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Let X1,..., X, be a random sample (i.i.d.) following Gamma(2, 3) for some unknown pa- rameter 3 > 0. (a) What is the MLE of 3? Derive it. Is the MLE of ß a consistent estimator for B? Prove/disprove it. (b) Hint: You may need continuous mapping theorem for convergence in probability. (c) ) What is the Method of Moments estimator of 3? Gamma(a, b) (d) for some a, b > 0. Derive the posterior distribution of 3 given (X1,... , X„) = (x1, ..., xn). Hint: For Gamma likelihood with known a (= 2) and unknown 3 (as is our case), the Now let's think like a Bayesian. Consider a prior distribution of 3 ~ %3| conjugate prior on 3 is a Gamma distribution. (e) of 3 assuming squared error loss? Using the same prior as in the previous sub-question, what is the Bayes estimator

Let X1,..., X, be a random sample (i.i.d.) following Gamma(2, 3) for some unknown pa- rameter 3 > 0. (a) What is the MLE of 3? Derive it. Is the MLE of ß a consistent estimator for B? Prove/disprove it. (b) Hint: You may need continuous mapping theorem for convergence in probability. (c) ) What is the Method of Moments estimator of 3? Gamma(a, b) (d) for some a, b > 0. Derive the posterior distribution of 3 given (X1,... , X„) = (x1, ..., xn). Hint: For Gamma likelihood with known a (= 2) and unknown 3 (as is our case), the Now let's think like a Bayesian. Consider a prior distribution of 3 ~ %3| conjugate prior on 3 is a Gamma distribution. (e) of 3 assuming squared error loss? Using the same prior as in the previous sub-question, what is the Bayes estimator

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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Let X1,..., Xn be a random sample (i.i.d.) following Gamma(2, 3) for some unknown pa- rameter 3 > 0. (а) What is the MLE of 3? Derive it. Is the MLE of Ba consistent estimator for 3? Prove/disprove it. (b) Hint: You may need continuous mapping theorem for convergence in probability. (c) ( What is the Method of Moments estimator of 3? (d) Gamma(a, b) Now let's think like a Bayesian. Consider a prior distribution of 3 for some a, b > 0. Derive the posterior distribution of 3 given (X1, . Hint: For Gamma likelihood with known a (= 2) and unknown 3 (as is our case), the X„) = (x1,. .. , xn). .... conjugate prior on 3 is a Gamma distribution. (e) of 3 assuming squared error loss? Using the same prior as in the previous sub-question, what is the Bayes estimator
Transcribed Image Text:Let X1,..., Xn be a random sample (i.i.d.) following Gamma(2, 3) for some unknown pa- rameter 3 > 0. (а) What is the MLE of 3? Derive it. Is the MLE of Ba consistent estimator for 3? Prove/disprove it. (b) Hint: You may need continuous mapping theorem for convergence in probability. (c) ( What is the Method of Moments estimator of 3? (d) Gamma(a, b) Now let's think like a Bayesian. Consider a prior distribution of 3 for some a, b > 0. Derive the posterior distribution of 3 given (X1, . Hint: For Gamma likelihood with known a (= 2) and unknown 3 (as is our case), the X„) = (x1,. .. , xn). .... conjugate prior on 3 is a Gamma distribution. (e) of 3 assuming squared error loss? Using the same prior as in the previous sub-question, what is the Bayes estimator
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