User:BeyondNormality/Aitchinson distribution

Aitchinson distribution

Notation	$\mathrm {Aitchinson} \left(a,b,\theta \right)$
Support	x ∈ { 0, 1, 2 , ... }
PMF	${\displaystyle }$
CDF	${\displaystyle }$
Mean	${\displaystyle }$
Median	${\displaystyle }$
Mode	${\displaystyle }$
Variance	${\displaystyle }$
Skewness	${\displaystyle }$
Excess kurtosis	${\displaystyle }$
Entropy	${\displaystyle }$
MGF	${\displaystyle }$
CF	${\displaystyle }$
PGF	${\displaystyle }$

The probability mass function of the Aitchinson distribution is given by

$\mathrm {Aitchinson} \left(x;a,b,\theta \right)\equiv \Pr(X=x)=e^{-a}\sum _{j=0}^{-2k+x}\sum _{k=0}^{\left[{\frac {x}{2}}\right]}{\frac {(bx)^{-j-2k+x}\left(-ae^{-b}(-1+\theta )\right)^{k}(a\theta )^{j}}{j!k!(-j-2k+x)!}}$

• $x=0,1,2,\dots$
• $a,b\geq 0$
• $0\leq \theta \leq 1$
• $\left[z\right]={\text{integer part of z}}$

Expected Value

$\operatorname {E} [X]=-a(b(\theta -1)+\theta -2)$

Variance

$\operatorname {Var} (X)=a(-b(b+5)(\theta -1)-3\theta +4)$

Moment Generating Function

$M_{X}(t)=\exp \left(-a\left((\theta -1)e^{b\left(e^{t}-1\right)+2t}-\theta e^{t}+1\right)\right)$

Characteristic Function

$\varphi _{X}(t)=\exp \left(-a\left((\theta -1)e^{b\left(e^{it}-1\right)+2it}-\theta e^{it}+1\right)\right)$

Probability Generating Function

$G(t)=\exp \left(-a\left((\theta -1)t^{2}e^{b(t-1)}-\theta t+1\right)\right)$

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
$\equiv$	Poisson $(a)$ $\vee$ modified displaced Poisson $\left(b,\theta \right)$	$P_{x}={\begin{cases}\theta &x=1\\{\frac {e^{-b}(1-\theta )b^{x-2}}{(x-2)!}}&x=2,3,\dots \\\end{cases}}\qquad b\geq 0\qquad 0\leq \theta \leq 1$
$\Leftarrow$	generalized Poisson family $\left(a,H(.)\right)$
$\leftarrow$	multiple Poisson $\left(n,a_{i}\right)$	$n\rightarrow \infty \qquad a_{i}={\begin{cases}a\theta &i=1\\{\frac {ab^{-2+i}e^{-b}(1-\theta )}{(-2+i)!}}&i=2,3,\dots \\\end{cases}}$
$\Rightarrow$	deterministic $\left(0\right)$	$a=0$
$\Rightarrow$	Hermite $\left(a',b'\right)$	$a=a'+b'\qquad \theta ={\frac {a'}{a'+b'}}\qquad b=0$
$\Rightarrow$	Hirata-Poisson $\left(a',b'\right)$	$\theta =1-b'\qquad b=0$
$\Rightarrow$	Poisson $\left(a\right)$	$\theta =1$

References

• Aitchinson, J. (1955). On the distribution of a positive random variable having a distribution probability mass at the origin J. of the American Statistical Association 50, 901-908
• Kupper, J. (1960-62). Wahrscheinlich-keitstheoretische Modelle in der Schadenversicherung. Teil I: Die Schadenzahl. Blätter der Deutschen Gesellschaft für Versicherungsmathematik 5, 451-503.
• Wimmer, G., Altmann. (1996a). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical J. 8, 995-1011.
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 7