In statistics and probability, the Neyman Type A distribution is a discrete probability distribution from the family of Compound Poisson distribution. First of all, to easily understand this distribution we will demonstrate it with the following example explained in *Univariate Discrete Distributions*; we have a statistical model of the distribution of larvae in a unit area of field (in a unit of habitat) by assuming that the variation in the number of clusters of eggs per unit area (per unit of habitat) could be represented by a Poisson distribution with parameter λ {\displaystyle \lambda } , while the number of larvae developing per cluster of eggs are assumed to have independent Poisson distribution all with the same parameter ϕ {\displaystyle \phi } . If we want to know how many larvae there are, we define a random variable Y as the sum of the number of larvae hatched in each group (given j groups). Therefore, Y = X1 + X2 + ... X j, where X1,...,Xj are independent Poisson variables with parameter λ {\displaystyle \lambda } and ϕ {\displaystyle \phi } .
History Jerzy Neyman was born in Russia in April 16 of 1894, he was a Polish statistician who spent the first part of his career in Europe. In 1939 he developed the Neyman Type A distribution to describe the distribution of larvae in experimental field plots. Above all, it is used to describe populations based on contagion, e.g., entomology (Beall[1940], Evans[1953]), accidents (Creswell i Froggatt [1963]), and bacteriology. The original derivation of this distribution was on the basis of a biological model and, presumably, it was expected that a good fit to the data would justify the hypothesized model. However, it is now known that it is possible to derive this distribution from different models (William Feller[1943]), and in view of this, Neyman's distribution derive as Compound Poisson distribution. This interpretation makes them suitable for modelling heterogeneous populations and renders them examples of apparent contagion. Despite this, the difficulties in dealing with Neyman's Type A arise from the fact that its expressions for probabilities are highly complex. Even estimations of parameters through efficient methods, such as maximum likelihood, are tedious and not easy to understand equations.
Definition
Probability generating function The probability generating function (pgf) G1(z), which creates N independent Xj random variables, is used to a branching process. Each Xj produces a random number of individuals, where X1, X2,... have the same distribution as X, which is that of X with pgf G2(z). The total number of individuals is then the random variable,
Y = S N = X 1 + X 2 + . . . + X N {\displaystyle Y=SN=X_{1}+X_{2}+...+X_{N}}
The p.g.f. of the distribution of SN is :
E [ z S N ] = E N [ E [ z S N | N ] ] = E N [ G 2 ( z ) ] = G 1 ( G 2 ( z ) ) {\displaystyle E[z^{SN}]=E_{N}[E[z^{SN}|N]]=E_{N}[G_{2}(z)]=G_{1}(G_{2}(z))}
One of the notations, which is particularly helpful, allows us to use a symbolic representation to refer to an F1 distribution that has been generalized by an F2 distribution is,
Y ∼ F 1 ⋀ F N {\displaystyle Y\sim F_{1}\bigwedge F_{N}}
In this instance, it is written as,
Y ∼ P o i s ( λ ) ⋀ P o i s ( ϕ ) {\displaystyle Y\sim \operatorname {Pois(\lambda )} \bigwedge \operatorname {Pois(\phi )} }
Finally, the probability generating function is,
G Y ( z ) = exp ( λ ( e ϕ ( z − 1 ) − 1 ) ) {\displaystyle G_{Y}(z)=\exp(\lambda (e^{\phi (z-1)}-1))}
From the generating function of probabilities we can calculate the probability mass function explained below.
Probability mass function Let X1,X2,...Xj be Poisson independent variables. The probability distribution of the random variable Y = X1 +X2+...Xj is the Neyman's Type A distribution with parameters λ {\displaystyle \lambda } and ϕ {\displaystyle \phi } .
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