EPFL Multivariate Statistics
Week 1
Exercise 1 Consider a bivariate Pareto density :
f(x, y) = c(x + y − 1)−p−2
, for x, y > 1, and p > 2.
1. Show that c is equal to p(p + 1).
2. Determine the marginal laws of this density and compute E [X].
3. Calculate the variance-covariance matrix Σ.
4. Consider a sample (X1, Y1)
0
, . . . ,(Xn, Yn)
0 of independent and identically distributed random vectors following
the Pareto density with parameters p. Estimate the parameter p using the maximum likelihood method.
Exercise 2 Let X1 and X2 be two independent Gamma random variables with common scale parameters :
X1 ∼ Gamma(α, λ) and X2 ∼ Gamma(β, λ). Define
Y1 = X1 + X2
Y2 =
X1
X1 + X2
1. Write the joint density of (X1, X2).
2. Determine the joint density of (Y1, Y2).
3. Deduce the marginal distributions of Y1 and Y2.
Exercise 3 Suppose that X1, . . . , Xn ∈ R
p are independent and identically distributed random vectors following a
multivariate Gaussian distribution Np(µ, Σ). We consider the sample mean
X¯ =
1
n
Xn
i=1
Xi
and the sample variance-covariance matrix
S =
1
n
Xn
i=1
Xi − X¯
Xi − X¯
0
1. Show that X¯ is an unbiased estimate of µ. (i.e. E
X¯
= µ).
2. Show that E [S] = n − 1
n
Σ. Propose another estimate of Σ which is not biased.
Exercise 4 We consider a matrix Σ ∈ R
p×p and we write
Σ =
Σ11 Σ12
Σ21 Σ22
and Σ
−1 = Ψ =
Ψ11 Ψ12
Ψ21 Ψ22
.
Show the following equations :
(a) Σ12Σ
−1
22 = −Ψ
−1
11 Ψ12
(b) Σ11 − Σ12Σ
−1
22 Σ21 = Ψ−1
11
