2.1. The following data are an i.i.d. sample from a Cauchy(, 1) distribution: 1.77, 0.23, 2.76, 3.80,

2.1. 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?