Exercise 1 (8 marks) Let Φ be the cumulative distribution function (cdf) of the standard normal distribution. The cdf of a two-parameters Birnbaum-Saunders random variable T is of the form:
where α, β > 0 are the shape and scale parameters, respectively. Such a random variable will be denoted by T ∼ BS(α, β).
1. Let f (t | α, β) be the probability density function (pdf) of T. Calculate f (t | α, β) and show that the quantile function of T isβ [ − √ 2]2
Plot the functions F, Q and f, for various values of (α, β). What is the shape of the Birnbaum- Saunders distribution when the shape parameter increases to ∞?
2. Let Z be a random variable defined by
Show that Z is a standard normal random variable. Give the distribution of the random variable
Show the relation
T = βψ(X) where ψ(x) := (x + √1 + x2)2 .
Use this transformation to generate 100 values from BS(2, 0.5) and BS(2, 1) distributions.
3. Calculate the mean and variance of T.
4. Show that if T ∼ BS(α, β) then 1 ∼ BS (α, 1 ) and use this result to generate 100 values from
BS(2, 2). Calculate the mean and variance of T−1.
5. Use a sample of 1000 values from BS(1.5, 10) to illustrate how the maximum likelihood method can be used to estimate the parameters of the Birnbaum-Saunders distribution. Plot the associated log-likelihood function and find the MLE of the parameters. Evaluate the efficiency in term of bias and variance of these MLE through 300 simulations. Illustrate the asymptotic normality of the obtained MLE.
Exercise 2 (8 marks) A random variable X follows an inverse Gaussian distribution with parame- ters µ > 0 and λ > 0 and we note X ∼ IG(µ, λ), if it has the pdf
f (x|µ, λ) = λ exp 2πx3
λ(x − µ)2 2µ2x, x > 0.
Let X1 and X2 two random variables such that
X ∼ IG(µ, λ) and 1 ∼ IG ( 1 , λ ) .
Denote by f1 and f2 the pdf of X1 and X2 respectively. Let Y be the mixture of X1 and X2 whose pdf is given by
fY (y) = 2 f1(y) + 2 f2(y).
1. Show that f2(x) = x f1(x), x > 0 and plot the function fY .
2. Show that when µ = β and λ = β , then Y ∼ BS(α, β).
3. Use the Accept-Reject algorithm to simulate a sample of 100 copies of X2, with instrumental density f1.
Hint: In order to be able to use the Accept-Reject algorithm, it is necessary do discretize the support of the densities in the sense that the equality obtained in question 2.1 is valid for x ∈
{x1, . . . , xN }, where the xi are generated following f1 and N is sufficiently large. Additionally,
clearly explain why the Accept-Reject algorithm is not directly applicable in this case.
4. Generate 100 realizations of the random variable Y.
Exercise 3 (4 marks) Read the paper [3] and :
1. Summarize the methodology (mainly Section 3.1).
2. Redo the simulations in section 4 : generate the model and calculate the MLEs to find the results reported in Tables 1 et 2 (columns MLE).
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