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Altham-multiplicative binomial Notation
A
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M
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l
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B
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n
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a
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{\displaystyle \mathrm {AlthamMultBin} (a,r,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 }
probability mass function of the Altham-multiplicative binomial distribution is
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≡
Pr
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X
=
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=
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n
x
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p
x
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1
−
p
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n
−
x
a
x
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n
−
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F
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n
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{\displaystyle \mathrm {AlthamMultBin} (x;a,n,p)\equiv \Pr(X=x)={\frac {{\binom {n}{x}}p^{x}(1-p)^{n-x}a^{x(n-x)}}{F(n)}}}
x
=
0
,
1
,
…
,
n
{\displaystyle x=0,1,\dots ,n}
a
≥
0
{\displaystyle a\geq 0}
n
∈
N
0
{\displaystyle n\in \mathbb {N} _{0}}
0
≤
p
≤
1
{\displaystyle 0\leq p\leq 1}
q
=
1
−
p
{\displaystyle q=1-p}
F
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n
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=
∑
j
=
0
n
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p
j
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1
−
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n
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a
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−
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{\displaystyle F(n)=\sum _{j=0}^{n}{\binom {n}{j}}p^{j}(1-p)^{n-j}a^{j(n-j)}}
Symbol
Meaning
∼
{\displaystyle \sim }
X
∼
Y
{\displaystyle X\sim Y}
≡
{\displaystyle \equiv }
the distribution in the title is identical with this distribution
⇐
{\displaystyle \Leftarrow }
the distribution in title is a special case of this distribution
⇒
{\displaystyle \Rightarrow }
this distribution is a special case of the distribution in the title
←
{\displaystyle \leftarrow }
this distribution converges to the distribution in the title
→
{\displaystyle \rightarrow }
the distribution in the title converges to this distribution
Relationship
Distribution
When
⇐
{\displaystyle \Leftarrow }
generalized power series family
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f
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{\displaystyle \left(a_{x},\theta ,f(.),T\right)}
a
x
=
n
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x
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a
x
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n
−
x
)
x
!
θ
=
p
q
f
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θ
)
=
∑
j
=
0
n
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n
j
)
θ
j
a
j
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n
−
j
)
T
=
{
0
,
…
,
n
}
{\displaystyle a_{x}={\frac {n_{(x)}a^{x(n-x)}}{x!}}\qquad \theta ={\frac {p}{q}}\qquad f(\theta )=\sum _{j=0}^{n}{\binom {n}{j}}\theta ^{j}a^{j(n-j)}\qquad T=\{0,\dots ,n\}}
⇒
{\displaystyle \Rightarrow }
binomial
(
n
,
p
)
{\displaystyle \left(n,p\right)}
a
=
1
{\displaystyle a=1}
⇒
{\displaystyle \Rightarrow }
deterministic
(
0
)
{\displaystyle \left(0\right)}
a
n
p
=
0
{\displaystyle anp=0}
⇒
{\displaystyle \Rightarrow }
deterministic
(
n
)
{\displaystyle \left(n\right)}
p
=
1
{\displaystyle p=1}
→
{\displaystyle \rightarrow }
Poisson
(
b
)
{\displaystyle \left(b\right)}
a
=
1
n
→
∞
p
→
0
n
p
→
b
{\displaystyle a=1\qquad n\rightarrow \infty \qquad p\rightarrow 0\qquad np\rightarrow 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
