User:BeyondNormality/Bhattacharya-negative binomial distribution

Bhattacharya-negative binomial

Notation	$\mathrm {BhattacharyaNB} ()$
Parameters	${\displaystyle }$
Support	${\displaystyle }$
PMF	${\displaystyle }$
CDF	${\displaystyle }$
Mean	${\displaystyle }$
Median	${\displaystyle }$
Mode	${\displaystyle }$
Variance	${\displaystyle }$
Skewness	${\displaystyle }$
Excess kurtosis	${\displaystyle }$
Entropy	${\displaystyle }$
MGF	${\displaystyle }$
CF	${\displaystyle }$
PGF	${\displaystyle }$

Also known as the generalized negative binomial, the probability mass function of the Bhattacharya-negative binomial distribution is given by

${\begin{aligned}\mathrm {BhattacharyaNB} (x;a,b,k)\equiv \Pr(X=x)&=\left(b^{a}(1+b)^{-a+k}(2+b)^{-k-x}\right)\sum _{j=0}^{\infty }\left({\frac {1}{2+b}}\right)^{j}{\binom {-1+a+j}{j}}{\binom {-1+j+k+x}{x}}\\&=b^{a}(b+1)^{k-a}(b+2)^{-k-x}{\binom {k+x-1}{x}}\,_{2}F_{1}\left(a,k+x;k;{\frac {1}{b+2}}\right)\end{aligned}}$

• $x=0,1,2,\dots$
• $a,k\geq 0$
• $b>0$

Expected Value

$\mathbb {E} (X)={\frac {a+bk}{b^{2}+b}}$

Variance

$\operatorname {Var} (X)={\frac {a(b(b+3)+1)+(b+2)b^{2}k}{b^{2}(b+1)^{2}}}$

Recurrence relation

$-\Pr(x-1)(a+b(k+2x-2)+k+3x-3)+(b+1)(b+2)x\Pr(x)+(k+x-2)\Pr(x-2)=0$

Moment Generating Function

$M_{X}(t)=\left(1-{\frac {e^{t}-1}{b}}\right)^{-a}\left(1-{\frac {e^{t}-1}{b+1}}\right)^{a-k}$

Characteristic Function

$\varphi _{X}(t)=\left(1-{\frac {e^{it}-1}{b}}\right)^{-a}\left(1-{\frac {e^{it}-1}{b+1}}\right)^{a-k}$

Probability Generating Function

$G(t)=\left(1-{\frac {t-1}{b}}\right)^{-a}\left(1-{\frac {t-1}{b+1}}\right)^{a-k}$

Interrelations

Symbol	Meaning
$\sim$	$X\sim Y$: the random variable X is distributed as the random variable Y
$\equiv$	the distribution in the title is identical with this distribution
$\Leftarrow$	the distribution in title is a special case of this distribution
$\Rightarrow$	this distribution is a special case of the distribution in the title
$\leftarrow$	this distribution converges to the distribution in the title
$\rightarrow$	the distribution in the title converges to this distribution

Relationship	Distribution	When
$\equiv$	negative binomial $\left(k+j,{\frac {b+1}{b+2}}\right){\underset {j}{\bigwedge }}$ negative binomial $\left(a,{\frac {b}{b+1}}\right)$
$\equiv$	negative binomial $\left(k-a,{\frac {b+1}{b+2}}\right)*\quad$ ${\displaystyle }$ negative binomial $\left(a,{\frac {b}{b+1}}\right)$	$k\geq a$
$\equiv$	Poisson $\left(\theta \right){\underset {\theta }{\bigwedge }}$ generalized exponential $f(y)$	$f(y)={\frac {b^{a}{(b+1)}^{k-a}}{\Gamma (k)}}y^{k-1}e^{-(b-1)y}\,_{1}F_{1}(a;k;y)\qquad y\in (0,\infty )\qquad k>0$
$\Leftarrow$	Kemp's binomial convolution $(u_{1},u_{2},Q_{1},Q_{2})$	$u_{1}=a-k\qquad u_{2}=-a\qquad Q_{1}={\frac {1}{b+1}}\qquad Q_{2}={\frac {1}{b+1}}$
$\Rightarrow$	deterministic (0)	$a=k=0$
$\Rightarrow$	2-shifted Flory	${\begin{cases}a=b=0\qquad k=2\\a=2\qquad b=1\qquad k=2\\\end{cases}}$
$\Rightarrow$	geometric $\left({\frac {1}{b+2}}\right)$	$a=0\qquad k=1$
$\Rightarrow$	geometric $\left({\frac {1}{b+1}}\right)$	$a=k=1$
$\Rightarrow$	negative binomial $\left(k,{\frac {b+1}{b+2}}\right)$	$a=0$
$\Rightarrow$	negative binomial $\left(k,{\frac {b}{b+1}}\right)$	$a=k$
$\Rightarrow$	Poisson-Lindley $(b)$	$a=2\qquad k=1$
$\rightarrow$	Poisson $\left(\theta \right)$	$a=0\qquad k\rightarrow \infty \qquad b\rightarrow \infty \qquad {\frac {k}{b+1}}\rightarrow \theta$