Question

Please give detailed steps. Thank you.

5. Let {X, : i-1..n^ denote a random sample of size n from a population described by a random varaible X following a Poisson(θ) distribution with PDF given by θ and var(X) θ (i.e. you do not You may take it as given that E(X) need to show these) a. Recall that an estimator is efficient, if it satisfies 2 conditions: 2) it achieves the Cramer-Rao Lower Bound (CLRB) for unbiased estimators: Show that the sample mean X a1. an unbiased estimator of θ n Σ x, is: a2. OPTIONAL (you will get full credit without this part): an efficient estimator of θ Hint: To find CRLB in part a2. follow the steps below: 1) find the expression for log f(x,θ) (the so-called log-likelihood of a single observation): θ logf(zi:0) ; θ) 0)1 (the information matrix equal- 2) diferentiate w.r.t. θ to find th e so-Called score Si^i 3) calculate varls(2%; θ)] or Elge log θ(z ity holds)

Answer 1

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