Question

Assume that the number of defects in a car has a Poisson distribution with parameter 𝜆. To estimate 𝜆 we obtain the random sample 𝑋₁,𝑋₂, … , 𝑋_n.

a. Find the Fisher information is a single observation using two methods.

b. Find the Cramer-Rao lower bound for the variance of an unbiased estimator of 𝜆.

c. Find the MLE of 𝜆 and show that the MLE is an efficient estimator.

Answer 1

Solution

a. Find the Fisher information is a single observation using two methods.

By definition, we have

                                 

Since

                                 

Therefore:

                                     

b. Find the Cramer-Rao lower bound for the variance of an unbiased estimator of 𝝀.

When 𝜆̂ achieves the Cramer-Rao lower bound then it is a minimum variance unbiased estimator such that

                                             

c. Find the MLE of 𝝀 and show that the MLE is an efficient estimator.

By using method of moment, we have 1st sample moment is 𝐸(𝑋)=𝑋̅ (1)

By the 1st population of moment 𝐸(𝑋)=𝜆 (2)

From (1) & (2) we get   

And 

                             

Which is equal to the lower bound of Cramer-Rao. Thus, it is efficient.

Therefore: 𝐭𝐡𝐞 𝐞𝐟𝐟𝐢𝐜𝐢𝐞𝐧𝐭 𝐞𝐬𝐭𝐢𝐦𝐚𝐭𝐨𝐫 𝐢𝐬 𝝀̂=𝑿