Hypothesis testing in mixture regression models - Columbia University

14 H.-T. Zhu and H. Zhang
(5) and the observed sample. Third, by looking at the coefficients of v 4 –v 8 , we find that the
estimators in the two latent groups have opposite signs, explaining why they are insignificant
in the ordinary logistic model. This underscores the importance of the two-component model.
The opposite signs of the estimators indicate reversed relationships between these risk factors
and preterm births in the two latent groups.
Acknowledgements
This research is supported in part by National Institutes of Health grants DA12468 and
AA12044. We thank the Joint Editor, an Associate Editor and two referees for valuable suggestions
which helped to improve our presentation greatly. We also thank Professor Michael
Bracken for the use of his data set.
Appendix A
Before we present all the assumptions, we need to define the following derivatives of f i .β, µ/: F i,3 .β, µ/ =
@
β 2 f i.β, µ/=f iÅ, F i,4 .µ/ = @ β @ µ f i .βÅ, µ/=f iÅ, F i,5 .µ/ = @ β @
µ 2 f i.βÅ, µ/=f iÅ and F i,7 .µ/ = @
µ 3 f i.βÅ, µ/=f iÅ:
The following assumptions are sufficient conditions to derive our asymptotic results. Detailed proofs of
theorems 1–3 can be found in Zhu and Zhang (2003).
A.1. Assumption 1 (identifiability)
As n →∞, sup ω∈Ω {n −1 |L n .ω/ − ¯L n .ω/|} → 0 in probability, where ¯L n .ω/ = E{L n .ω/}. For every δ > 0,
we have
where ¯ω n is the maximizer of ¯L n .ω/ and
for every δ > 0.
lim inf
n→∞ .n−1 [ ¯L n . ¯ω n / − sup { ¯L n .ω/}]/ >0,
ω∈Ω=Ω Å,δ
ΩÅ,δ = {ω : ||β − βÅ|| δ, ||µ 1 − µÅ|| δ, α||µ 2 − µÅ|| δ} ∩ Ω
A.2. Assumption 2
For a small δ 0 > 0, let B δ0 = {.β, µ/ : ||β − βÅ|| δ 0 and ||µ|| M} ∩ Ω,
{∣∣ ∣∣∣ ∣∣∣ 1
sup
.β,µ/∈B δ0
n
∣∣ ∣∣ }
n∑
∣∣∣ ∣∣∣
F i,1 .β, µ/
∣∣ + 1 n∑
∣∣∣ ∣∣∣
F i,3 .β, µ/
n
∣∣ + 1 n∑
F i,2 .µ/
i=1
n ∣∣ + ∑ 6 1 n∑
∣∣
F i,k .µ/
i=1
k=4 n ∣∣
= o p .1/,
i=1
{∣∣ }
∣∣∣ ∣∣∣ 1 n∑
sup √n F i,k .µ/
∣∣
= O p .1/, k = 4, 6, 7,
||µ||M
i=1
sup
.β,µ/∈B δ0
{ 1
n
i=1
}
n∑
∑
||F i,1 .β, µ/|| 3 +||F i,3 .β, µ/|| 3 +||F i,2 .µ/|| 3 + 7 ||F i,k .µ/|| 3 = O p .1/:
i=1
Moreover, max 1in sup .β,µ/∈Bδ0 {||F i,1 .β, µ/||+||F i,2 .µ/|| 2 } = o p .n 1=2 /:
k=4
A.3. Assumption 3
.W n .·/, J n .·// ⇒ .W.·/, J.·//, where these processes are indexed by ||µ|| M, and the stochastic process
{.W.µ/, J.µ// : ||µ|| M} has bounded continuous sample paths with probability 1. Each J.µ/ is
a symmetric matrix and ∞ > sup ||µ||M [λ max {J.µ/}] inf ||µ||M [λ min {J.µ/}] > 0 holds almost surely.