[Solved] AMS 597-Homework 7

AMS597

1. Estimate the bias, standard error, standard normal bootstrap CI, basic bootstrap CI and bootstrap t-interval at 95% of the sample median for x where

> set.seed(123)

> x <- rnorm(50)

2. Another test statistic for two group comparison, i.e., H₀ : F = G vs H_a : F 6= G in the univariate case is the Cramer-von Mises statistic

where F_n is the ecdf of x₁,,x_n and G_m is the ecdf of y₁,,y_m.

Implement the two group comparison using Cramer-von Mises test statistic but obtaining the p-value from permutation test.

3. Implement the bivariate Spearman rank correlation test as a permutation test. Compare the achieved significance level of the permutation test with the p-value reported by test on the following samples:

> set.seed(123)

> x <- rnorm(50)

> y <- 0.2*x+rnorm(50)

4. (a) Derive the Newton-Raphson iteration for solving

e^2x = x + 6

Implement your iteration as a R function and report the solution to the equation.

(b) Compare your answer to the solution using uniroot function (i.e., Brents method).

5. (a) Let x₁,,x_n be a random sample from a Poisson(). Find the maximum likelihood estimator of using the function optimize in R. Apply this to

set.seed(123) lambda.true <- 10 x <- rpois(1000,lambda.true)

(b) Let x₁,,x_n be a random sample from a Beta(a,b). Find the maximum likelihood estimator of = (a,b) using the function optim and optimx in R. Apply this to

set.seed(123) a.true <- 5

b.true <- 2

x <- rbeta(1000,shape1=a.true,shape2=b.true)