Methods of Moments Estimator

The method of moments is a statistical technique for estimating unknown parameters of a probability distribution. The idea is to match the sample moments of the data to the corresponding theoretical moments of the distribution. In this case, the sample mean is x=n1​∑i=1n​xi​, and the theoretical mean of a Poisson distribution with parameter $\lambda$ is E[xi​]=λ. Therefore, the method of moments estimator for $\lambda$ is simply the sample mean:

λ^=x

Unbiasedness

An estimator is said to be unbiased if the expected value of the estimator is equal to the true value of the parameter being estimated. In this case, the expected value of the method of moments estimator is

E[λ^]=E[x]=E[n1​i=1∑n​xi​]=n1​i=1∑n​E[xi​]=n1​i=1∑n​λ=λ

Therefore, the method of moments estimator is unbiased.

R Function

Here is an R function that calculates the method of moments estimator for the mean of a Poisson distribution:

mm_estimate <- function(x) {
  mean(x)
}

MM Estimate of $\lambda$

Using the R function mm_estimate, we can calculate the MM estimate of $\lambda$ as follows:

set.seed(1)
x <- rpois(100, 3.6)
mm_estimate(x)