Note that the true distribution of the pooled sample is
$\bar{\mathbb{P}}=\kappa\mathbb{P}_{1}+(1-\kappa)\mathbb{P}_{2}$, where
$\mathbb{P}_{i}$ denotes the distribution of the $i$th sample. In particular,
$\mathbb{P}$ does not correspond to a logistic-Cox model. Let
$Q_{\Lambda,\theta}$ be the corresponding distribution of a logistic-Cox model
with parameters $(\Lambda,\theta)$ and corresponding log-likelihood

$\displaystyle l(\delta,y,x,z;\Lambda,\theta)$
$\displaystyle=\delta\left\\{\log\phi(\gamma^{T}x)+\log
f_{u}(y|z;\Lambda,\beta)\right\\}$
$\displaystyle\qquad+(1-\delta)\log\left\\{1-\phi(\gamma^{T}x)+\phi(\gamma^{T}x)S_{u}(y|z;\Lambda,\beta)\right\\}$

By assumption (I2), the event times on the pooled sample happen on
$[0,\bar{\tau}_{0}]$, where $\bar{\tau}_{0}=\max\\{\tau_{0,1},\tau_{0,2}\\}$.
Hence $S_{u}(t|z;\Lambda,\beta)=0$ for $t\geq\bar{\tau}_{0}$ which corresponds
to $\Lambda$ being defined on $[0,\bar{\tau}_{0})$. In addition,
$\bar{\mathbb{P}}({\Delta}=1,{Y}=\bar{\tau}_{0})>0$ by assumptions (A3) and
(I3). Hence, we can restrict on distributions $Q_{\Lambda,\theta}$ that have a
positive mass at $\bar{\tau}_{0}$, meaning that
$\lim_{t\to\bar{\tau}_{0}}\Lambda(t)<+\infty$ and we can denote the limit by
$\Lambda(\bar{\tau}_{0})$. To reflect the existence of the jump in the
likelihood, for the terms with $\Delta=1$ and ${Y}=\bar{\tau}$ we have
$f_{u}(\bar{\tau}|z;\Lambda,\beta)=S_{u}(\bar{\tau}-|z;\Lambda,\beta)=\exp(-\Lambda(\bar{\tau}_{0})e^{\beta^{T}z})$,
instead of the usual expression
$f_{u}(t|z;\Lambda,\beta)={\lambda_{u}(t|z;\beta)}S_{u}(\bar{\tau}-|z;\Lambda,\beta).$
Here $\lambda_{u}$ denotes the hazard function corresponding to $S_{u}$ and
the baseline hazard function corresponding to $\Lambda$ will be denoted by
$\lambda$. Define

$(\bar{\Lambda},\bar{\theta})=\mathop{\mathrm{argmax}}_{\Lambda,\theta}\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]=\mathop{\mathrm{argmin}}_{\Lambda,\theta}\mathrm{KL}(\bar{\mathbb{P}}|Q_{\Lambda,\theta})$
(II.18)

where $KL(\cdot|\cdot)$ denotes the Kullback-Leibler divergence between two
distributions.

######  Lemma 1.

The argmax defined in (II.18) exists.

###### Proof.

We show that the argmax can be restricted on a bounded set, from which the
existence follows because of continuity. In the three steps below we deal
consequently with $\beta$, $\Lambda$ and $\gamma$.

Step 1. First we show that for any $K>0$ there exists $\bar{c}>0$ such that
for any $c\geq\bar{c}$ we have $\inf_{\tilde{\beta}\in
S^{q-1}}\bar{\mathbb{P}}(c\,|\tilde{\beta}^{T}{Z}|>K)>0$, where $S^{q-1}$ is
the unit circle in $\mathbb{R}^{q}$. Suppose by contradiction that there
exists $K$ such that for any $c$ we have $\inf_{\tilde{\beta}\in
S^{q-1}}\bar{\mathbb{P}}(c\,|\tilde{\beta}^{T}{Z}|>K)=0$. Note that the
infimum is actually a minimum because $S^{q-1}$ is compact and the function is
continuous. Hence, it means that for any $c$ there exists $\tilde{\beta}\in
S^{q-1}$ for which $\bar{\mathbb{P}}(c\,|\tilde{\beta}^{T}{Z}|\leq K)=1$.
Equivalently, for any $\epsilon>0$, there exists $\tilde{\beta}\in S^{q-1}$
for which $\bar{\mathbb{P}}(|\tilde{\beta}^{T}{Z}|\leq\epsilon)=1$. The closed
subsets of $S^{q-1}$ defined by $B_{m}=\\{\tilde{\beta}\in
S^{q-1}\,\big{|}\,\bar{\mathbb{P}}(|\tilde{\beta}^{T}{Z}|\leq\frac{1}{m})=1\\}$
are non-empty for all $m$ and $B_{m}\downarrow B=\cap_{m}B_{m}$. $B$ cannot be
empty because then $(B_{m}^{c})_{m}$ form an open covering of the compact
$S^{q-1}$ and there would exists a finite sub-covering, which is impossible
since all $B_{m}$ are non-empty. It follows that $B$ is not empty, which is
equivalent to saying that there exists $\tilde{\beta}\in S^{q-1}$ for which
$\bar{\mathbb{P}}(|\tilde{\beta}^{T}{Z}|=0)=1$. This contradicts the
assumption that $Var({Z})$ has full rank.
Next, let
$\eta=\frac{1}{2}\Lambda({\bar{\tau}_{0}})\inf_{z}\bar{\mathbb{P}}({Y}=\bar{\tau}_{0}|{\Delta}=1,{Z}=z)$
and choose $K$ such that $x\leq\eta e^{x}$ for all $x\geq K$. Let
$\beta=c{\tilde{\beta}}$ with $\tilde{\beta}\in S^{q-1}$, $c>\bar{c}$. We will
show that, as $c$ increases, the expectation in (II.18) becomes arbitrarily
small. For fixed $\gamma$ and $\Lambda$, we can write

