Calculating methods of moment estimator for Poisson Distribution

Question

BRIEF:
A statistical modelling team is considering using a compound Poisson distribution to model the total number of motor insurance claims from accidents on the road in a certain area. More precisely, the total number of yearly claims in that area is modelled as a random variable X constructed as follows:

$$X = \sum_{i=1}^{N} Z_i \ $$ where $N$~Poisson$(\lambda)$ with $\lambda > 0,\space$ and$\space$ {${{Z_i}}$}$^N_{i=1}$ is an independent and identically distributed sample of size N from a Poisson distribution with mean $\theta$.

QUESTION
Assuming that $\lambda$ is known, use the properties of the expectatiojn operator to derive the method of moments estimator for $\theta$, denoted henceforth by ${\hat{\theta}}(\underline{X})$, where $\underline{X}=(X_1 , ...,X_m)$ is an i.i.d compound Poisson ample of size $m$.

I have no idea where to even begin for this question... any help on how to start would be great, thank you!

Answer 1

Hint: Compute $$\mathbb{E}[X | N=n]$$ and calculate $\mathbb{E}[X]$ by the formula $$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X|N]]= \sum_{n=0}^\infty \mathbb{E}[X | N=n]\mathbb{P}(N=n).$$ Once you know $\mathbb{E}[X]$ it shouldn't be too hard to find the moment estimator for $\theta$.

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Step 1:

\begin{align*}\mathbb{E}[X | N = n] &= \mathbb{E}[\sum_{k=1}^n Z_k] = n \theta \end{align*}

Step 2:

$$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X|N]] = \mathbb{E}[N\theta] = \lambda \theta.$$

Step 3:

Equating the theoretical moment with the empirical moment and solving for $\theta$ we then get $$\hat{\theta}(X_1,\dots, X_m) = \frac{1}{\lambda} \bar{X}$$