PS4 10 - t that produces an unbiased estimate of . 2. Show...

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University of Minnesota Dept. of Electrical and Computer Engineering EE 8581 DETECTION AND ESTIMATION THEORY Spring 2010 Problem Set 4 Assigned: February 23, 2010 Due: March 2, 2010 Readings : Read Levy Sections 5.1- 5.2, Chapter 4. Problems : Solve problems 4.10 and 4.14 in Chapter 4 of Levy’s book and the following problem: Problem 3: We discussed in class the notion of a minimal sufficient statistic. Recall that t(r) is a sufficient statistic for the parameter θ if the density of r given t is independent of . t is minimal if it is a function of all other sufficient statistics. 1. We shall say that a sufficient statistic is complete if E( W(t) | ) = 0 for all implies that W(t) =0 for all with probability 1. Show that there is only one function of a complete sufficient statistic
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Unformatted text preview: t that produces an unbiased estimate of . 2. Show that a complete statistic is minimal. 3. Assume that in a given problem n independent vector samples r m , m = 1, 2, … , n , are available for processing. Each vector has a distribution of the form ? ( ± ² | ³ ) = ´ ( ³ ) µ ( ± ² )exp ⁡ ( ∑ ? ¶ ( ³ ) · ¶ ( ± ² )). ¸ ¶ =1 The functions p i ( ) and t i ( r m ) are scalar functions of θ and r m respectively. Show that the statistic T( r ) = [T 1 ( r ) T 2 ( r ) … T k ( r )] where ¹ ¶ ( ± ) = ∑ · ¶ ( º » ) ¼ ² =1 is sufficient and complete for . Note that the dimension of the statistic is k and is independent of the number n of vector samples available for processing....
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This note was uploaded on 06/04/2010 for the course EE 8581 taught by Professor Staff during the Spring '08 term at Minnesota.

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