Neyman Type A distribution

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],{{cite journal |last=Beall|first=G.|date= 1940 |title= The fit and significance of contagious distributions when applied to observations on larval insects|journal = Ecology | volume = 21 |issue=4 | pages = 460–474|doi=10.2307/1930285 |jstor=1930285 |bibcode=1940Ecol...21..460B }} Evans[1953]{{cite journal |last=Evans|first=D.A.|date= 1953 |title= Experimental evidence concerning contagious distributions in ecology |journal = Biometrika|volume=40 |issue=1–2 |pages=186–211 |doi=10.1093/biomet/40.1-2.186 }}), accidents (Creswell i Froggatt [1963]),{{cite book |last=Creswell|first=W.L.|author2=Froggatt, P.|date= 1963 |title= The Causation of Bus Driver Accidents| location= London | publisher= Oxford University Press}} 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]),{{cite journal |last=Feller|first=W.|date= 1943 |title= On a general class of "contagious" distributions |journal = Annals of Mathematical Statistics|volume= 14|issue=4 |page=389|doi=10.1214/aoms/1177731359 }} 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.

The probability generating function (pgf) G₁(z), which creates N independent X_j random variables, is used to a branching process. Each X_j 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,

:$Y\; =\; SN\; =\; X\_1\; +\; X\_2\; +\; ...\; +\; X\_N$

The p.g.f. of the distribution of SN is :

:$E[z^\{SN\}]\; =\; E\_N[E[z^\{SN\}|N]]\; =\; E\_N[G\_2(z)]\; =\; G\_1(G\_2(z))$

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,

:$Y\backslash sim\; F\_1\backslash bigwedge\; F\_N$

In this instance, it is written as,

:$Y\backslash sim\; \backslash operatorname\{Pois(\backslash lambda)\}\backslash bigwedge\; \backslash operatorname\{Pois(\backslash phi)\}$

Finally, the probability generating function is,

:$G\_Y(z)\; =\; \backslash exp(\backslash lambda(e^\{\backslash phi\; (z-1)\}-1))$

From the generating function of probabilities we can calculate the probability mass function explained below.

Let X₁,X₂,...X_j be Poisson independent variables. The probability distribution of the random variable Y = X₁ +X₂+...X_j is the Neyman's Type A distribution with parameters $\backslash lambda$ and $\backslash phi$.

:$p\_x\; =\; P(Y=x)\; =\; \backslash frac\{e^\{-\backslash lambda\}\backslash phi^x\}\{x!\}\; \backslash sum\_\{j=0\}^\{\backslash infty\}\; \backslash frac\{(\backslash lambda\; e^\{-\backslash phi\})^j\; j^x\}\{j!\}~~$$~~x\; =\; 1,2,...$

Alternatively,

:$p\_x\; =\; P(Y=x)\; =\; \backslash frac\{e^\{-\backslash lambda\; +\; \backslash lambda\; e^\{-\backslash phi\}\}\backslash phi^x\}\{x!\}\; \backslash sum\_\{j=0\}^\{x\}S(x,j)\backslash lambda^j\; e^\{-\backslash phi^j\}$

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

:$G(z)\; =\; e^\{-\backslash lambda\; +\; \backslash lambda\; e^\{-\backslash phi\}\}\backslash sum\_\{j=0\}^\{\backslash infty\}\backslash frac\{\backslash lambda^j\; e^\{-j\backslash phi\}(e^\{\backslash phi\; z\}-1)^j\}\{j|\}$

:$=\; e^\{-\backslash lambda\; +\; \backslash lambda\; e^\{-\backslash phi\}\}\backslash sum\_\{j=0\}^\{\backslash infty\}\; (\backslash lambda\; e^\{-\backslash phi\})^j\; \backslash sum\_\{x=j\}^\{\backslash infty\}\backslash frac\{S(x,j)\backslash phi^x\; z^x\}\{x!\}$

:$=\; \backslash frac\{e^\{-\backslash lambda\; +\; \backslash lambda\; e^\{-\backslash phi\}\}\backslash phi^x\}\{x!\}\; \backslash sum\_\{j=0\}^\{x\}S(x,j)\backslash lambda^j\; e^\{-\backslash phi^j\}$

Another form to estimate the probabilities is with recurring successions,{{cite book |last=Johnson|first=N.L|author2=Kemp, A.W|author3=Kotz, S|author-link= Norman Lloyd Johnson|date= 2005 |title= Univariate Discret Distributions | location= Canada|publisher= Wiley-Intersciencie|page= 404}}

