Week_4_Problems - Week 4 Problems 1. The Pareto...

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Week 4 Problems 1. The Pareto distribution is often used to describe data such as income which have a long right tail. Suppose we can model 12 , ,..., n y yy by n independent copies of the random variable Y with probability density function 1 (:) , 1 f y θ + = > where 0 > is a parameter to be estimated. a) Verify that the integral from 1 to of the pdf is 1. b) Find the MLE of in terms of , ,..., n y c) Find the relative likelihood function () R d) The median α is defined by ( ) 0.5 PY = . Find the MLE of . 2. The following data (available in the file q2week4. txt) can be described by the model ~ ( , ), 1,. ..,20, independent ii YGx i β σ = x y x y 0.64 0.96 0.08 0.10 0.94 1.38 0.01 -0.03 0.29 0.08 0.49 0.78 0.08 -0.05 0.90 1.27 0.41 0.60 0.06 -0.07 0.38 0.46 0.13 -0.01 0.42 0.54 0.40 0.50 0.82 1.27 0.87 1.26 0.30 0.72 0.23 0.09 0.77 0.85 0.94 1.21 a) Interpret and . b) Find algebraic expressions for the MLEs of , βσ in terms of the ( , ) x y c) Is the MLE of the same as for the simple linear regression model discussed in class? 3. In a test to compare the usability of two software programs A and B, 12 employees are selected and divided at random into two groups of 6, one group for each program. A series of tasks is constructed and the time in minutes to
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This note was uploaded on 03/20/2011 for the course STATISTICS 231 taught by Professor Mckay during the Spring '10 term at Waterloo.

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Week_4_Problems - Week 4 Problems 1. The Pareto...

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