2. Suppose now X₁,..., Xn are iid exponential random variables with mean 0, so the common PDF is, for x,00, given by 1 = e-x/0 0 fo(x): (a) Determine the maximum likelihood estimator ML (X). (b) Determine the Bayes estimator flat (X) under squared-error loss using the weight function w(0) 1 (the "flat prior"). (c) Determine the Bayes estimator conj (X) under squared-error loss using the conjugate prior w (0): = for > 0. (d) Determine the risk R(0|d) of the estimator d(X)= = 100e-A0/0 fao+¹(ao)' l + Σ₁=1 Xi n+k under squared-error loss and hence also determine the limiting (rescaled) risk limn→ nR(Old). (e) Determine the risk R(0|d) and limiting (rescaled) risk limn→∞ nR(0|d) where d is replaced by each of the 3 estimators in the questions (a)-(c) above.
2. Suppose now X₁,..., Xn are iid exponential random variables with mean 0, so the common PDF is, for x,00, given by 1 = e-x/0 0 fo(x): (a) Determine the maximum likelihood estimator ML (X). (b) Determine the Bayes estimator flat (X) under squared-error loss using the weight function w(0) 1 (the "flat prior"). (c) Determine the Bayes estimator conj (X) under squared-error loss using the conjugate prior w (0): = for > 0. (d) Determine the risk R(0|d) of the estimator d(X)= = 100e-A0/0 fao+¹(ao)' l + Σ₁=1 Xi n+k under squared-error loss and hence also determine the limiting (rescaled) risk limn→ nR(Old). (e) Determine the risk R(0|d) and limiting (rescaled) risk limn→∞ nR(0|d) where d is replaced by each of the 3 estimators in the questions (a)-(c) above.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:2. Suppose now X₁,..., Xn are iid exponential random variables with mean 0, so the common PDF
is, for x, > 0, given by
fo(x) = -1/e-x/0
(a) Determine the maximum likelihood estimator ML (X).
(b)
Determine the Bayes estimator flat (X) under squared-error loss using the weight function
w(0) = 1 (the "flat prior").
(c) Determine the Bayes estimator conj(X) under squared-error loss using the conjugate prior
w(0)
for 0 > 0.
(d) Determine the risk R(0|d) of the estimator
10⁰e-10/0
Aαo+¹(ao)'
l + Ei=₁ Xi
n+k
d(X)
under squared-error loss and hence also determine the limiting (rescaled) risk limn→∞ nR(0|d).
(e) Determine the risk R(0|d) and limiting (rescaled) risk limn→∞ nR(0|d) where d is replaced
by each of the 3 estimators in the questions (a)-(c) above.
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