User:BeyondNormality/Beall-Rescia distribution

Beall-Rescia

Notation	$\mathrm {BeallRescia} (a,b,n)$
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	${\displaystyle }$

• $a,b,n\geq 0$

Moment Generating Function

$M_{X}(t)=\exp \left(a\left(ne^{b\left(e^{t}-1\right)}\left(b\left(e^{t}-1\right)\right)^{-n}\left(\Gamma (n)-\Gamma \left(n,b\left(-1+e^{t}\right)\right)\right)-1\right)\right)$

Characteristic Function

$\varphi _{X}(t)=\exp \left(a\left(ne^{b\left(e^{it}-1\right)}\left(b\left(e^{it}-1\right)\right)^{-n}\left(\Gamma (n)-\Gamma \left(n,b\left(-1+e^{it}\right)\right)\right)-1\right)\right)$

Probability Generating Function

${\begin{aligned}G(t)&=\exp(-a)\exp \left(a\Gamma (n+1)\sum _{j=0}^{\infty }{\frac {b^{j}(-1+t)^{j}}{\Gamma (1+j+n)}}\right)\\&=exp(-a)\,_{0}F_{0}\left(a\,_{1}F_{1}(1;n+1;b(t-1))\right)\\&={\frac {\,_{0}F_{0}\left(a\,_{1}F_{1}(1;n+1;b(t-1))\right)}{\,_{0}F_{0}(a)}}\\&=\exp \left(ane^{b(t-1)}(b(t-1))^{-n}(\Gamma (n)-\Gamma (n,b(t-1)))-a\right)\end{aligned}}$

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
$\equiv$	Kemp-Dacey-hypergeometric $\vee$ Kemp-hypergeometric
$\equiv$	Poisson $(a)\vee$ {[ binomial $(n,p){\underset {n}{\wedge }}$ Poisson $(b)$] ${\underset {p}{\wedge }}$ beta $(1,n)$}	$n>0$
$\equiv$	Poisson $(a)\vee$ [ Poisson $(bp){\underset {p}{\wedge }}$ beta $(1,n)$)	$n>0\qquad \Leftarrow$ generalized Poisson family $[a,H(.)]$
$\Leftarrow$	Gurland (type 1) $\left(a,b,\alpha ,n\right)$	$\alpha =1$
$\leftarrow$	Gurland (type 3) $\left(b,k,P,\alpha ,n\right)$	$\alpha =1\qquad k\rightarrow \infty \qquad P\rightarrow \infty \qquad kP\rightarrow a$
$\leftarrow$	multiple Poisson $\left(n',a_{i}\right)$	$n'\rightarrow \infty \qquad {\begin{aligned}a_{i}&=a\Gamma (n+1)\sum _{j=0}^{\infty }{\binom {i+j}{j}}{\frac {b^{i+j}(-1)^{j}}{\Gamma (1+i+j+n)}}\\&=a\Gamma (n+1){\frac {b^{i}}{\Gamma (i+n+1)}}\,_{1}F_{1}(i+1;i+n+1;-b)\end{aligned}}$
$\Rightarrow$	Cernuschi-Castagnetto-Poisson $\left(a',b'\right)$	$a=b'\qquad b={\frac {a'e^{b'}}{b'}}\qquad n=0$
$\Rightarrow$	deterministic $\left(0\right)$	$a=0{\text{ or }}b=0$
$\Rightarrow$	Kupper-Poisson $\left(a',b'\right)$	$a=b'\qquad b={\frac {a'}{1-e^{-b'}}}\qquad n=0$
$\Rightarrow$	Neyman (Type A) $\left(a,b\right)$	$n=0$
$\Rightarrow$	Neyman (Type B) $\left(a,b\right)$	$n=1$
$\Rightarrow$	Neyman (Type C) $\left(a,b\right)$	$n=2$
$\rightarrow$	Poisson-geometric $\left(a,\theta \right)$	$n\rightarrow \infty \qquad b\rightarrow \infty \qquad {\frac {b}{n}}\rightarrow {\frac {\theta }{1-\theta }}$
$\rightarrow$	Pólya-Aeppli$\left(a',p\right)$	$n\rightarrow \infty \qquad b\rightarrow \infty \qquad {\frac {b}{n}}\rightarrow {\frac {q}{p}}\qquad {\frac {1}{2}}<p\leq 1$

References

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• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 15