EPFL Problem Set 1

MATH-449 - Biostatistics
EPFL
Problem Set 1
1. A survival time T is exponentially distributed with rate parameter β > 0 if its survival function,
S(t) = P(T > t), takes the form S(t) = e
−βt for t ≥ 0.
a) Find the density function f(t) = −
d
dtS(t).
b) Find the hazard function and the cumulative hazard function.
c) A waiting time T is memoryless if P(T > t + s|T > t) = P(T > s) for all t, s ≥ 0, i.e.
if the waiting time distribution does not depend on how much time has already elapsed.
Show that an exponentially distributed waiting time is memoryless.
2. a) Find E[T] when T is a Weibull distributed variable, i.e. when the hazard function of T
is α(t) = λktk−1
for λ, k > 0
‡
b) (Exercise 1.3 in ABG 2008) Suppose T is a survival time with finite expectation. Show
that†
E[T] = Z ∞
0
P(T > s)ds.
3. (Exercise 2.1 in ABG 2008) Let Mn be a discrete time martingale with respect to the filtration
Fn, for n ∈ {0, 1, 2, · · · }. By definition of M being a martingale we have that E[Mn|Fn−1] =
Mn−1 for all n ≥ 1. Show that this is equivalent to E(Mn|Fm) = Mm whenever n ≥ m ≥ 0.
4. (Exercise 2.4 in ABG 2008) Let M be as in the previous problem, and suppose M0 = 0.
Prove that M2 − hMi is a martingale with respect to the filtration F, that is, that E(M2
n −
hMin|Fn−1) = M2
n−1 − hMin−1
5. Suppose we have n independent survival times {Ti}
n
i=1, where Ti corresponds to the time of
death of individual i. Suppose we somehow could observe each individual from t = 0 up to
his/her time of death.
In the lectures you learned that a counting process {N(t)}t≥0 is an increasing right-continuous
integer-valued stochastic process such that N(0) = 0. Write down the counting process Nc
i
(that ”counts” the death of individual i) in terms of Ti
.
You will also learn about the intensity process λ of a counting process N with respect to a
filtration F. It is informally defined through the relationship λ(t)dt = E[dN(t)|Ft].
In general, if the intensity λ(t) of a counting process N(t) with respect to Ft can be written
on the form
λ(t) = α(t) · Y (t),
where α is an unknown deterministic function and Y is an Ft-predictable§
function that does
not depend on α, N(t) is said to satisfy the multiplicative intensity model ∗
.
6. (Exercise 1.10 in ABG 2008) Consider the scenario in Exercise 5, and let F
c
t be the filtration
generated by {Nc
i
(s), s ≤ t, i = 1, · · · , n}. In the lectures we will see that the intensity of Nc
i
with respect to F
c
in this case is λ
c
i
(t) = E[dNc
i
(t)|Fc
t
] = αi(t)Yi(t), where αi(t) is the hazard
function of individual i and Yi(t) = I(Ti ≥ t). Consider the aggregated counting process
Nc
(t) = Pn
i=1 Nc
i
(t).
‡Hint: Express the solution using the gamma function, which is given by Γ(z) = R ∞
0
x
z−1e−xdx.
†Hint: Write T =
R ∞
0
I(T > u)du.
§Recall that this holds when Y is left-continuous and adapted to F, i.e. that all the information needed to know
the value of Y at time t is contained in Ft.
∗We will later derive estimators for the unknown function α under the multiplicative intensity model.
i) Let {ηi(t)}
n
i=1 be known, positive, continuous functions. Find the intensity process of Nc
with respect to F
c
t when αi take the following forms:
a) αi(t) = α(t)
b) αi(t) = ηi(t)α(t)
c) αi(t) = α(t) + ηi(t)
ii) For which of the three cases in i) does Nc
satisfy the multiplicative intensity model?