$\begin{split}&\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]\\\
&=\mathbb{E}_{\bar{\mathbb{P}}}[{\Delta}\beta^{T}{Z}{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}-{\Delta}\Lambda({Y})e^{\beta^{T}{Z}}]+R_{1}\\\
&=\mathbb{E}_{\bar{\mathbb{P}}}\left[\left\\{{\Delta}c\tilde{\beta}^{T}{Z}{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}-{\Delta}\Lambda({Y})e^{c\tilde{\beta}^{T}{Z}}\right\\}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}>0\\}}\mathds{1}_{\\{c\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]\\\
&+\mathbb{E}_{\bar{\mathbb{P}}}\left[\left\\{{\Delta}c\tilde{\beta}^{T}{Z}{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}-{\Delta}\Lambda({Y})e^{c\tilde{\beta}^{T}{Z}}\right\\}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}<0\\}}\mathds{1}_{\\{c\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]+R_{2},\end{split}$

where $R_{1}$ and $R_{2}$ denote terms that are bounded in absolute value.
Using
$\mathbb{E}_{\bar{\mathbb{P}}}[\Lambda({Y})|{Z},{\Delta}=1]>\Lambda({\bar{\tau}_{0}})\inf_{z}\bar{\mathbb{P}}({Y}=\bar{\tau}_{0}|{\Delta}=1,{Z}=z)=2\eta>0$,
we obtain the following bound

$\begin{split}\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]&\leq\mathbb{E}_{\bar{\mathbb{P}}}\left[\left\\{{\Delta}c\tilde{\beta}^{T}{Z}-2{\Delta}\eta
e^{c\tilde{\beta}^{T}{Z}}\right\\}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}>0\\}}\mathds{1}_{\\{c\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]\\\
&+c\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}\tilde{\beta}^{T}{Z}{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}<0\\}}\mathds{1}_{\\{c\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]+R_{2}\\\
&\leq-\eta\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}e^{c\tilde{\beta}^{T}{Z}}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}>0\\}}\mathds{1}_{\\{c\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]\\\
&-c\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}|\tilde{\beta}^{T}{Z}|{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}<0\\}}\mathds{1}_{\\{c\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]+R_{2}\\\
&\leq-\eta\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}e^{c\tilde{\beta}^{T}{Z}}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}>0\\}}\mathds{1}_{\\{\bar{c}\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]\\\
&-c\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}|\tilde{\beta}^{T}{Z}|{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}\mathds{1}_{\\{\tilde{\beta}^{T}{Z}<0\\}}\mathds{1}_{\\{\bar{c}\,|\tilde{\beta}^{T}{Z}|>K\\}}\right]+R_{2}\\\
\end{split}$

This further leads to

$\begin{split}&\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]\\\
&\leq-\eta
e^{cK/\bar{c}}\bar{\mathbb{P}}\left({\Delta}=1,\tilde{\beta}^{T}{Z}>0,\bar{c}\,|\tilde{\beta}^{T}{Z}|>K\right)\\\
&-c\frac{K}{\bar{c}}\bar{\mathbb{P}}\left({\Delta}=1,{Y}<\bar{\tau}_{0},\tilde{\beta}^{T}{Z}<0,\bar{c}\,|\tilde{\beta}^{T}Z|>K\right)+R_{2}\\\
&\leq-c\frac{K}{\bar{c}}\bar{\mathbb{P}}\left({\Delta}=1,{Y}<\bar{\tau}_{0},\bar{c}\,|\tilde{\beta}^{T}{Z}|>K\right)+R_{2}\\\
&\leq-c\frac{K}{\bar{c}}\inf_{z}\bar{\mathbb{P}}(\Delta=1,{Y}<\bar{\tau}_{0}|{Z}=z)\inf_{\tilde{\beta}\in
S^{q-1}}\bar{\mathbb{P}}\left(\bar{c}\,|\tilde{\beta}^{T}{Z}|>K\right)+R_{2}\end{split}$

Since both infimums are strictly positive,
$\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]$ can be
made arbitrarily small for $c$ sufficiently large (and how large $c$ should be
does not depend on $\tilde{\beta}$). Hence, we can restrict the argmax on a
bounded set for $\beta$.

Step 2. Next we show that, there exists $M>0$ such that it suffices to search
for the maximizer among $\Lambda$ that are bounded by $M$. Let $\Lambda$ be
such that $\Lambda(\bar{\tau}_{0})>M$. We can construct
$\tilde{\Lambda}(t)=c\Lambda(t)$ with $c=M/\Lambda(\bar{\tau}_{0})\in(0,1)$.
We have $\tilde{\Lambda}(\bar{\tau}_{0})=M$ and $\tilde{\lambda}=c\lambda$. We
show that

$\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]<\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\tilde{\Lambda},\theta)].$

Indeed we have

$\begin{split}&\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]-\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\tilde{\Lambda},\theta)]\\\
&=\mathbb{E}_{\bar{\mathbb{P}}}\left[-{\Delta}\log
c{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}-{\Delta}\Lambda({Y})e^{\beta^{T}{Z}}+{\Delta}c\Lambda({Y})e^{\beta^{T}{Z}}\right.\\\
&\qquad\left.+(1-{\Delta})\log\frac{1-\phi(\gamma^{T}{X})+\phi(\gamma^{T}{X})S_{u}({Y}|{Z};\Lambda,\beta)}{1-\phi(\gamma^{T}{X})+\phi(\gamma^{T}{X})S_{u}({Y}|{Z};\tilde{\Lambda},\beta)}\right]\end{split}$

