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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.