Statistics 255
Ex: Parametric Estimation of S(t)
Suppose that Ti ∼ Exp(λ) and Ci ∼ Unif(a, b) where a and b are known constants with λ an unknown parameter to
estimate. Further assume that the Ti ’s and Ci ’s are totally independent.
Goal: Find the MLE of λ based on Xi ≡ min(Ti , Ci ) and δi = 1[Ti =Xi ] and it’s asymptotic distribution. From this,
find the MLE of ST (t; λ) = Pr[T > t] and it’s asymptotic distribution.
Solution: For notational convenience, let Ti ∼ F (·) and Ci ∼ G(·). Now to write down the likelihood consider the
following cases:
δi = 1
⇒
Ti = Xi , Ci > Xi
⇒ likelihood contribution is fT (Xi )[1 − GC (Xi )]
δi = 0
⇒
Ci = Xi , Ti > Xi
⇒ likelihood contribution is gC (Xi )[1 − FT (Xi )]
So, we have that
Li (λ; Xi , δi ) = δi {fT (Xi )[1 − GC (Xi )]} + (1 − δi ) {gC (Xi )[1 − FT (Xi )]}
1−δi
δ
= {fT (Xi )[1 − GC (Xi )]} i {gC (Xi )[1 − FT (Xi )]}
δi 1−δi
= λe−λXi [1 − GC (Xi )]
gC (Xi )e−λXi
= λδi e−λXi [1 − GC (Xi )]δi [gC (Xi )]1−δi
`i (λ) = δi log λ − λXi + k
Ui (λ) =
δi
− Xi
λ
δi
Ii (λ) = −E − 2
λ
From this, the MLE satisfies
U(λ̂) =
n
X
Ui (λ̂) = 0
i=1
Pn
⇒
i=1 δi
λ̂
−
n
X
Xi = 0
i=1
Pn
δi
δ̄
⇒ λ̂ = Pni=1
=
X
X̄
i=1 i
Since this is a regular problem, we know from large sample MLE theory that
√
n(λ̂ − λ) →d N (0, I1−1 (λ)).
Now, by the invariance property of MLE’s, the MLE of S(t; λ) is
b
b = e−λt
b = S(t; λ)
S(t)
,
1
t > 0.
(∝ fT (Xi )δi ST (Xi )1−δi )
So, taking g(x) = e−xt (ie. g 0 (x) = −te−xt ), from the δ-method we have
i
√ h
b − S(t; λ) →d N 0, t2 e−2λt I −1 (λ) ,
n S(t; λ)
1
or
b ∼N
S(t; λ)
˙
S(t; λ), t2 e−2λt n−1 I1−1 (λ) .
**Notes:
A. Within the asymptotic variance we have
Pr[Ti < Ci ]
δi
,
I1 (λ) = E 2 =
λ
λ2
and
Z
b
Z
[1 − FT (c)]gC (c)dc = 1 −
Pr[Ti < Ci ] = 1 − Pr[Ti > Ci ] = 1 −
a
a
b
e−λc
e−λb − e−λa
dc = 1 +
.
b−a
λ(b − a)
In real applications, we don’t generally know the censoring distribution, so we estimate it’s impact on the information
by using the observed information
n
n
X
X
∂
δi
b
− Ui (λ) =
I(λ)
=
.
∂λ
λ2
i=1
i=1
B. In general, under the assumption that censoring is independent of failure times, we specify the i-th observations
contribution to the likelihood function as
Li (λ; Xi , δi ) = fT (Xi )δi ST (Xi )1−δi
2