SIMULATION AND THE MONTE CARLO METHOD ASSIGNMENT

Suppose that I draw a sample of size n IID from a density f with cumulative distribution function F. Let U = max{X₁, . . . , X_n} and let V = median {X₁, . . . , X_n}. In the calculation of the median, assume that n is odd. Then the density of U and the density of V is given as

f_U(u) = nf(u)F(u)^n-1,

and the density of V is given as

f_V(v) = n!/((n-1/2)!(n-1/2)!)(f(v)F(v)^(n-1)/2(1 - F(v))^(n-1)/2.

Note that the support is driven by the original density f.

(a) Find f_U and f_V when the density f is the Uniform [0, 1].

(b) Find f_U when the density f is the Uniform [0, θ].

(c) Draw a sample of size n = 25 from the Uniform [0, 1] distribution and calculate the statistic t to be the median of the sample. Use the nonparametric bootstrap to find the bootstrap distribution of t with B = 10000. Plot a histogram of the result, and compare this with the sampling distribution of t (superimpose it on the histogram). What do you notice?

(d) Draw a sample of size n = 25 from the Uniform [0, 1] distribution and calculate the statistic t to be the maximum of the sample. Use the nonparametric bootstrap to find the bootstrap distribution of t with B = 10000. Plot a histogram of the result, and compare this with the sampling distribution of t (superimpose it on the histogram). What do you notice?

(e) Suppose that you draw a sample from a Uniform [0, θ] distribution. Let the maximum of the sample be your estimate of the unknown θ (in fact, this is the maximum likelihood estimator).

Draw a sample of size n = 25 from the Uniform [0, θ] distribution and calculate the statistic t to be the maximum of the sample. Use θ = 1; but this will be the "unknown" parameter (like the mean and the standard deviation when we did the normal examples before).

Use the parametric bootstrap to find the bootstrap distribution of t with B = 10000. Plot a histogram of the result, and compare this with the sampling distribution of t (superimpose it on the histogram). What do you notice?

The course is called Simulation and Monte Carlo methods using R.

Attachment:- Assignment Files.rar