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A3-433-2011-W

Course: ACTSC 433, Winter 2012
School: Waterloo
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Word Count: 270

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3 Assignment ACTSC 433/833, Winter 2012 (Due at the beginning of the class on March 1) 1. You are given the following ages at time of death for 10 individuals: 38, 45, 52, 52, 56, 59, 68, 70, 74, and 82. Determine the kernel density estimate for the probability that an individual will survive at least to age 60 using (a) the uniform kernel with bandwidth b = 5; [5 marks] (b) and the uniform kernel with...

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3 Assignment ACTSC 433/833, Winter 2012 (Due at the beginning of the class on March 1) 1. You are given the following ages at time of death for 10 individuals: 38, 45, 52, 52, 56, 59, 68, 70, 74, and 82. Determine the kernel density estimate for the probability that an individual will survive at least to age 60 using (a) the uniform kernel with bandwidth b = 5; [5 marks] (b) and the uniform kernel with bandwidth b = 15. [5 marks] 2. You are given the following loss amounts for 5 insurance portfolios: 120, 190, 241, 442, and 578. Determine the kernel density estimate for the probability that the loss in a portfolio will exceed 220 using (a) the triangular kernel with bandwidth b = 75; [5 marks] (b) and the triangular kernel with bandwidth b = 100. [5 marks] 3. Let X1 , ..., Xn (n > 2) be a random sample from the exponential density function f (x) = ex , x > 0, > 0. (a) Determine the MLE for . (b) Show that MLE the is consistent for . [3 marks] [7 marks] 4. Let X1 , ..., Xn be a random sample from the normal distribution N (, 2 ), where the parameters < < and > 0. (a) Suppose that the parameter is known. i. Determine the MLE 2 for 2 . [3 marks] ii. Show that the MLE 2 is consistent for 2 . [5 marks] (b) Suppose that both the parameters and 2 are unknown. i. Determine the MLE for and the MLE 2 for 2 . [6 marks] ii. Verify if the MLE is consistent for . [4 marks] 2 2 iii. Verify if the MLE is consistent for . [7 marks] 5. Let X1 , ..., Xn be a random sample from the uniform distribution U ( 1, + 1), where the parameter > 1. (a) Determine the MLE of . (b) Verify if X(1) +X(n) 2 [3 marks] is an MLE for , where X(1) = min{X1 , ..., Xn } and X(n) = max{X1 , ..., Xn }. (c) Verify if X(1) +X(n) 2 [2 marks] is a consistent estimator of . [10 marks] [Total: 70 marks] 1
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