Since $S_{u}({Y}|{Z};\tilde{\Lambda},\beta)>S_{u}({Y}|{Z};\Lambda,\beta)$ the
ratio is smaller than 1 and as a result the $(1-{\Delta})$ term in the
expectation is negative. Hence

$\begin{split}&\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]-\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\tilde{\Lambda},\theta)]\\\
&<\mathbb{E}_{\bar{\mathbb{P}}}\left[-{\Delta}\log
c{\mathds{1}_{\\{{Y}<\bar{\tau}_{0}\\}}}-{\Delta}\Lambda({Y})e^{\beta^{T}{Z}}+{\Delta}c\Lambda({Y})e^{\beta^{T}{Z}}\right]\\\
&=-\bar{\mathbb{P}}({\Delta}=1,{{Y}<\bar{\tau}_{0}})\log
c-(1-c)\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}\Lambda({Y})e^{\beta^{T}{Z}}\right]\\\
&\leq-\bar{\mathbb{P}}({\Delta}=1,{{Y}<\bar{\tau}_{0}})\log
c-(1-c)\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}\Lambda({Y})e^{\beta^{T}{Z}}\mathds{1}_{\\{{Y}=\bar{\tau}_{0}\\}}\right]\\\
&=-\bar{\mathbb{P}}({\Delta}=1,{{Y}<\bar{\tau}_{0}})\log
c-(1-c)\Lambda(\bar{\tau}_{0})\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}e^{\beta^{T}{Z}}\mathds{1}_{\\{{Y}=\bar{\tau}_{0}\\}}\right].\end{split}$

Since we are restricting $\beta$ on a compact and $Z$ is assumed to have
bounded support, there exist $c_{2}>0$ such that $e^{\beta^{T}{Z}}>c_{2}$
a.s.. It follows that

$\begin{split}&\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]-\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\tilde{\Lambda},\theta)]\\\
&<-\bar{\mathbb{P}}({\Delta}=1,{{Y}<\bar{\tau}_{0}})\log
c-(1-c)\Lambda(\bar{\tau}_{0})c_{2}\bar{\mathbb{P}}({\Delta}=1,{Y}=\bar{\tau}_{0})\\\
&=\bar{\mathbb{P}}({\Delta}=1,{{Y}<\bar{\tau}_{0}})(\log\Lambda(\bar{\tau}_{0})-\log
M)-(\Lambda(\bar{\tau}_{0})-M)c_{2}\bar{\mathbb{P}}({\Delta}=1,{Y}=\bar{\tau}_{0})\\\
&<\bar{\mathbb{P}}({\Delta}=1,{{Y}<\bar{\tau}_{0}})(\Lambda(\bar{\tau}_{0})-M)\frac{1}{M}-(\Lambda(\bar{\tau}_{0})-M)c_{2}\bar{\mathbb{P}}({\Delta}=1,{Y}=\bar{\tau}_{0})\\\
&=(\Lambda(\bar{\tau}_{0})-M)\left\\{\frac{1}{M}\bar{\mathbb{P}}({\Delta}=1,{{Y}<\bar{\tau}_{0}})-c_{2}\bar{\mathbb{P}}({\Delta}=1,{Y}=\bar{\tau}_{0})\right\\}<0\end{split}$

for large enough $M$ since $\bar{\mathbb{P}}({\Delta}=1,{Y}=\bar{\tau})>0$ by
assumption. Hence we conclude that, there exists $M$ such that it is
sufficient to search for the maximizer among $\Lambda$’s bounded by $M$.

Step 3. We can also restrict the argmax on a bounded set for $\gamma$ because
as $\|\gamma\|\to\infty$, for fixed values of $\beta$ and $\Lambda$, the
expectation converges to $-\infty$. Indeed we have

$\begin{split}&\mathbb{E}_{\bar{\mathbb{P}}}[l(\Delta,{Y},{X},{Z};\Lambda,\theta)]\\\
&=\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}\log\phi(\gamma^{T}{X})\mathds{1}_{\\{\gamma^{T}{X}>0\\}}\right]+\mathbb{E}_{\bar{\mathbb{P}}}\left[{\Delta}\log\phi(\gamma^{T}{X})\mathds{1}_{\\{\gamma^{T}{X}\leq
0\\}}\right]+R_{3},\end{split}$

where $R_{3}$ denotes terms bounded in absolute value. The first term is
bounded and using the same reasoning as with $\beta$ it can be shown that the
second term converges to $-\infty$.

