Midterm-2009 - Detection and Estimation Midterm 2009 1(35...

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Detection and Estimation Midterm, 2009 1. (35%) Assume we have observed N samples x[0], x[1], …, x[N-1], whose model is expressed as 1 - N ..., 1, , 0 ], [ ] [ = + = n n w A n x . where A is known and w[n] is white Poisson noise defined as = = otherwise 0 0,1,2,. .. w[n] ])! [ ( ]) [ ( ] [ n w e n w p n w λ (Remark: E{w[n]} = Var{w[n]} = ) ( Case 1 ) Assume we want to estimate the parameter λ based on {x[0], …, x[N-1]}. (a) Find the CRLB for λ . ( 5 % ) (b) Find the Minimum Variance Unbiased (MVU) Estimator for λ . (5%) (c) Find the minimal sufficient statistics for the estimation of λ . (5%) (d) Find the Maximum Likelihood Estimator (MLE) for λ . ( 5 % ) ( Case 2 ) Assume k = 1/ λ and we want to estimate k based on {x[0], …, x[N-1]}. (e) Find the Maximum Likelihood Estimator (MLE) for k. (5%) (f) F i n d t h e C R L B f o r k . ( 5 % ) (g) Is this MLE estimator an MVU estimator for k? Please state your reasoning. (5%) 2. (30%) Assume we have observed N samples x[0], x[1], …, x[N-1], whose model is
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Midterm-2009 - Detection and Estimation Midterm 2009 1(35...

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