1
AMA531
Tutorial 1
1. Consider the inverse Gaussian distribution with density given by
(
1/2
2 )
θ
θ x−µ
f (x) =
exp −
, x > 0.
2πx3
2x
µ
(a) Show that
n
X
(xi − µ)2
i=1
where x̄ = (1/n)
Pn
i=1
xi
=µ
2
n X
1
i=1
1
−
xi x̄
+
n
(x̄ − µ)2
x̄
xi .
(b) For a random sample at values (x1 , . . . , xn ), show that the maximium likelihood
estimates of µ and θ are respectively
µ̂ = x̄
and
n
θ̂ = P n
1
i=1
xi
−
1
x̄
.
2. Suppose that X1 , . . . , Xn are independent and normally distributed with mean E(Xi ) =
µ and V ar(Xi ) = (θmi )−1 , where mi ’s are known positive constant. Show that the
maximum likelihood estimators of µ and θ are
µ̂ = X̄
and
θ̂ = n
( n
X
)−1
mi (Xi − X̄)2
,
i=1
respectively, where X̄ = (1/m)
Pn
i=1
mi Xi and m =
Pn
i=1
mi .
3. Suppose X1 , . . . , Xn are i.i.d. with a distribution from the linear exponential family,
with density
p(x)er(θ)x
.
f (x) =
q(θ)
Prove that the maximum likelihood estimator of the mean is the sample mean. In
other words, if θ̂ is the MLE of θ, then
d = µ(θ̂) = X̄.
µ(θ)
(µ(θ) = E(X) =
q 0 (θ)
.)
r0 (θ)q(θ)