We conclude that we can restrict the argmax on a bounded set, from which the
existance of the argmax follows as the criteria is continuous with respect to
the parameters. ∎

In what follows, we assume that the maximizer $(\bar{\Lambda},\bar{\theta})$
is unique. It will also be useful to characterize it as the solution of the
score equation defined similarly to [29]. As in the proof of Theorem 3,
consider $\mathcal{H}_{m}=\\{h=(h_{1},h_{2})\in
BV[0,\bar{\tau}_{0}]\times\mathbb{R}^{p+q}:\|h_{1}\|_{v}+\|h_{2}\|_{1}\leq
m\\}$, where $m<\infty$ and $\|h_{1}\|_{v}$ is the absolute value of
$h_{1}(0)$ plus the total variation of $h_{1}$ on the interval
$[0,\bar{\tau}_{0}]$. Define the functions

$\begin{split}&\psi_{(\Lambda,\theta),h}(\delta,y,x,z)=\delta\left[h_{1}(y)+h_{21}^{T}x+h^{T}_{22}z\right]-\left\\{\phi(\gamma^{T}x)-(1-\delta)g(y,\Lambda,\theta)\right\\}h_{21}^{T}x\\\
&\qquad\qquad\qquad-\left\\{\delta+(1-\delta)g(y,\Lambda,\theta)\right\\}\left\\{e^{\beta^{T}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)+e^{\beta^{T}z}\Lambda(y)h^{T}_{22}z\right\\},\end{split}$

where

$g(t,\Lambda,\theta)=\frac{\phi(\gamma^{T}x)\exp\left(-\Lambda(t)\exp\left(\beta^{T}z\right)\right)}{1-\phi(\gamma^{T}x)+\phi(\gamma^{T}x)\exp\left(-\Lambda(t)\exp\left(\beta^{T}z\right)\right)}.$

We will denote by $\mathbb{P}^{\pi}_{n_{i},i}\psi_{(\Lambda,\theta),h}$ the
score function for the i-th permuted sample

$\begin{split}\mathbb{P}^{\pi}_{n_{i},i}\psi_{(\Lambda,\theta),h}&=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\Delta^{\pi}_{ij}\left[h_{1}({Y}_{j}^{\pi})+h_{21}^{T}X_{ij}^{\pi}+h^{T}_{22}Z_{ij}^{\pi}\right]\\\
&\quad-\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\left\\{\phi(\gamma^{T}X_{ij}^{\pi})-(1-\Delta_{ij}^{\pi})g_{ij}^{\pi}(Y_{ij}^{\pi},\Lambda,\theta)\right\\}h_{21}^{T}X_{ij}^{\pi}\\\
&\quad-\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\left\\{\Delta_{ij}^{\pi}+(1-\Delta_{ij}^{\pi})g_{ij}^{\pi}(Y_{ij}^{\pi},\Lambda,\theta)\right\\}\\\
&\quad\qquad\qquad\quad\times\left\\{e^{\beta^{T}Z_{ij}^{\pi}}\int_{0}^{Y_{ij}^{\pi}}h_{1}(s)\mathrm{d}\Lambda(s)+e^{\beta^{T}Z_{ij}^{\pi}}\Lambda(Y_{ij}^{\pi})h^{T}_{22}Z_{ij}^{\pi}\right\\},\end{split}$

where $h=(h_{1},h_{2})=(h_{1},h_{21},h_{22})\in\mathcal{H}_{m}$ and

$g_{ij}^{\pi}(t,\Lambda,\theta)=\frac{\phi(\gamma^{T}X_{ij}^{\pi})\exp\left(-\Lambda(t)\exp\left(\beta^{T}Z_{ij}^{\pi}\right)\right)}{1-\phi(\gamma^{T}X_{ij}^{\pi})+\phi(\gamma^{T}X_{ij}^{\pi})\exp\left(-\Lambda(t)\exp\left(\beta^{T}Z_{ij}^{\pi}\right)\right)}.$

Similarly, $\mathbb{P}_{n_{1}+n_{2}}\psi_{(\Lambda,\theta),h}$ and
$\bar{\mathbb{P}}\psi_{(\Lambda,\theta),h}$ are defined using the empirical
distribution of the pooled sample or the true distribution of the pooled
sample $\bar{\mathbb{P}}=\kappa\mathbb{P}_{1}+(1-\kappa)\mathbb{P}_{2}$
respectively. By definition $(\bar{\Lambda},\bar{\theta})$ is the solution of
$\bar{\mathbb{P}}\psi_{(\Lambda,\theta),h}\overset{!}{=}0$.

######  Lemma 2.

Assume the maximizer $(\bar{\Lambda},\bar{\theta})$ defined in (II.18) is
unique. The pooled maximum likelihood estimator
$(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}})$ is a (weakly)
consistent estimator of $(\bar{\Lambda},\bar{\theta})$.

###### Proof.

We will pursue similar ideas as [29] in the proofs of his Lemma 2 and Theorem
2. Comparing the arguments in [29] that lead to the maximum likelihood
estimator in Display (12) of that paper, it is evident that the pooled
estimator $\bar{\Lambda}_{n_{1}+n_{2}}$ must exhibit a similar structure. In
particular,

$\bar{\Lambda}_{n_{1}+n_{2}}(t)=\int_{0}^{t}\frac{dN(s)}{\sum_{j=1}^{n_{1}+n_{2}}R_{j}(s)\exp(\bar{\beta}_{n_{1}+n_{2}}^{T}Z_{j})\\{\Delta_{j}+(1-\Delta_{j})\bar{g}_{j}(Y_{j};\bar{\theta}_{n_{1}+n_{2}},\bar{\Lambda}_{n_{1}+n_{2}})\\}},$

where $N=N_{1}+N_{2}$ is the pooled counting process,
$R_{j}(s)=\mathds{1}_{\\{Y_{j}\geq s\\}}$ denotes the at-risk process of the
$j$-th pooled individual and

$\bar{g}_{i}(t;\Lambda,\theta)=\frac{\phi(\gamma^{T}X_{i})\exp(-\Lambda(t)\exp(\beta^{T}Z_{i}))}{1-\phi(\gamma^{T}X_{i})+\phi(\gamma^{T}X_{i})\exp(-\Lambda(t)\exp(\beta^{T}Z_{i}))}.$

