Georgia Institute Of Technology

MATH 4261

Problem 1
X1; : : : ; Xn is an i.i.d. sample from Gamma distribution with the p.d.f of Xi:
f(x; α; β) =
8><>:
1
Γ (α)βα xα-1e-x=β; x > 0;
0 otherwise:
Let β = θ > 0 be unknown parameter and α = 4.
(a) (1 point) Find MLE θ ^ of θ;
(b) (1 point) Check if θ ^ is unbiased;
(c)* (1 point) Check if θ ^ is consistent (Hint: recall the sufficient conditions for consistency that
involve Var(θ ^), discussed on Nov. 17th lecture.)
(d) (1 point) What is the asymptotic distribution of pn(θ ^- θ)? (Hint: apply the CLT.)
Solution (for general α > 0):
(a) The log-likelihood function for the considered model and its derivative read as follows
‘(θ) = -n log(Γ (α)) - nα log(θ) - 1
θ
nX i
=1
Xi + (α - 1)
nX i
=1
log(Xi);
@
@θ ‘(θ) = -nα θ + θ1 2 X n
i=1
Xi:
Solving the equation @θ @ ‘(θ) = 0 w.r.t. θ > 0, we find the MLE

1
nX i
=1
(b) Using that Eθ(Xi) = αθ, we obtain
Eθ(θ ^) = θ;
hence θ ^ is unbiased estimate of parameter θ.
(c) By the i.i.d. property of the random sample,
Varθ(θ ^) = 1
α2n Varθ(X1) ! 0; n ! +1;
hence using the sufficient conditions for consistency of a point estimate (or Chebyshev’s inequality),
we conclude that
θ ^ = θ ^n ! P θ; n ! +1;
i.e. θ ^ is a consistent estimate of parameter θ.
(d) The CLT and the properties Eθ(Xi) = αθ, Varθ(Xi) = αθ2 imply
θ ^- Eθ(θ ^)
qVarθ(θ ^) =
pn(θ ^- θ)
pθ2=α
d
! N (0; 1); n ! +1;
Page 1 of 4