Question

Just part d. Using R programming, struggling with the code.

Let Y1,…,Yn be independent Poisson random variables with mean θ.

a) Derive the method of moments estimator for θ.

b) Derive the maximum likelihood estimator θ̂n for θ. How does this compare to what you found in part a.

c) Provide the asymptotic sampling distribution for θ̂n.

d) With the following data values for y given in R below, what are the chances that the data could have been generated from a Poisson with parameter θ=5? Or θ=6? Or θ=7?

y <- c(3, 5, 6, 5, 2, 6, 6, 7, 8, 8, 7, 8, 0, 5, 7, 6, 6, 10, 6, 5, 6, 7, 6,  
    9, 8, 4, 8, 7, 11, 9, 4, 4, 7, 9, 8, 6, 5, 6, 12, 10, 7, 13, 8, 12, 9, 4, 
    10, 8, 4, 5)

plz help

Answer 1

For a Poisson random variable Y, the probability mass function is given by:

and the likelihood of observed data is the product of all individual which is,

You can see that the likelihoods are very low as in order of 10^-58, 10^-53 and 10^-52

So, for practical purposes we compare the values of log likelihood which are given below likelihood. From there you can observe that log likelihood for observed data is highest for theta = 7. That is, the given observations of Y is more likely to come from a Poisson distribution with theta = 7 as compare to theta = 5 and theta = 7.