Similarly, we define

$\tilde{\Lambda}_{n_{1}+n_{2}}(t)=\int_{0}^{t}\frac{dN(s)}{\sum_{j=1}^{n_{1}+n_{2}}R_{j}(s)\exp(\bar{\beta}^{T}Z_{j})\\{\Delta_{j}+(1-\Delta_{j})\bar{g}_{j}(Y_{j};\bar{\theta},\bar{\Lambda})\\}}.$

We have already noticed that
$\sup_{n_{1},n_{2}}\tilde{\Lambda}_{n_{1}+n_{2}}(\tau)<\infty$ a.s.; also,
following the lines of Lemma 2 (ii) in [29], there exists a non-negative and
integrable function $\eta:[0,\bar{\tau}_{0}]\to(0,\infty)$ bounded away from 0
such that, for each $\omega\in\Omega$, there exists a subsequence
$(n_{1,k_{1}}(\omega)+n_{2,k_{2}}(\omega))$ such that
$\sup_{t\in(0,\bar{\tau}_{0}]}|\frac{\mathrm{d}\bar{\Lambda}_{n_{1,k_{1}}+n_{2,k_{2}}}}{\mathrm{d}\tilde{\Lambda}_{n_{1,k_{1}}+n_{2,k_{2}}}}(t)-\eta(t)|\to
0$. For notational convenience, we will write
$\bar{n}_{k}=n_{1,k_{1}}+n_{2,k_{2}}$ from now on. Arguing for a fixed
$\omega$ and along subsequences had similarly been done by [33] and [42] based
on Helly’s theorem.

To prove the desired consistency of the pooled estimators, we will show that
the difference of the log-likelihoods, say
$\bar{\ell}_{\bar{n}_{k}}(\bar{\Lambda}_{\bar{n}_{k}},\bar{\theta}_{\bar{n}_{k}})$
and $\bar{\ell}_{\bar{n}_{k}}(\tilde{\Lambda}_{\bar{n}_{k}},\bar{\theta})$
converges to zero. Clearly,

$\displaystyle 0\leq$ $\displaystyle\
\bar{\ell}_{\bar{n}_{k}}(\bar{\Lambda}_{\bar{n}_{k}},\bar{\theta}_{\bar{n}_{k}})-\bar{\ell}_{\bar{n}_{k}}(\tilde{\Lambda}_{\bar{n}_{k}},\bar{\theta})$
$\displaystyle=$
$\displaystyle\frac{1}{\bar{n}_{k}}\sum_{i=1}^{\bar{n}_{k}}\Big{[}\Delta_{i}\log{\frac{\bar{g}_{i}(Y_{i};\bar{\Lambda}_{\bar{n}_{k}},\bar{\theta}_{\bar{n}_{k}})}{\bar{g}_{i}(Y_{i},\tilde{\Lambda}_{\bar{n}_{k}},\bar{\theta})}}+\Delta_{i}\log\frac{\Delta\bar{\Lambda}_{\bar{n}_{k}}(Y_{i})}{\Delta\tilde{\Lambda}_{\bar{n}_{k}}(Y_{i})}+\Delta_{i}(\bar{\beta}_{n_{k}}-\bar{\beta})^{T}Z_{i}$
$\displaystyle\qquad\qquad+\log\frac{\bar{S}_{i}(Y_{i};\bar{\Lambda}_{\bar{n}_{k}},\bar{\theta}_{\bar{n}_{k}})}{\bar{S}_{i}(Y_{i};\tilde{\Lambda}_{\bar{n}_{k}},\bar{\theta})}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{\bar{n}_{k}}\sum_{i=1}^{\bar{n}_{k}}\Big{[}\Delta_{i}\log\frac{\phi(\bar{\gamma}^{T}_{n_{k}}X_{i})}{\phi(\bar{\gamma}^{T}X_{i})}-\Delta_{i}\bar{\Lambda}_{n_{k}}(Y_{i})\exp(\bar{\beta}^{T}_{n_{k}}X_{i})+\Delta_{i}\tilde{\Lambda}_{n_{k}}(Y_{i})\exp(\bar{\beta}^{T}X_{i})$
$\displaystyle\qquad+\Delta_{i}\log\frac{\Delta\bar{\Lambda}_{\bar{n}_{k}}(Y_{i})}{\Delta\tilde{\Lambda}_{\bar{n}_{k}}(Y_{i})}+\Delta_{i}(\bar{\beta}_{n_{k}}-\bar{\beta})^{T}Z_{i}+(1-\Delta_{i})\log\frac{\bar{S}_{i}(Y_{i};\bar{\Lambda}_{\bar{n}_{k}},\bar{\theta}_{\bar{n}_{k}})}{\bar{S}_{i}(Y_{i};\tilde{\Lambda}_{\bar{n}_{k}},\bar{\theta})}\Big{]}$

where and
$\bar{S}_{i}(t;\Lambda,\theta)=1-\phi(\gamma^{T}X_{i})+\phi(\gamma^{T}X_{i})\exp(-\Lambda(t)\exp(\beta^{T}Z_{i}))$.

