## Description

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?