3714-exercise 5 - Lecturer: Andrew Baczkowski PDF

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Lecturer: Andrew Baczkowski...

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School of Mathematics MATH3714/5714: Linear Regression and Robustness Exercise 5

Q1 Section 7.4. For each of objective functions (least squares, Huber’s t-function, Hampel’s function and Tukey’s bisquare) draw a sketch of ρ, ψ and the weight function w. Q2 Section 7.4, 7.5. Using the data in http://www.maths.leeds.ac.uk/˜sta6ajb/math3714/math3714-15.rr use R to compute the Least Median of Squares (LMS) estimate, and the Least Trimmed Squares (LTS) estimate, with the latter evaluated for h = 14, . . . , 25. Compare these estimates with (a) each other (over all h (LTS) and vs LMS) (b) the ordinary least squares solution (c) the M-estimates already computed in Example 23 in section 7.4. You may also find useful commands in Example 24 of section 7.5 and the corresponding R file http://www.maths.leeds.ac.uk/˜sta6ajb/math3714/math3714-17.rr Q3 Section 7.4, 7.5. Using the data of Question 2, examine the final weights of each observation for each method used. Compare the final weights of each observation used for (a) LTS (with h = 23) vs bisquare (b) LMS vs Hampel Q4 Section 7.6. Carry out a simulation with 10,000 samples of size 100 from a standard normal distribution. For each sample, compute the mean and the median. What is the ratio of the standard deviation of the 10,000 sample means to the standard deviation of the 10,000 medians? Compare your answer to the theory....