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Stat 411 – Homework 02 Due: Friday 01/27 1. Let X 1 ,...,X n be iid Unif (0 ,θ ), where θ > 0 is unknown. (a) Let ˆ θ n = X ( n ) , the sample maximum. Find the CDF of ˆ θ n . (Hint: This is similar to Problem #2 on Homework 01.) (b) Show that ˆ θ n is a consistent estimator of θ . (Hint: θ is always greater than ˆ θ n (why?); this makes it easy to use part (a) above.) 2. Same setup as in Problem #1 above. (a) Show that ˆ θ n is not an unbiased estimator of θ . (Hint: Get the PDF of ˆ θ n by diﬀerentiating the CDF in Problem #1a above, then calculate expected value as usual. Alternatively, see the hint to Problem #4b below.) (b) Use the work in part (a) to construct a new estimator ˜ θ n , a linear function ˆ θ n , which is an unbiased estimator of θ . Show that ˜ θ n is also consistent. 3. Suppose X ∼ Bin ( n = 10 ,θ ) where θ ∈ (0 , 1) is unknown. Consider the following two estimators of θ : ˆ θ 1 = X/ 10 and ˆ θ 2 = ( X + 1) / 12.
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.
- Spring '08