User:BeyondNormality/Ahuja-negative binomial distribution

Ahuja-negative binomial distribution

Notation	$\mathrm {AhujaNB} \left(k,n,p\right)$
Support	x ∈ { n, n+1, n+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	${\displaystyle }$

Also known as the distribution of the sum of independent decapitated negative binomial variables, n-fold convolution of the zero-truncated negative binomial distribution, associated Lah distribution, the probability mass function of the Ahuja-negative binomial distribution is given by

${\begin{aligned}\mathrm {AhujaNB} (x;k,n,p)\equiv \Pr(X=x)&=\sum _{r=1}^{n}(-1)^{n-r}q^{x}{\binom {n}{r}}\left(p^{-k}-1\right)^{-n}{\binom {kr+x-1}{x}}\\&=\sum _{r=1}^{n}(-1)^{n-r}q^{x}{\binom {n}{r}}p^{kn}\left(1-p^{k}\right)^{-n}{\binom {kr+x-1}{x}}\end{aligned}}$

• $x=n,n+1,n+2,\dots$
• $q=1-p$
• $N\in \mathbb {N}$
  • $k\neq 0$
  • $0<p<1$
• $N\in \mathbb {N} _{0}$
  • $k\geq 0$
    • $0<p\leq 1$
  • $-1\leq k<0$
    • $0\leq p\leq 1$

Expected Value

$\mathbb {E} (X)={\frac {knq}{p\left(1-p^{k}\right)}}$

Variance

$\operatorname {Var} (X)={\frac {knq\left(1-(1+kq)p^{k}\right)}{p^{2}\left(p^{k}-1\right)^{2}}}$

Moment Generating Function

$M_{X}(t)=\left({\frac {(1-q)^{k}\left(1-\left(1-qe^{t}\right)^{-k}\right)}{(1-q)^{k}-1}}\right)^{n}$

Characteristic Function

$\varphi _{X}(t)=\left({\frac {(1-q)^{k}\left(1-\left(1-qe^{it}\right)^{-k}\right)}{(1-q)^{k}-1}}\right)^{n}$

Probability Generating Function

${\begin{aligned}G(t)&={\left({\frac {t\,_{2}F_{1}(k+1,1;2;qt)}{\,_{2}F_{1}(k+1,1;2;q)}}\right)}^{n}\\&=\left({\frac {1-p^{k}(1-qt)^{-k}}{p^{k}-1}}\right)^{n}\qquad k\neq 0,0<p<1\end{aligned}}$

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
$\equiv$	positive negative binomial ${\left(k,p\right)}^{{n}^{*}}$
$\Leftarrow$	Cacoullos-Charalambides $\left(n,r,s,\theta \right)$	$r=1\qquad s=k\qquad \theta =q$
$\Leftarrow$	generalized C $\left(k,n,\beta ,\theta \right)$	$\beta =1\qquad \theta =q$
$\Leftarrow$	generalized power series $\left(a,\theta ,f(.),T\right)$	$a_{x}=\sum _{r=1}^{n}(-1)^{n-r}{\binom {n}{r}}{\binom {kr+x-1}{x}}\qquad f(\theta )=\left((1-\theta )^{-k}-1\right)^{n}\qquad \theta =q\qquad T=\{n,n+1,\dots \}$
$\Leftarrow$	n-displaced generalized power series $\left(a,\theta ,f(.)\right)$	$a_{x}=\sum _{r=1}^{n}(-1)^{n-r}{\binom {n}{r}}{\binom {kr+x+n-1}{x+n}}\qquad f(\theta )=\left({\frac {(1-\theta )^{-k}-1}{\theta }}\right)^{n}\qquad \theta =q\qquad n\in \mathbb {N}$
$\Rightarrow$	deterministic (0)	$n=0$
$\rightarrow$	Stirling (type 2) $\left(n,\theta \right)$	$k\rightarrow \infty \qquad q\rightarrow 0\qquad kq\rightarrow \theta$

References

• Ahuja, J.C. (1971a). Distribution of the sum of independent decapitated negative binomial variables Annals of Mathematical Statistics 42, 383-384
• Ahuja, J.C., Enneking, E.A. (1972) Recurrence relation for the minimal variance unbiased estimator of the parameter of a left truncated Poisson distribution. J. of the American Statistical Association 67, 232-233.
• Berg, S. (1975). A note on the connection between factorial series distribution and zero-truncated power series distribution. Scandinavian Actuarial J. 233-237.
• Cacoullos, T. (1975) Multiparameter Stirling and C-type distributions. In: Patil, G.P., Kotz, S. Ord, J.K. (eds), Statistical Distributions in Scientific Work, Vol. 1: 19-30. Dordrecht: Reidel.
• Cacoullos, T., Charalambides, C. (1975) On minimum variance unbiased estimation for truncated binomial and negative binomial distributions. Annals of the Institute of Statistical Mathematics 27, 235-244.
• Charalambides, Ch.A. (1977a). A new kind of numbers appearing in the n-fold convolution of truncated binomial and negative binomial distributions. SIAM J. of Applied Mathematics 33, 279-288.
• Charalambides, Ch.A. (1984). Probabilities and moments of generalized discrete distributions using finite difference equations. Communications in Statistics - Theory and Methods 13, 3225-3241.
• Gupta, R.C. (1974a). Modified power series distributions and some of its applications. Sankhyā B, 35, 288-298.
• Gupta, R.P., Kabe, D.G. (1974). Distributions of sums of truncated discrete variables with applications. Communications in Statistics 3, 1161-1170.
• Johnson, N.L., Kotz, S., Kemp, A.W. (1992). Univariate Discrete Distributions. New York: Wiley
• Medhi, J. (1975). On the convolutions of left-truncated generalised negative binomial and Poisson variables. Sankhyā B, 37, 293-299.
• Saleh, A.K.M.E., Rahim, M.A. (1972). Distributions of the sum of variates from truncated discrete populations. Canadian Mathematical Bulletin 15, 395-398.
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 6