Question

3. Method of moments estimators Bookmark this page For each of the following distributions, give the method of moments estimator in terms of the sample averages Xn and X, assuming we have access to n i.i.d. observations X1,...,xn. In other words, express the parameters as functions of E [X1 and E[X] and then apply these functions to Xn and X

Answer 1

we have given the

$X_{i} \sim Poisson\left ( \lambda \right ) ,\lambda >0$

The Probability Mass Function of X is defines as

$P_{\lambda }\left ( X=k \right )=\frac{e^{-\lambda}\lambda ^{k}}{k!} ; k=0,1,2,....$

we know that Population mean of Poisson distribution is $\lambda$

$E\left ( X \right )=\lambda$

we know that the first moment is sample mean

$\mu_{1}= \bar{x}$

$\mu_{1}= E\left ( X\right )$

$\mu _{1}=E\left ( X \right )=\lambda$

$\mu _{1}=E\left ( X \right )=\bar{x}=\lambda$

that is  $\bar{x}=\lambda$

that is $\widehat{\lambda}=\bar{x}$

$\widehat{\lambda}=\frac{\sum x_{i}}{n}$

Hence, the method of moments estimator of $\widehat{\lambda}$ is the sample mean.

Result : the method of moments estimator of$\widehat{\lambda}$ is the sample mean.