:$p\_x\; =\; P(Y=x)\; =\; \backslash frac\{\backslash lambda\backslash phi\; e^\{-\; \backslash phi\}\}\{x\}\; \backslash sum\_\{r=0\}^\{x-1\}\; \backslash frac\{\backslash phi^r\}\{r!\}\; p\_\{x-r-1\}~~$,$~~p\_0\; =\; \backslash exp(-\backslash lambda\; +\; \backslash lambda\; e\; ^\{-\; \backslash phi\})$

Although its length varies directly with n, this recurrence relation is only employed for numerical computation and is particularly useful for computer applications.

where

• x = 0, 1, 2, ... , except for probabilities of recurring successions, where x = 1, 2, 3, ...
• $j\; \backslash leq\; x$
• $\backslash lambda$, $\backslash phi>0$.
• x! and j! are the factorials of x and j, respectively.
• one of the properties of Stirling numbers of the second kind is as follows:{{cite book |last=Johnson|first=N.L|author2=Kemp, A.W|author3=Kotz, S|author-link= Norman Lloyd Johnson|date= 2005 |title= Univariate Discret Distributions | location= Canada|publisher= Wiley-Intersciencie|section=1.1.3}}

:$(e^\{\backslash phi\; z\}\; -\; 1)^j\; =\; j!\; \backslash sum\_\{x=j\}^\backslash infty\; \backslash frac\{S(x,j)\{\backslash phi\; z\}^x\}\{x!\}$

:$Y\backslash \; \backslash sim\; \backslash operatorname\{NA\}(\backslash lambda,\backslash phi)\backslash ,$

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 $\backslash operatorname\{NA(\backslash lambda,\backslash phi)\}$, the moment generating function exists and is equal to

:$M(t)\; =\; G\_Y(e^t)\; =\; \backslash exp(\backslash lambda(e^\{\backslash phi\; (e^t-1)\}-1))$

The cumulant generating function is the logarithm of the moment generating function and is equal to

:$K(t)\; =\; \backslash log(M(t))\; =\; \backslash lambda(e^\{\backslash phi\; (e^t-1)\}-1)$

In the following table we can see the moments of the order from 1 to 4

The skewness is the third moment centered around the mean divided by the 3/2 power of the standard deviation, and for the $\backslash operatorname\{NA\}$ distribution is,

:$\backslash gamma\_1\; =\; \backslash frac\{\backslash mu\_3\}\{\backslash mu\_2^\{3/2\}\}\; =\; \backslash frac\{\backslash lambda\; \backslash phi(1\; +\; 3\backslash phi\; +\; \backslash phi^2)\}\{(\backslash lambda\; \backslash phi(1\; +\; \backslash phi))^\{3/2\}\}$

The kurtosis is the fourth moment centered around the mean, divided by the square of the variance, and for the $\backslash operatorname\{NA\}$ distribution is,

:$\backslash beta\_2=\; \backslash frac\{\backslash mu\_4\}\{\backslash mu\_2^2\}\; =\; \backslash frac\{\backslash lambda\; \backslash phi(1\; +\; 7\backslash phi\; +\; 6\backslash phi^2\; +\; \backslash phi^3)\; +\; 3\backslash lambda^2\backslash phi^2(1\; +\; \backslash phi)^2\}\{\backslash lambda^2\; \backslash phi^2(1\; +\; \backslash phi)^2\}\; =\; \backslash frac\{1+7\backslash phi+6\backslash phi^2+\backslash phi^3\}\{\backslash phi\backslash lambda\; (1+\backslash phi)^2\}\; +\; 3$

The excess kurtosis is just a correction to make the kurtosis of the normal distribution equal to zero, and it is the following,

:$\backslash gamma\_2=\; \backslash frac\{\backslash mu\_4\}\{\backslash mu\_2^2\}-3\; =\; \backslash frac\{1+7\backslash phi+6\backslash phi^2+\backslash phi^3\}\{\backslash phi\backslash lambda\; (1+\backslash phi)^2\}$

• Always $\backslash beta\_2\; >3$, or $\backslash gamma\_2\; >0$ the distribution has a high acute peak around the mean and fatter tails.

In a discrete distribution the characteristic function of any real-valued random variable is defined as the expected value of $e^\{itX\}$, where i is the imaginary unit and t ∈ R

:$\backslash phi(t)=\; E[e^\{itX\}]\; =\; \backslash sum\_\{j=0\}^\backslash infty\; e^\{ijt\}P[X=j]$

This function is related to the moment generating function via $\backslash phi\_x(t)\; =\; M\_X(it)$. Hence for this distribution the characteristic function is,

:$\backslash phi\_x(t)\; =\; \backslash exp(\backslash lambda(e^\{\backslash phi\; (e^\{it\}-1)\}-1))$

• Note that the symbol $\backslash phi\_x$ is used to represent the characteristic function.