The space of bounded, increasing functions with discontinuities only at
$\tau_{0,1}$ and $\tau_{0,2}$ is separable with respect to the supremum norm.
Also, Euclidean spaces are separable. Denote by
$(\Lambda_{l},\theta_{l})_{l\in\mathbb{N}}$ a countable subset that is dense
in the product of the just-described spaces. For each $l\in\mathbb{N}$, by the
strong law of large numbers,

$\begin{split}\frac{1}{\bar{n}_{k}}\sum_{i=1}^{\bar{n}_{k}}&\Big{[}\Delta_{i}\log\phi(\gamma_{l}^{T}X_{i})-\Delta_{i}\Lambda_{l}(t)\exp(\beta^{T}_{l}Z_{i})+\Delta_{i}\log\mathrm{d}\Lambda_{l}(Y_{i})+\Delta_{i}\beta_{l}^{T}Z_{i}\\\
&\quad+(1-\Delta_{i})\log\bar{S}_{i}(Y_{i};\Lambda_{l},\theta_{l})\Big{]}\end{split}$

converges a.s. to its expectation, i.e., for all $\omega\in\Omega_{l}$ with
$\bar{\mathbb{P}}(\Omega_{l})=1$. From now on, we restrict $\omega$ to be in
the intersection $\bigcap_{l\in\mathbb{N}}\Omega_{l}$ which also has
probability 1. Consequently, also due to the continuity of the likelihoods in
$\Lambda$ and $\theta$,

$\displaystyle 0\leq$ $\displaystyle\
\bar{\ell}_{\bar{n}_{k}}(\bar{\Lambda}_{\bar{n}_{k}},\bar{\theta}_{\bar{n}_{k}})-\bar{\ell}_{\bar{n}_{k}}(\tilde{\Lambda}_{\bar{n}_{k}},\bar{\theta})$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\bar{\mathbb{P}}}\Big{[}\Delta_{i}\log\frac{\phi(\gamma^{*^{T}}X_{i})}{\phi(\bar{\gamma}^{T}X_{i})}-\\{\Lambda^{*}(t)\exp(\beta^{*^{T}}Z_{i})-\bar{\Lambda}(t)\exp(\bar{\beta}^{T}Z_{i})\\}+\Delta_{i}\log\frac{\mathrm{d}\Lambda^{*}(Y_{i})}{\mathrm{d}\bar{\Lambda}(Y_{i})}$
$\displaystyle\qquad\qquad+\Delta_{i}(\beta^{*}-\bar{\beta})^{T}Z_{i}+\log\frac{\bar{S}_{i}(Y_{i};\Lambda^{*},\theta^{*})}{\bar{S}_{i}(Y_{i};\bar{\Lambda},\bar{\theta})}\Big{]}+o(1)$

where the expectation is taken with respect to ${X},{Y},{Z}$ and
$\Lambda^{*}$, $\beta^{*}$, $\gamma^{*}$ are fixed (depending on $\omega$).
For a.e. $\omega$, the conditional expectation in the previous display
represents a negative KL-divergence of the logistic-Cox model specified by
$(\bar{\Lambda},\bar{\theta})$ from the model specified by
$(\Lambda^{*}(\omega),\theta^{*}(\omega))$. As a consequence, it must be 0,
i.e.,
$\bar{\ell}(\Lambda^{*},\theta^{*})=\bar{\ell}(\bar{\Lambda},\bar{\theta})$
$\bar{\mathbb{P}}$-a.e.. We use this fact to identify all model components,
one by one; every equality below is to be understood $\bar{\mathbb{P}}$-a.s..

We first consider ${\Delta}=0$ and ${Y}\geq\bar{\tau}_{0}$, for which
$\bar{S}(Y;\Lambda^{*},\theta^{*})=\phi(\gamma^{*T}X)$ and
$\bar{S}(Y;\bar{\Lambda},\bar{\theta})=\phi(\bar{\gamma}^{T}X)$. From this we
can identify $\gamma^{*}=\bar{\gamma}$ a.s. for the logistic model. Next, for
$\Delta=0$ and $Y<\bar{\tau}_{0}$, we obtain
$\bar{S}(Y;\Lambda^{*},\theta^{*})=\bar{S}(Y;\bar{\Lambda},\bar{\theta})$,
hence

$\displaystyle\exp(-\Lambda^{*}(Y)\exp(\beta^{*T}Z))=\exp(-\bar{\Lambda}(Y)\exp(\bar{\beta}^{T}Z))$

Upon inserting different combinations of $Y$ and $Z$, we conclude that
$\beta^{*}=\bar{\beta}$ and $\Lambda^{*}=\bar{\Lambda}$ a.s..

∎

The following lemma establishes the consistency of randomly permuted
Z-estimators. Since the proof does not make use of the specific underlying
model structure, it is clear that this result holds more generally, i.e., also
beyond logistic-Cox cure models.

######  Lemma 3.

Assume the maximizer $(\bar{\Lambda},\bar{\theta})$ defined in (II.18) is
unique. The permutation estimators
$(\hat{\Lambda}_{n_{1},1}^{\pi},\hat{\theta}_{n_{1},1}^{\pi})$ and
$(\hat{\Lambda}_{n_{2},2}^{\pi},\hat{\theta}_{n_{2},2}^{\pi})$ converge in
probability to $(\bar{\Lambda},\bar{\theta})$.

###### Proof.

First, we would like to point out that conditional convergence in probability
(given a $\sigma$-algebra) is equivalent to the unconditional convergence in
probability; a variant of Fact 1 in the Supporting Information of [16]
similarly holds for the present setting. That is why we do not distinguish
between conditional and unconditional consistency.

To prove the consistency of the permuted estimators, we are going to employ
the permutation version of the score equations, i.e.,
$\mathbb{P}_{n_{i},i}^{\pi}\psi_{(\Lambda,\theta),h}\stackrel{{\scriptstyle!}}{{=}}0$
for all indexing $h$, $i=1,2$. So far, we know by definition that
$\mathbb{P}_{n_{i},i}^{\pi}\psi_{(\hat{\Lambda}^{\pi}_{n_{i},i},\hat{\theta}^{\pi}_{n_{i},i}),h}=0$
and that
$\mathbb{P}_{n_{1}+n_{2}}\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}=0$.

