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100BHW3 - STAT 100B Homework III Problem 1 Please derive...

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STAT 100B Homework III Problem 1. Please derive the exponential distribution and interpret the meaning of the param- eter λ . Calculate E ( X ), E ( X 2 ), and Pr( X > t ). Problem 2. Continue. Let X 1 , X 2 , ..., X n be independent observations from Exp( λ ). Derive the estimators of λ from the following estimating equations: (1) E λ ( X ) = ( X 1 + ... + X n ) /n . (2) E λ ( X 2 ) = ( X 2 1 + ... + X 2 n ) /n . (3) Pr( X > t ) = n i =1 1 X i >t /n for fixed t (let’s use t = 1). Suppose X 1 , X 2 , ..., X n are 0.5399 1.4874 1.5651 0.5659 0.8786 0.0296 1.5420 0.8868 0.3808 1.5744, where n = 10. Please calculate the estimates ˆ λ using the above three estimators. Problem 3. Continue. I generate 5 data sets from Exp(1), each data set consists of 10 observa- tions: (1) 0.5399 1.4874 1.5651 0.5659 0.8786 0.0296 1.5420 0.8868 0.3808 1.5744 (2) 0.8592 0.5451 0.9681 0.2304 1.1874 0.0100 0.4408 0.2950 1.5485 0.4989 (3) 0.6626 0.2740 0.2442 2.8271 0.1343 0.2372 1.1393 1.3170 0.1753 0.4622 (4) 1.0968 0.6352 0.3844 0.5061 4.1991 0.8240 0.0407 0.8212 0.4640 0.9930 (5) 0.8372 0.4455 0.7742 2.9904 0.2640 0.6965 0.3193 0.0689 2.0116 0.5531 For each data set, calculate ˆ λ using the three estimators in Problem 2. For each estimator, let ˆ λ 1 , ..., ˆ λ 5 be the estimates calculated from the five data sets respectively. Please calculate 5 k =1 ( ˆ λ k - 1) 2 / 5 for each estimator. Which estimator has the best performance?
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