The cumulative distribution function is,

:$\backslash begin\{align\}$

F(x;\lambda,\phi)& = P(Y \leq x)\\

&= e^{-\lambda} \sum_{i=0}^{x} \frac{\phi^i}{i!} \sum_{j=0}^{\infty} \frac{(\lambda e^{-\phi})^j j^i}{j!}\\

&= e^{-\lambda} \sum_{i=0}^{x} \sum_{j=0}^{\infty} \frac{\phi^i(\lambda e^{-\phi})^j j^i}{i!j!} \end{align}

• The index of dispersion is a normalized measure of the dispersion of a probability distribution. It is defined as the ratio of the variance $\backslash sigma^2$ to the mean $\backslash mu$,{{cite web |url= https://statologos.com/indice-de-dispersion/ |title= Índice de dispersión (Varianza a Razón media)|date=April 18, 2022 |website= Statologos}}

:$d\; =\; \backslash frac\{\backslash sigma^2\}\{\backslash mu\}\; =\; 1\; +\; \backslash phi$

• From a sample of size N, where each random variable Y_i comes from a $\backslash operatorname\{NA(\backslash lambda,\backslash phi)\}$, where Y₁, Y₂, .., Y_n are independent. This gives the MLE estimator as,{{cite book |last= Shenton|first= L.R|author2 = Bowman, K.O |date= 1977|title=Maximum Likelihood Estimation in Small Samples |location= London |publisher=Griffin |section= 9.6.1}}

:$\backslash sum\_\{i=1\}^\{N\}\; \backslash frac\{x\_i\}\{N\}\; =\; \backslash bar\{x\}\; =\; \backslash bar\{\backslash lambda\}\backslash bar\{\backslash phi\}\; ~~~$ where $\backslash mu$ is the poblational mean of $\backslash bar\{x\}$

• Between the two earlier expressions We are able to parametrize using $\backslash mu$ and $d$,

: $$

\begin{cases}

\mu = \lambda\phi \\

d = 1 + \phi

\end{cases}

\longrightarrow

\quad\!

\begin{aligned}

\lambda = \frac{\mu}{d-1} \\

\phi = d -1

\end{aligned}

The mean and the variance of the NA($\backslash lambda,\backslash phi$) are $\backslash mu\; =\; \backslash lambda\backslash phi$ and $\backslash lambda\backslash phi(1\; +\; \backslash phi)$, respectively. So we have these two equations,{{cite journal |last1= Shenton |first1= L.R |date= 1949| title = On the efficiency of the method of moments and Neyman Type A Distribution|url=https://www.jstor.org/stable/2332680 |journal= Biometrika |volume=36 |issue= 3–4 |pages= 450–454 |doi= 10.1093/biomet/36.3-4.450 |jstor= 2332680 |pmid= 15402077 }}

: $$

\begin{cases}

\bar{x} = \lambda\phi \\

s^2 = \lambda\phi(1 + \phi)

\end{cases}

• $s^2$ and $\backslash bar\{x\}$ are the mostral variance and mean respectively.

Solving these two equations we get the moment estimators $\backslash hat\{\backslash lambda\}$ and $\backslash hat\{\backslash phi\}$ of $\backslash lambda$ and $\backslash phi$.

:$\backslash bar\{\backslash lambda\}\; =\; \backslash frac\{\backslash bar\{x\}^2\}\{s^2\; -\; \backslash bar\{x\}\}$

:$\backslash bar\{\backslash phi\}\; =\; \backslash frac\{s^2\; -\; \backslash bar\{x\}\}\{\backslash bar\{x\}\}$

Calculating the maximum likelihood estimator of $\backslash lambda$ and $\backslash phi$ involves multiplying all the probabilities in the probability mass function to obtain the expression $\backslash mathcal\{L\}(\backslash lambda,\backslash phi;x\_1,\backslash ldots,x\_n)$.

When we apply the parameterization adjustment defined in "Other Properties," we get $\backslash mathcal\{L\}(\backslash mu,\; d;X)$. We may define the Maximum likelihood estimation based on a single parameter if we estimate the $\backslash mu$ as the $\backslash bar\{x\}$ (sample mean) given a sample X of size N. We can see it below.

:$\backslash mathcal\{L\}(d;X)=\; \backslash prod\_\{i=1\}^n\; P(x\_i;d)$

• To estimate the probabilities, we will use the p.m.f. of recurring successions, so that the calculation is less complex.

