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Suppose I have this small data set and it is required to fit a compound poisson distribution based on the data [with respect to the the real blood corpuscular (x) per cell] available.

\begin{array}{| c | c |} \hline \\\text{x} & \text{Number of Cells} \\ \\\hline 1& f_1 \\\\\hline 2& f_2\\\\\hline 3& f_3\\\\\hline 4& f_4\\\\\hline 5 & f_{5}\\\\\hline \end{array}

Also $\sum_{i=1}^5 f_i=N$

There is a result that if a random variable $\text{X}\sim \mathcal{Poisson}(\lambda)$ then the compound poisson distribution of $\text{X}\sim \mathcal{Neg \ Bin }(q,p)$ .

Suppose, if it is assumed that $\lambda = \frac{\sum\mathbb{f_ix_i}}{\sum\mathbb{f_i}}$, then I need to calculate $\text{g}(\lambda) = \begin{cases} \frac{ae^{-a\lambda}\lambda^{v-1}}{\Gamma(V)} & \mathrm{ } \lambda>0, a>0, V>0 \\ 0 & \mathrm{ } \lambda \leq 0 \end{cases}.$, where $a$ & $V$ should be determined. Once they are determined then I can find out $$\mathbb{P}\mathrm{(X=x)=\binom{-V}{x}} \textrm{p^V q^x}, \ \ \textrm{ where } \mathrm{p=\frac{a}{1+a}, \ q=1-p,} \ \ \textrm{ x=0,1,2,..,5}$$

Now applying newton-raphson iteration the expected frequencies $e_i$'s can be computed.

Now here are my questions-

  1. how to evaluate $a$ & $V$ while finding out $\text{g}(\lambda)$ on the basis of assumed value of $\lambda$?
  1. Is the column - 'Number of Cells' alone insufficient to fit a compound poisson distribution?
  1. if my approach is incorrect then what is the example or the general structure to fit a compound poisson distribution based on such small data?

Here is one partially related problem but yet it is different. Any help is highly valued and appreciated.

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