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, . . . , Xn} and let V = median{X1, . . . , Xn}. In the calculation of the median, assume that n is odd. Then the density of U and the density of V is given as
fU(u) = nf(u)F(u)n-1,
and the density of V is given as
fV(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 fU and fV when the density f is the Uniform[0, 1].
(b) Find fU 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 likleihood 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?
Attachment:- Assignment Files.rar