When $\backslash operatorname\{NA(\backslash lambda,\backslash phi)\}$ 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:

: $$

\begin{cases}

H_0: d=1 \\

H_1: d> 1

\end{cases}

The likelihood-ratio test statistic for $\backslash operatorname\{NA\}$ is,

:$W\; =\; 2(\backslash mathcal\{L\}(X;\backslash mu,d)-\backslash mathcal\{L\}(X;\backslash mu,1))$

Where likelihood $\backslash mathcal\{L\}()$ is the log-likelihood function. W does not have an asymptotic $\backslash chi\_1^2$ 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 $\backslash chi\_1^2$ have a 50:50 mixture. For this mixture, the $\backslash alpha$ upper-tail percentage points are the same as the $2\backslash alpha$ upper-tail percentage points for a $\backslash chi\_1^2$

The poisson distribution (on { 0, 1, 2, 3, ... }) is a special case of the Neyman Type A distribution, with

::$\backslash operatorname\{Pois\}(\backslash lambda)\; =\; \backslash operatorname\{NA\}(\backslash lambda,\backslash ,\; 0).\backslash ,$

From moments of order 1 and 2 we can write the population mean and variance based on the parameters $\backslash lambda$ and $\backslash phi$.

:$\backslash mu\; =\; \backslash lambda\backslash phi$

:$\backslash sigma^2\; =\; \backslash lambda\backslash phi\; (1\; +\; \backslash phi)$

In the dispersion index d we observe that by substituting $\backslash mu$ for the parametrized equation of order 1 and $\backslash sigma$ for the one of order 2, we obtain $d\; =\; 1\; +\; \backslash phi$. Our variable Y is therefore distributed as a Poisson of parameter $\backslash lambda$ when d approaches to 1.

Then we have that,

$\backslash lim\_\{d\backslash rightarrow\; 1\}\backslash operatorname\{NA\}(\backslash lambda,\backslash ,\; \backslash phi)\; \backslash rightarrow\; \backslash operatorname\{Pois\}(\backslash lambda)$

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]{{cite book |last=Archibald|first=E.E.A.|title= Plant populations I: A new application of Neyman's contagious distribution, Annals of Botany| location= London | publisher= Oxford University Press| pages=221–235}} 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).{{cite journal |last=Skellam|first=J.G.|date= 1958 |title= On the derivation and applicability of Neyman's type A distribution|journal = Biometrika| volume = 45 |issue=1–2 | pages = 32–36|doi=10.1093/biomet/45.1-2.32 }} 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:

1. Each driver is susceptible to "spells," the number of which is Poissonian for any given amount of time, with the same parameter $\backslash lambda$ for all drivers.
2. A driver's performance during a spell is poor, and he is likely to experience a Poissonian number of collisions, with the same parameter $\backslash phi$ for all drivers.
3. Each driver behaves independently.
4. No accidents can occur outside of a spell.

These assumptions lead to a Neyman type A distribution via the $\backslash operatorname\{Pois(\backslash lambda)\}$ and $\backslash operatorname\{Pois(\backslash phi)\}$ 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:

:$\backslash operatorname\{P(K\; =\; k\backslash phi)\}\; =\; \backslash frac\{e^\{-\backslash lambda\}\; \backslash lambda^k\}\{k!\}$

• taking values $~\; 0,~\; \backslash phi,\; ~\; 2\backslash phi,\; ~\; 3\backslash phi,\; ~\; ...$

and that a driver with proneness k$\backslash phi$ has X accidents where:

:$~\backslash operatorname\{P(X\; =\; x|k\backslash phi)\}\; =\; \backslash frac\{e^\{-k\backslash phi\}(k\backslash phi)^x\}\{x!\}$

This is the $~\backslash operatorname\{Pois(k\backslash phi)\}\; \backslash wedge\; \backslash operatorname\{Pois(\backslash lambda)\}$ 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.

• The reference of the usage history.

• The code below simulates 5000 instances of $\backslash operatorname\{NA(\backslash lambda,\backslash phi)\}$,

rNeymanA <- function(n,lambda, phi){

r <- numeric()

for (j in 1:n) {

k = rpois(1,lambda)

r[j] <- sum(rpois(k,phi))

}

return(r)

}

• The mass function of recurring probabilities is implemented in R in order to estimate theoretical probabilities; we can see it below,

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 $\backslash lambda\; =\; 2$ and $\backslash phi=\; 1$. It is displayed in the following table,

{{reflist}}

{{ProbDistributions|discrete-infinite}}

Category:Discrete distributions

Category:Poisson distribution

Category:Compound probability distributions