User:BeyondNormality/Altham-multiplicative binomial distribution

Altham-multiplicative binomial
Altham-multiplicative binomial
Notation	AlthamMultBin(a,r,n)
Parameters
Support
PMF
CDF
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF
CF
PGF

The probability mass function of the Altham-multiplicative binomial distribution is

A l t h a m M u l t B i n (x; a, n, p) \equiv Pr (X = x) = \frac{(\binom{n}{x}) p^{x} (1 - p)^{n - x} a^{x (n - x)}}{F (n)}

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
$\to$	the distribution in the title converges to this distribution

Relationship	Distribution	When
$\Leftarrow$	generalized power series family $(a_{x}, θ, f (.), T)$	$a_{x} = \frac{n_{(x)} a^{x (n - x)}}{x!} θ = \frac{p}{q} f (θ) = \sum_{j = 0}^{n} (\binom{n}{j}) θ^{j} a^{j (n - j)} T = {0, \dots, n}$
$\Rightarrow$	binomial $(n, p)$	$a = 1$
$\Rightarrow$	deterministic $(0)$	$a n p = 0$
$\Rightarrow$	deterministic $(n)$	$p = 1$
$\to$	Poisson $(b)$	$a = 1 n \to \infty p \to 0 n p \to b$

Altham, P. (1978). Two generalizations of the binomial distribution. Applied Statistics 27, 162-167
Altmann-Fitter (1994). Iterative Anpassung diskreter Wahrscheinlichkeitsverteilungen. Lüdenscheid:RAM-Verlag.
Haseman, J.K. Kupper, L.L. (1979). Analysis of dichotomous response data from certain toxicological experiments. Biometrics 35, 281-293
Johnson, N.L., Kotz, S., Kemp, A.W. (1992). Univariate Discrete Distributions. New York: Wiley. pg 149f
Makuch, R.W., Stephens, M.A., Escobar, M. (1989). Generalized binomial models to examine the historical control assumption in active control equivalence studies. The Statistician 38, 61-70.
Paul, S.R. (1982). Analysis of proportions of affected foetuses in teratological experiments. Biometrics 38, 361-370
Rudolpher, S.M. (1990). A Markov chain model of extrabinomial variations. Biometrika 77, 255-264
Tarone, R.F. (1979). Testing the goodness of fit of the binomial distribution. Biometrika 66, 585-590
Wilcox, R.R. (1981). A review of the beta-binomial model and its extensions. J. of Educational Statistics 6, 3-32
Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 8