AMS597

1. Using only random uniform generator, generate a random sample of size 1000 from the F distribution with 5 and 10 degrees of freedom. (Hint: If X ²_a and Y ²_b, X and Y are independent, then follows F distribution with a and b degrees of freedom.)
2. One disadvantage of Box-Muller algorithm is that it requires computing sine and cosine functions. By some transformation, one can show that the following closely related algorithm also generates independent standard normal random variables X and Y .
   • Generate U₁ U(0,1) and U₂ U(0,1)
   • Set V₁ = 2U₁ 1 and.
   • If S > 1, return to (a).
   • Otherwise,

This method is called the polar method. Use this method to generate 10000 standard normal random variables.

3. A t-distributed random variable of k degrees of freedom is defined as

where Z N(0,1) and W ²_k, Z and W are independent. Using only the uniform number generator in R (runif), generate 100 random samples X, where X is a mixture t distribution, i.e., X 0.3t₃ + 0.35t₅ + 0.35t₇.

4. Write your function rmultivarNorm(n,mu,Sigma) which will generate n multivariate normal random variables using only runif.
5. Write your own function ls() which performs the least square estimation for a continuous response variable y regressed on two predictors x₁ which is a numeric predictor and x₂ which is a categorical predictor. You may assume that your model contains an intercept. Test your function on the ChickWeight dataset in R, where y is weight, x₁ is

Time (assumed to be continuous) and x₂ is Diet, i.e., check if your function can reproduce the estimated beta coefficients from

data(ChickWeight) fit <- lm(ChickWeight$weight~ChickWeight$Time+ChickWeight$Diet) fit$coef

6. Use the mcycle data in the MASS package for this problem (see Homework 3 Qn 5).

Instead of fitting polynomial regression, fit an additive model using splines to this dataset.

Compare the MSE of the spline model to the polynomial regression model, where MSE = is the observed response variable and Y_i is the predicted response

variable.