1. Math
  2. Statistics And Probability
  3. please give detailed steps thank you...

Question: please give detailed steps thank you...

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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)b. Find the maximum likelihood estimator (MLE) of θ? Is the MLE the same as the method of moments (MOM) estimator of θ? Explain. Is this estimator consistent (no proof required, just explain why or why not)? Hint: MLE maximizes the sum of the individual log-likelihoods, i.e. it solves: 2-1 You have the log-likelihood of a random draw. I(xī.0) a2 (if you opted to solve it) log f(xi; θ from Part c. The skweness of the Poisson(0) distribution is given by: y Show that the estimator γ-+ is consistent for γ. - °Is this estimator unbiased? Can you tell the direction of its bias? Hint Apply Jensens inequality to find the direction of the bias

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