HW11_Problem_Set

# HW11_Problem_Set - N ≥ K lottery tickets. There are...

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ORIE 3500/5500 Fall Term 2008 Assignment 11 Note that the due date is Friday, December 5 , at 1:00 pm you need to mention your section number on the top of your answer script. 1. [5 + 5 + 5 = 15 pts] If X 1 , . . . , X n are i.i.d. Uniform [ - θ, θ ], then ﬁnd the MLE of θ . Find its bias and MSE. 2. [5 + 5 = 10 pts] If X 1 , . . . , X n are i.i.d. Poisson ( λ ), ﬁnd the MLE for λ . Show that the MLE is consistent. 3. [10 pts] Suppose that T 1 , T 2 , T 3 are independent estimators of an unknown pa- rameter θ . If all of them are unbiased and var ( T 1 ) = 2 var ( T 2 ) = var ( T 3 ) / 2, which one among ( T 1 +2 T 2 + T 3 ) / 4 , (2 T 1 +3 T 2 + T 3 ) / 6 , ( T 1 + 3 T 2 + 4 T 3 ) / 8 should be preferred as an estimator of θ and why? 4. [5 + 5 = 10 pts] If X 1 , . . . , X n are i.i.d. Geometric ( p ), ﬁnd the ML estimator of p . Is the estimator unbiased? 5. [10 pts] Cornell University holds a Thanksgiving lottery at which students can win turkeys. Suppose that there are K turkeys and

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Unformatted text preview: N ≥ K lottery tickets. There are exactly K winning tickets. Tom wins x turkeys. Provide a ML estimator for the number of tickets that he purchased. 1 6. [(5 + 5) + 5 = 15 pts] There are N lottery tickets with numbers 1 , 2 , . . . , N . However, N is unknown. In an unobserved moment, Uncle Ezra takes a ticket at random and memorizes its number. He puts it back, and repeats this experiment n times. (a) Provide Uncle Ezra’s maximum-likelihood estimator T for N . Is the estimator unbiased? (Hint: If X is a random variable with values in N , then E ( X ) = ∑ k ≥ 1 P ( X ≥ k ).) (b) Calculate approximately for large N the normalized expected value E N ( T ) /N . (Hint: Think about what you have learned in calculus about Riemann sums.) 2...
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## This homework help was uploaded on 02/20/2009 for the course ORIE 3500 taught by Professor Weber during the Fall '08 term at Cornell.

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HW11_Problem_Set - N ≥ K lottery tickets. There are...

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