The individual claim amount random variable, X, follows a gamma distribution with
PDF:
f(x) =
1
\gamma \beta
4
x
3e
−
x
\beta x > 0
(i) State the mean and variance of this distribution.
[1]
(ii) The maximum likelihood estimator of \beta (based on a random sample X1
, ... , Xn) is:
A
∑Xi
4
B
4n
∑Xi
C XÌ…
D
XÌ…
4
[3]
The second derivative of the log-likelihood with respect to \beta is:
d
2
d\beta
2
ln L(\beta ) =
4n\beta − 2∑xi
\beta
3
(iii) The Cramer Rao lower bound for estimators of is:
A
4n
\beta 2
B
\beta
2
4n
C
4n−8
\beta 2
D
\beta
2
4n−8
[3]
The last 10 claims totalled £17,885.
(iv) Calculate an approximate 95% confidence interval for the true value of \beta using
the asymptotic distribution of the maximum likelihood estimator of \beta .
[2]
A confidence interval for \beta can also be obtained using the result
2
\beta
∑Xi~\chi 8n
2
n
i=1
where X1
, ... , Xn is a random sample of claims.
(v) An exact 95% confidence interval for \beta is:
A (304.8, 693.1)
B (320.2,659.5)
C (335.6,625.9)
D (351.0,592.3)