Also, since
$n_{1}\mathbb{P}_{n_{1},1}^{\pi}\psi_{(\Lambda,\theta),h}+n_{2}\mathbb{P}_{n_{2},2}^{\pi}\psi_{(\Lambda,\theta),h}=(n_{1}+n_{2})\mathbb{P}_{n_{1}+n_{2}}\psi_{(\Lambda,\theta),h}$,
we have that
$n_{1}\mathbb{P}_{n_{1},1}^{\pi}\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}=-n_{2}\mathbb{P}_{n_{2},2}^{\pi}\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}$.
Thus, because both of these permuted expressions are connected, we will only
focus on the index $i=1$ from now on.

Furthermore, upon integrating out all permutations, it is easy to see that the
(conditional) expectation is
$\mathbb{E}[\mathbb{P}_{n_{1},1}^{\pi}\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}\
|\ Y_{i},\Delta_{i},X_{i},Z_{i}:i=1,\dots,n_{1}+n_{2}]=0.$ Additionally,
straightforward and standard algebra for permuted linear statistics for the
conditional variance leads to

$Var[\mathbb{P}_{n_{1},1}^{\pi}\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}\
|\
Y_{i},\Delta_{i},X_{i},Z_{i}:i=1,\dots,n_{1}+n_{2}]=O_{p}((n_{1}+n_{2})^{-1}).$

Consequently, Chebychev’s inequality (applied to the conditional distribution)
verifies that the permutation-based score equations evaluated at the point
$(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}})$ all converge to 0
in probability. Similar convergences in probability (not necessarily to zero)
also hold for other evaluation points.

Hence, the pooled estimator is asymptotically a solution to the permutation-
based score equations. Now, since
$(\hat{\Lambda}^{\pi}_{n_{1},1},\hat{\theta}^{\pi}_{n_{1},1})$ is another
(finite sample) solution and the “true” solution
$(\bar{\Lambda},\bar{\theta})$ is assumed to be unique, the permuted estimator
must approach the pooled estimator in probability as the sample size goes in
infinity. Anything else would contradict the continuity of the map
$(\Lambda,\theta)\mapsto\bar{\mathbb{E}}_{\bar{\mathbb{P}}}(\bar{\ell}(\Lambda,\theta))$.

∎

######  Lemma 4.

Assume the maximizer $(\bar{\Lambda},\bar{\theta})$ defined in (II.18) is
unique. Conditionally on the observations, the process

$\langle
n_{i}^{1/2}(\hat{\Lambda}^{\pi}_{n_{i},i}-\bar{\Lambda}_{n_{1}+n_{2}}),n_{i}^{1/2}(\hat{\theta}_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}})\rangle,\qquad
i=1,2$

defined as in (II.11) and indexed by $h\in\mathcal{H}_{m}$ converges weakly in
$l^{\infty}(\mathcal{H}_{m})$ to a tight Gaussian process $G^{*}_{i}$ in
$l^{\infty}(\mathcal{H}_{m})$, in outer probability.

###### Proof.

We will apply Theorem 5. We need to show that the sample specific estimators
$(\hat{\Lambda}^{\pi}_{n_{i},i},\hat{\theta}^{\pi}_{n_{i},i})$ satisfy the
conditions of Theorem 3.3.1 in [49]. Consistency of the estimators was shown
in Lemmas 2 and 3 above. Verification of the other conditions can be done as
in [29]. We omit the other details here since the proof goes along the same
lines as points a)-c) below.

a) To verify condition (II.27) of Theorem 5, it suffices to show that for any
sequence $\epsilon_{n_{i}}\to 0$,

$\sup_{\begin{subarray}{c}\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}\leq\epsilon_{n_{i}},\\\
\,\|\beta-\bar{\beta}_{n_{1}+n_{2}}\|\leq\epsilon_{n_{i}},\\\
\|\gamma-\bar{\gamma}_{n_{1}+n_{2}}\|\leq\epsilon_{n_{i}}\end{subarray}}\frac{\left|(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{(\Lambda,\theta),h}-(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}\right|}{n_{i}^{-1/2}\vee\|\beta-\bar{\beta}_{n_{1}+n_{2}}\|\vee\|\gamma-\bar{\gamma}_{n_{1}+n_{2}}\|\vee\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}}$
(II.19)

converges to zero in probability given the data. For simplicity, we can write

$\displaystyle(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{(\Lambda,\theta),h}-(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}$
$\displaystyle=\sum_{j=1}^{6}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})a_{j,h}$

where

$\displaystyle a_{1,h}(\delta,y,x,z)$
$\displaystyle=-h_{21}^{T}x\left\\{\phi(\gamma^{T}x)-\phi(\bar{\gamma}^{T}_{n_{1}+n_{2}}x)\right\\}$
(II.20) $\displaystyle a_{2,h}(\delta,y,x,z)$
$\displaystyle=(1-\delta)h_{21}^{T}x\left\\{g(y,\Lambda,\theta)-g(y,\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}})\right\\}$
$\displaystyle a_{3,h}(\delta,y,x,z)$
$\displaystyle=\delta\left\\{e^{\beta^{T}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)-e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\bar{\Lambda}_{n_{1}+n_{2}}(s)\right\\}$
$\displaystyle a_{4,h}(\delta,y,x,z)$ $\displaystyle=\delta
h^{T}_{22}z\left\\{e^{\beta^{T}z}\Lambda(y)-e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\bar{\Lambda}_{n_{1}+n_{2}}(y)\right\\}$
$\displaystyle a_{5,h}(\delta,y,x,z)$
$\displaystyle=(1-\delta)\left\\{g(y,\Lambda,\theta)e^{\beta^{T}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)\right.$
$\displaystyle\quad\qquad\qquad-\left.g(y,\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}})e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\bar{\Lambda}_{n_{1}+n_{2}}(s)\right\\}$
$\displaystyle a_{6,h}(\delta,y,x,z)$
$\displaystyle=(1-\delta)h^{T}_{22}z\left\\{g(y,\Lambda,\theta)e^{\beta^{T}z}\Lambda(y)\right.$
$\displaystyle\quad\qquad\qquad\qquad-\left.g(y,\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}})e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\bar{\Lambda}_{n_{1}+n_{2}}(y)\right\\}.$

