Compound Poisson-family discrete probability distribution
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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 Discret 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 formula_5, while the number of larvae developing per cluster of eggs are assumed to have independent Poisson distribution all with the same parameter formula_6. 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" = "X"1 + "X"2 + ... "X" j, where "X"1...,"X"j are independent Poisson variables with parameter formula_7 and formula_8.
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) "G"1("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 "G"2("z"). The total number of  individuals is then the random variable,
formula_9
The p.g.f. of the distribution of "SN" is :
formula_10
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,
formula_11
In this instance, it is written as,
formula_12
Finally, the probability generating function is,
formula_13
From the generating function of probabilities we can calculate the probability mass function explained below.
Probability mass function.
Let "X"1,"X"2..."X"j be Poisson independent variables. The probability distribution of the random variable "Y" = "X"1 +"X"2+..."X"j is the Neyman's Type A distribution with parameters formula_5 and formula_15.
formula_16formula_17
Alternatively,
formula_18
In order to see how the previous expression develops, we must bear in mind that the probability mass function is calculated from the probability generating function, and use the property of Stirling Numbers. Let's see the development
formula_19
formula_20
formula_21
Another form to estimate the probabilities is with recurring successions,
formula_22,formula_23
Although its length varies directly with "n", this recurrence relation is only employed for numerical computation and is particularly useful for computer applications.
where
formula_27
formula_28
Properties.
Moment and cumulant generating functions.
The moment generating function of a random variable "X" is defined as the expected value of "e""t", as a function of the real parameter "t". For an formula_29, the moment generating function exists and is equal to
formula_30
The cumulant generating function is the logarithm of the moment generating function and is equal to 
formula_31
In the following table we can see the moments of the order from 1 to 4
Skewness.
The skewness is the third moment centered around the mean divided by the 3/2 power of the standard deviation, and for the formula_32 distribution is,
formula_33
Kurtosis.
The kurtosis is the fourth moment centered around the mean, divided by the square of the variance, and for the formula_32 distribution is,
formula_35
The excess kurtosis is just a correction to make the kurtosis of the normal distribution equal to zero, and it is the following,
formula_36
Characteristic function.
In a discrete distribution the characteristic function of any real-valued random variable is defined as the expected value of formula_39, where "i" is the imaginary unit and "t" ∈ "R"
formula_40
This function is related to the moment generating function via formula_41. Hence for this distribution the characteristic function is,
formula_42
Cumulative distribution function.
The cumulative distribution function is,
formula_44
formula_47
formula_49 where formula_50 is the poblational mean of formula_51
 formula_54
Parameter estimation.
Method of moments.
The mean and the variance of the NA(formula_55) are formula_56 and formula_57, respectively. So we have these two equations,
 formula_58
Solving these two equations we get the moment estimators formula_61 and formula_62 of formula_7 and formula_8.
formula_65
formula_66
Maximum likelihood.
Calculating the maximum likelihood estimator of formula_7 and formula_8 involves multiplying all the probabilities in the probability mass function to obtain the expression formula_69.
When we apply the parameterization adjustment defined in "Other Properties," we get formula_70. We may define the Maximum likelihood estimation based on a single parameter if we estimate the formula_50 as the formula_51 (sample mean) given a sample "X" of size "N". We can see it below.
formula_73
Testing Poisson assumption.
When formula_29 is used to simulate a data sample it is important to see if the Poisson distribution fits the data well. For this, the following Hypothesis test is used:
 formula_75
Likelihood-ratio test.
The likelihood-ratio test statistic for formula_32 is,
formula_77
Where likelihood formula_78 is the log-likelihood function. "W" does not have an asymptotic formula_79 distribution as expected under the null hypothesis since "d" = 1 is at the parameter domain's edge. In the asymptotic distribution of "W," it can be demonstrated that the constant 0 and formula_79 have a 50:50 mixture. For this mixture, the formula_81 upper-tail percentage points are the same as the formula_82 upper-tail percentage points for a formula_79
Related distributions.
The poisson distribution (on { 0, 1, 2, 3, ... }) is a special case of the Neyman Type A distribution, with
formula_84
From moments of order 1 and 2 we can write the population mean and variance based on the parameters formula_7 and formula_8.
formula_56
formula_88
In the dispersion index "d" we observe that by substituting formula_52 for the parametrized equation of order 1 and formula_90 for the one of order 2, we obtain formula_91. Our variable "Y" is therefore distributed as a Poisson of parameter formula_5 when "d" approaches to 1.
Then we have that,
formula_93
Applications.
Usage History.
When the Neyman's type A species' reproduction results in clusters, a distribution has been used to characterize the dispersion of plants. This typically occurs when a species develops from offspring of parent plants or from seeds that fall close to the parent plant. However, Archibald (1948] observed that there is insufficient data to infer the kind of reproduction from the type of fitted distribution. While Neyman type A produced positive results for plant distributions, Evans (1953] showed that Negative binomial distribution produced positive results for insect distributions. Neyman type A distributions have also been studied in the context of ecology, and the results that unless the plant clusters are so compact as to not lie across the edge of the square used to pick sample locations, the distribution is unlikely to be applicable to plant populations. The compactness of the clusters is a hidden assumption in Neyman's original derivation of the distribution, according to Skellam (1958). The results were shown to be significantly impacted by the square size selection.
In the context of bus driver accidents, Cresswell and Froggatt (1963) derived the Neyman type A based on the following hypotheses:
These assumptions lead to a Neyman type A distribution via the formula_96 and formula_97 model. In contrast to their "short distribution," Cresswell and Froggatt called this one the "long distribution" because of its lengthy tail. According to Irwin(1964), a type A distribution can also be obtained by assuming that various drivers have different levels of proneness, or "K". with probability:
formula_98
and that a driver with proneness "k"formula_8 has "X" accidents where:
formula_101
This is the formula_102 model with mixing over the values taken by "K".
Distribution was also suggested in the application for the grouping of minority tents for food from 1965 to 1969. In this regard, it was predicted that only the clustering rates or the average number of entities per grouping needed to be approximated, rather than adjusting distributions on d very big data bases.
Calculating Neyman Type A probabilities in R.
 rNeymanA ;- function(n,lambda, phi){
 r ;- numeric()
 for (j in 1:n) {
 k = rpois(1,lambda)
 r[j] ;- sum(rpois(k,phi)) 
 return(r)
 dNeyman.rec ;- function(x, lambda, phi){
 p ;- numeric()
 p[1] ;- exp(-lambda + lambda*exp(-phi))
 c ;- lambda*phi*exp(-phi)
 if(x == 0){
 return(p[1])
 else{
 for (i in 1:x) {
 suma = 0
 for (r in 0:(i-1)) {
 suma = suma + (phi^r)/(factorial(r))*p[i-r]
 p[i+1] = (c/(i))*suma # +1 per l'R
 res ;- p[i+1]
 return(res)
We compare results between the relative frequencies obtained by the simulation and the probabilities computed by the p.m.f. . Given two values for the parameters formula_104 and formula_105. It is displayed in the following table,
References.
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