Let {X, X,..., Xn} be an independent random sample from Poisson (2), where is an unknown...

Transcribed Image Text:

Let {X₁, X₂,..., Xn} be an independent random sample from Poisson (2), where is an unknown parameter. Suppose that the sample size n is large. (a) (b) (c) (d) (e) Show that both the sample mean X and the sample variance S² are unbiased estimator of 2. Calculate the Cramer-Rao lower bound of unbiased estimators of 2. Which one of the estimators in part (a) should be preferred and why? What is the large sample size distribution of X? Using part (d), or otherwise, derive an approximate 1 - a confidence interval for 1. Hint: leads to -2뜰 X - λ Za =1-a P((λ-Au)(-A) < 0) ≈ 1-α, where > < >, are the roots of the quadratic equation 2²-( 28 + n λ + X² = 0. Let {X₁, X₂,..., Xn} be an independent random sample from Poisson (2), where is an unknown parameter. Suppose that the sample size n is large. (a) (b) (c) (d) (e) Show that both the sample mean X and the sample variance S² are unbiased estimator of 2. Calculate the Cramer-Rao lower bound of unbiased estimators of 2. Which one of the estimators in part (a) should be preferred and why? What is the large sample size distribution of X? Using part (d), or otherwise, derive an approximate 1 - a confidence interval for 1. Hint: leads to -2뜰 X - λ Za =1-a P((λ-Au)(-A) < 0) ≈ 1-α, where > < >, are the roots of the quadratic equation 2²-( 28 + n λ + X² = 0.