MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS
Problems on Estimation
1. Suppose {(ui , vi ), i = 1, 2, . . . , n} is a random sample from
f (u, v) =
ua+b−1 v −(1+b) (1 − v)b−1 exp {−u/(λv)}
.
λa+b Γ (a) Γ (b)
This distribution is due to Nadarajah (2009). A bivariate distribution with gamma and beta
marginals with application to drought data. Journal of Applied Statistics, Volume 36, Issue
3, pages 277-301.
If a and b are assumed known find the mle of λ.
2. Show that the loglikelihood of a random sample (x1 , y1 ), . . . , (xn , yn ) from
f (x, y) = c (x + y − 1)−(p+2) ,
x > 1, y > 1
(where p > 0) is given by
n−1 log L = log p(p + 1) − (p + 2)ᾱ,
where
nᾱ =
n
X
log (xi + yi − 1) .
i=1
Show further that the maximum likelihood estimator of p is given by
1
1
p̂ = − +
ᾱ 2
r
1
1
+ .
ᾱ2 4
3. Find the maximum likelihood estimates of the 2 × 1 mean vector µ and the 2 × 2 covariance
matrix Σ based on the random sample
"
X =
3 4 5 4
6 4 7 7
#
from a bivariate normal population.
4. Let X1 , . . . , X20 be a random sample of size n = 20 from an N6 (µ, Σ) population. Specify
each of the following completely.
(i) the distribution of (X1 − µ)T Σ−1 (X1 − µ).
√
(ii) the distribution of X̄ and n(X̄ − µ).
(iii) the distribution of (n − 1)S.
5. Let X1 , . . . , X75 be a random sample of size n = 75 from a population distribution with mean
µ and covariance matrix Σ. Specify the approximate distribution of the following completely.
(i) X̄.
1
(ii) (X̄ − µ)T S−1 (X̄ − µ).
6. Show that
n
X
(xj − x̄) (x̄ − µ)T
j=1
and
n
X
(x̄ − µ) (xj − x̄)T
j=1
are both p × p matrices of zeros.
7. For a random sample x1 , . . . , xn from the joint pdf


p−1
X

i=0
f (x) = λ1−p exp −


p−i
(xi+1 − xi ) ,

λ
xi > 0
find the mle of λ.
8. A genetic example presented by Fisher (1970) has four outcomes with probabilities (2 + θ)/4,
(1 − θ)/4, (1 − θ)/4 and θ/4, respectively. If in n trials one observes xi results for outcome i,
P
where xi = n, then x = (x1 , x2 , x3 , x4 )T has a multinomial distribution. Find the mle of θ.
2
T
9. If x is distributed as N√
p (µ, σ I), where µ is known to lie on the unit sphere µ µ = 1 show
that the mle of µ is x/ xT x.
10. Suppose x1 , . . . , xn constitute a sample from Np (µ, Σ). If µ = kµ0 with both µ0 and Σ
known then find the mle of k. Show that it is unbiased and find its variance.
11. Suppose x1 , . . . , xn constitute a sample from Np (µ, Σ). If Rµ = r with R, r and Σ known
then find the mle of µ.
12. Suppose x1 , . . . , xn constitute a sample from Np (µ, Σ). If Σ = kΣ0 with both µ and Σ0
known then find the mle of k.
13. Suppose x1 , . . . , xn constitute a sample from Np (µ, Σ). If Σ = kΣ0 with Σ0 known but µ
unknown then find the mle of k and µ.
14. If A and Σ be partitioned into q and p − q rows and columns
A=
A11 A12
A21 A22
Σ=
Σ11 Σ12
Σ21 Σ22
!
and
!
.
If A is distributed according to Wn (· | Σ) then show that A11 is distributed according to
Wn (· | Σ11 ).
2
15. If A and Σ be partitioned into p1 , . . . , pq rows (p1 + · · · + pq = p) and columns
A11 · · · A1q
 ..
.. 
A= .
. 
Aq1 · · · Aqq


and
Σ11 · · · Σ1q
 ..
..  .
Σ= .
. 
Σq1 · · · Σqq


If Σij = 0 for i 6= j and if A is distributed according to Wn (· | Σ) then show A11 , · · · , Aqq
are independently distributed according to Wn (· | Σjj ).
16. Consider (x1 , x2 , . . . , xp ) is from Weinman’s p-variate exponential distribution with the joint
pdf

p−1
Y
f (x) = 



p−1
X

i=0
 exp −
λ−1
i
i=0

xi+1 − xi 
(p − i)
.

λi
Find the mles of λi , i = 0, 1, . . . , p − 1.
17. Suppose x = (x1 , x2 , . . . , xp ) has the joint pdf
"
Γ((n + p)/2)
(x − µ)T Σ−1 (x − µ)
f (x) =
1
+
n
Γ(n/2)(nπ)p/2 | Σ |1/2
#−(n+p)/2
.
This is the pdf the multivariate t distribution - see Kotz and Nadarajah (2004). Multivariate
t Distributions and Their Applications. Cambridge University Press.
Find the mle of µ assuming Σ is known.
18. In problem 17, find the mle of k if Σ = kΣ0 and µ and Σ0 are assumed known.
19. In problem 17, find the mle of k if µ = kµ0 and µ0 and Σ are assumed known.
3