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Statistical Computing and Simulation Spring 2006 Assignment 6, Due June 6/2006 1. Use the bisection, false positions, and secant methods to find the roots of: (a) f ( x) x3 2 x 2 x 1 ; e( x y ) (b) f ( x, y ) 2. 0.5 ; 1 x2 y2 (c) f ( x, y, z ) 4 1 / x 1 / y 1 / z 3 / xyz . You need to specify the starting points, convergence criterion, and number of iterations. Consider a multinomial observation X ( x1 , x2 , x3 , x4 ) with class probabilities given by ( p1 , p2 , p3 , p4 ) ( 2 1 1 , , , ) , where 0 1. The sample 4 4 4 4 size is n xi and the parameter is to be estimated from the observed frequencies (1997, 906, 904, 32), i.e., sample size 3839. Use the secant, Ridder’s (or Brent’s), and Newton-Raphosn methods to find the MLE (via l ' ( ) ). You may choose your own starting points and convergence criterion (preferred 3. 10 6 or smaller). Try at least three different methods to find the estimates of B and C for the Gompertz model, x BC x , x 0, given the data of Example 6 in class. You 4. 5. could count “nlminb” as one of the method (for replacing Newton’s method). Also, similar to what we saw in the class, discuss what is the influence of starting points to the number of iterations. Experiment with as many variance reduction techniques as you can think of to apply the problem of evaluating P( X 1) for X ~ Cauchy 1 Evaluate sin 2 (1 / x) dx by both numerical and Monte Carlo integration, and 0 compare their errors with respect to the numbers of observations used. Also, 6. propose at least three simulation methods to reduce the variance of Monte Carlo integration and compare their variances. First, simulate 100 observations from Beta(2,3) and then use 3 density estimating methods to smooth the observations. You need to specify the parameters in the smoothing methods, and compare the results. 7. Let x be 100 equally spaced points on [0,2] and let yi sin xi i with i ~ N (0,0.04) . Apply 3 linear smoothers and compare the differences.