12. A smoothing spline is to be fit to the points (0, 3), (1, 2), and (3, 6).
The candidate function is:
( )
( ) ( )
( )( ) ( ) ( )( )
3
2 3
2.6 4/15 4/15 , 0 1
2.6 8/15 1 0.8 1 2/15 1 1 3
x x x
f x
x x x x
⎧ = ⎪⎨ − + ≤ ≤
+ − + − − − ≤ ≤ ⎪⎩
Determine the value of S, the squared norm smoothness criterion.
(A) Less than 2.35
(B) At least 2.35, but less than 2.50
(C) At least 2.50, but less than 2.65
(D) At least 2.65, but less than 2.80
(E) At least 2.80
Exam C: Fall 2005 -13- GO ON TO NEXT PAGE
13. You are given the following about a Cox proportional hazards model for mortality:
(i) There are two covariates: : z1 = 1 for smoker and 0 for non-smoker, and z2 =1 for
male and 0 for female.
(ii) The parameter estimates are 1
βˆ = 0.05 and 2
βˆ = 0.15 .
(iii) The covariance matrix of the parameter estimates, 1
ˆβ
and 2
ˆβ
, is:
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
0.0001 0.0003
0.0002 0.0001
Determine the upper limit of the 95% confidence interval for the relative risk of a female
non-smoker compared to a male smoker.
(A) Less than 0.6
(B) At least 0.6, but less than 0.8
(C) At least 0.8, but less than 1.0
(D) At least 1.0, but less than 1.2
(E) At least 1.2
Exam C: Fall 2005 -14- GO ON TO NEXT PAGE
14. You are given:
(i) Fifty claims have been observed from a lognormal distribution with unknown
parameters μ and σ .
(ii) The maximum likelihood estimates are 􀀃 μ = 6.84 and σ􀀃 = 1.49 .
(iii) The covariance matrix ofμˆ and σˆ is:
0 0444 0
0 0 0222
.
.
LN M
OQ P
(iv) The partial derivatives of the lognormal cumulative distribution function are:
F φ (z)
μ σ
∂ −
=
∂
and
∂
∂
=
F −z× z
σ
φ
σ
b g
(v) An approximate 95% confidence interval for the probability that the next claim
will be less than or equal to 5000 is:
[PL, PH]
Determine PL.
(A) 0.73
(B) 0.76
(C) 0.79
(D) 0.82
(E) 0.85
Exam C: Fall 2005 -15- GO ON TO NEXT PAGE
15. For a particular policy, the conditional probability of the annual number of claims given
Θ = θ , and the probability distribution of Θ are as follows:
Number of Claims 0 1 2
Probability 2θ θ 1−3θ
θ 0.10 0.30
Probability 0.80 0.20
One claim was observed in Year 1.
Calculate the Bayesian estimate of the expected number of claims for Year 2.
(A) Less than 1.1
(B) At least 1.1, but less than 1.2
(C) At least 1.2, but less than 1.3
(D) At least 1.3, but less than 1.4
(E) At least 1.4
Exam C: Fall 2005 -16- GO ON TO NEXT PAGE
16. You simulate observations from a specific distribution F(x), such that the number of
simulations N is sufficiently large to be at least 95 percent confident of estimating
F(1500) correctly within 1 percent.
Let P represent the number of simulated values less than 1500.
Determine which of the following could be values of N and P.
(A) N = 2000 P = 1890
(B) N = 3000 P = 2500
(C) N = 3500 P = 3100
(D) N = 4000 P = 3630
(E) N = 4500 P = 4020
Exam C: Fall 2005 -17- GO ON TO NEXT PAGE
17. For a survival study, you are given:
(i) Deaths occurred at times s 1 2 9 y < y < … < y .
(ii) The Nelson-Aalen estimates of the cumulative hazard function at 3 y and 4 y are:
^ ^
H(y3)=0.4128 and H(y4)=0.5691
(iii) The estimated variances of the estimates in (ii) are:
^
Var[H(y3)] = 0.009565
∧
and ^
Var[H(y4 )] = 0.014448
∧
Determine the number of deaths at y4 .
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
Exam C: Fall 2005 -18- GO ON TO NEXT PAGE
18. A random sample of size n is drawn from a distribution with probability density function:
2 ( ) , 0 , 0
( )
f x x
x
θ
θ
θ
= < < ∞ >
+
Determine the asymptotic variance of the maximum likelihood estimator of θ .
(A) n
3θ 2
(B) 3 2
1
nθ
(C) 2
3
nθ
(D) 3 2
n
θ
(E) 3 2
1
θ
Exam C: Fall 2005 -19- GO ON TO NEXT PAGE