Exercises

Monte Carlo methods.

1. The density f (x, y) of bi-dimensional vector (X, Y ) is inversely proportional to the 1 + x² + x²y² + y² on the square x, y ∈ [-2, 2], i.e.

f (x, y) = (c/(1 + x² + x²y² + y²)) , x, y ∈ [-2, 2].

1. Construct the algorithm for simulation of the vector (X, Y ). Plot the function f (x, y). Simulate 10000 points of (X, Y ) and construct the corresponding histogram. Compare the density profile with the histogram.

2. Using Monte Carlo method calculate the probability P(|X - Y| ≤ 0.5). Check empirically the convergence for the true value.

3. Using Monte Carlo method calculate the integral

I = ₀∫²dx _-2∫¹ dy.(xy/(1 + x² + x²y² + y²))

Check empirically the convergence.

2. Consider the problem of comparing means of two populations: let (x₁, . . . , x_n) i.i.d samples from X ∼ U [-1, 1] distribution and suppose (y₁, . . . , y_n) are i.i.d. samples from Y ∼ U [-1 + θ, 1 + θ], where θ is positive. Let n = 10.

1. In order to test the hypothesis H₀ : µ_x = µ_y versus H₁ : µ_x ≠ µ_y use the t-Student statistics (variances are equals):

t = (x¯ - y¯)/(σ_P√(2/n)), with σ_P = √(σˆ²_x(n - 1) + σˆ²_y(n - 1))/(2n -2)   

where

σˆ²_x = 1/(n-1)Σ_i=1ⁿ(x_i - x¯)², σˆ²_y = 1/(n-1)Σ_i=1ⁿ(y_i - y¯)²,

Using simulation experiment construct the curve the power of the test based on the points of θ: t=seq(0,1,by=0.05), considering the α = 0.05:

(a) for any θ_i ∈ t simulate N times samples (x₁, . . . , x₁₀)j, (y₁, . . . , y₁₀)j, j = 1, . . . , N . For example choose N = 1000;

(b) for any (x₁, . . . , x₁₀)j, (y₁, . . . , y₁₀)j calculate t_j statistics (1) and compare it with t_1-α/2,2n-2 quantile of t-Student distribution with 2n - 2 degree of freedom;

(c) accept the estimation of error of type II as M_i/N , where M_i is the number of how many time we accept the H₀, when H₀ is wrong (because θ_i > 0) i.e. how many |t_j| ≤ t_1-α/2,2n-2; thus, the power will be 1 - M_i/N.

2. Suggest and describe the bootstrap test hypothesis for the same problem and the same α = 0.05 (use the same statistics (1)). Using simulation experiment construct the curve of the power of the test based on the points of θ: t=seq(0,1,by=0.05), considering the α = 0.05:

(a) for any θi ∈ t simulate N times samples (x₁, . . . , x₁₀)j, (y₁, . . . , y₁₀)j, j = 1, . . . , N . For example choose N = 1000;

(b) for any (x₁, . . . , x₁₀)j, (y₁, . . . , y₁₀)j calculate t_j statistics (1) and then using the bootstrapping calculate pb-value: see slides how to estimate pb-value; here wee need to run the bootstrap; choose B as 1000, for example; remember, we accept H0 is pb-value is greater then α and reject H0 otherwise;

(c) accept the estimation of error of type II as M_i/N , where M_i is the number of how many time we accept the H₀, when H₀ is wrong (because θ_i > 0) i.e. how many |t_j| ≤ t_1-α/2,2n-2; thus, the power will be 1 - M_i/N.

Plot both power curves and compare.

3. Choose the instrumental function g(x) = e^-x and evaluate the integral by importance sampling:

I = ₀∫^∞ 1/(1 + e^x) dx.

1. Find the value of integral I.

2. Plot the convergence of importance sampling estimator Iˆ_n to the true value.

3. Is the variance of Iˆ_n finite?