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distribution: 1.77, −0.23, 2.76, 3.80...

The following data are an i.i.d. sample from a Cauchy(θ, 1) distribution: 1.77, −0.23, 2.76, 3.80, 3.47, 56.75, −1.34, 4.24, −2.44, 3.29, 3.71, −2.40, 4.53, −0.07, −1.05, −13.87, −2.53, −1.75, 0.27, 43.21.a. Graph the log likelihood function. Find the MLE for θ using the Newton–Raphson method. Try all of the following starting points: −11, −1, 0, 1.5, 4, 4.7, 7, 8, and 38. Discuss your results. Is the mean of the data a good starting point?b. Apply the bisection method with starting points −1 and 1. Use additional runs to illustrate manners in which the bisection method may fail to find the global maximum.c. Apply fixed-point iterations as in (2.29), starting from −1, with scaling choices of α = 1, 0.64, and 0.25. Investigate other choices of starting values and scaling factors. d. From starting values of (θ (0) , θ (1) ) = (−2, −1), apply the secant method to estimate θ. What happens when (θ (0) , θ (1) ) = (−3, 3), and for other starting choices?e. Use this example to compare the speed and stability of the Newton–Raphson method, bisection, fixed-point iteration, and the secant method. Do your conclusions change when you apply the methods to a random sample of size 20 from a N(θ, 1) distribution?

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