{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW3 - ORIE6700 Homework 3 Fall 2010 1 Prove that if T(X is...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ORIE6700 Homework 3 Fall 2010 1. Prove that if T (X ) is a suﬃcient statistic for θ, then the posterior density for θ depends on x only through T (x). 2. Suppose S is an unbiased estimator of θ and b = 0. Why is S + b inadmissible? 3. C & B 7.6 (a)-(b) on p. 355 4. C & B 7.7 on p. 355 5. C & B 7.10 on p. 356 1 1 1 6. Let X1 , . . . , Xn , be a sample from U (θ − 2 , θ + 1 ), the uniform distribution on (θ − 2 , θ + 2 ). Find an 2 MLE of θ. Is the MLE unique? 7. Let X1 , . . . , Xn , n ≥ 2, be iid with common density f (x|θ) = 1 exp{−(x − µ)/σ }, σ x ≥ µ, where θ = (µ, σ 2 ), −∞ < µ < ∞, σ 2 > 0. (a) Find the MLE of µ and σ 2 . (b) Say we know that µ = 0; give the MLE for σ 2 . Then ﬁnd the MLE of Pθ [X ∗ ≥ t] for t ≥ µ, where we still assume that µ = 0 and where X ∗ ∼ f (x|θ) is a hypothetical new observation (this is an example of prediction, where we use what we have learned from the ﬁrst n observations to predict the distribution of the next observation) 8. C & B 7.33 on p. 361 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online