(l-QS)]-a G(s) = (1 qs) ,NB- Gamma G(s) = 1 + f3log - , NB-Beta Exponential G(s) =
A+log - P
p
(a-1) . (a-1) . c L~o j (-1)1 . . AaL~o j (-1)1 -( -) (1 qs) . ,NB- Exponentiated exponential G(s) = (1 qs) . , NB-Inverse B a.b log - +c(b+]) log - +Ar(1+])
p p
1
(
a¢+lo/-QS)2 .! ( 1- )
Gaussian G(s) = IT P e¢(2a)2. 2K_~.,j 2¢ (a¢ + log pqS) where Kr(w) is a Bessel
function of the third kind.

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