Next we consider the third term. The other terms can be handled similarly.
First note that, by Lemma 2, $\bar{\Lambda}_{n_{1}+n_{2}}$ and
$\bar{\theta}_{n_{1}+n_{2}}$ are consistent estimates of $\bar{\Lambda}$ and
$\bar{\theta}$, and as a result they are bounded on a set of probability
converging to one. From a Taylor expansion we have

$\begin{split}&(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})a_{3,h}\\\
&=(\beta-\bar{\beta}_{n_{1}+n_{2}})^{T}\int z\delta
e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)\\\
&\quad+\int\delta
e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\int_{0}^{y}h_{1}(s)\mathrm{d}(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(s)\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)\\\
&\quad+o^{*}_{P}\left(\int\delta
e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\int_{0}^{y}h_{1}(s)\mathrm{d}(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(s)\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)\right)\\\
&\quad+o^{*}_{P}(\|\beta-\bar{\beta}_{n_{1}+n_{2}}\|)\\\ \end{split}$

For the first term, since the class of functions that we are integrating is
Donsker and uniformly bounded, by Theorem 3.7.2 in [49] it follows that,
conditionally on the observations

$\sup_{\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}\leq\epsilon_{n_{i}}}\int
z\delta
e^{\bar{\beta}^{T}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)=o^{*}_{P}(1).$
(II.21)

The second term can be rewritten as

$\int_{0}^{\bar{\tau}_{0}}D_{n}(s)h_{1}(s)\mathrm{d}(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(s)$

where

$D_{n}(s)=\int\delta\mathds{1}_{\\{y>s\\}}e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z).$

By integration by parts and the chain rule we have

$\begin{split}&\int_{0}^{\bar{\tau}_{0}}D_{n}(s)h_{1}(s)\mathrm{d}(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(s)\\\
&=D_{n}(\bar{\tau}_{0})h_{1}(\bar{\tau}_{0})(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(\bar{\tau}_{0})-\int_{0}^{\bar{\tau}_{0}}(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})\mathrm{d}\left[D_{n}(s)h_{1}(s)\right]\\\
&=D_{n}(\bar{\tau}_{0})h_{1}(\bar{\tau}_{0})(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(\bar{\tau}_{0})-\int_{0}^{\bar{\tau}_{0}}(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(s)D_{n}(s)\mathrm{d}h_{1}(s)\\\
&\quad+\int\delta(\Lambda-\bar{\Lambda})(y)h_{1}(y)e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)\\\
&\quad+\int\delta(\bar{\Lambda}-\bar{\Lambda}_{n_{1}+n_{2}})(y)h_{1}(y)e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)\end{split}$

Again, by Theorem 3.7.2 in [49], it follows that, conditionally on the
observations $D_{n}=o^{*}_{P}(1)$. Since $h_{1}$ is bounded, it follows

$\sup_{\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}\leq\epsilon_{n_{i}}}\frac{\left|D_{n}(\bar{\tau}_{0})h_{1}(\bar{\tau}_{0})(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(\bar{\tau}_{0})\right|}{\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}}=o^{*}_{P}(1).$

In addition we also have that conditionally on the observations
$\sup_{s\in[0,\tau_{0}]}|D_{n}(s)|=o^{*}_{P}(1)$ and since $h_{1}$ is of
bounded variation

$\begin{split}&\sup_{\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}\leq\epsilon_{n_{i}}}\frac{\left|\int_{0}^{\bar{\tau}_{0}}(\Lambda-\bar{\Lambda}_{n_{1}+n_{2}})(s)D_{n}(s)\mathrm{d}h_{1}(s)\right|}{\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}}\\\
&\quad\leq\sup_{t\in[0,\bar{\tau}_{0}]}|D_{n}(s)|\int_{0}^{\bar{\tau}_{0}}\,\left|\mathrm{d}h(s)\right|=o_{P}^{*}(1).\end{split}$

Since $\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}\leq\epsilon_{n_{i}}$
implies $\|\Lambda-\bar{\Lambda}\|_{\infty}\leq\tilde{\epsilon}_{n_{i}}$ for
some $\tilde{\epsilon}_{n_{i}}\to 0$, the class
$\\{g_{\Lambda}(y,\delta,z)=\delta(\Lambda-\bar{\Lambda})(y)h_{1}(y)e^{\bar{\beta}^{T}z}:\,\|\Lambda-\bar{\Lambda}\|_{\infty}\leq\tilde{\epsilon}_{n_{i}}\\}$
is a Donsker class (product of bounded variation functions, uniformly bounded)
and

$\mathbb{E}_{\bar{\mathbb{P}}}\left[\Delta(\Lambda-\bar{\Lambda})(Y)^{2}h_{1}(Y)^{2}e^{2\bar{\beta}^{T}Z}\right]=O(\tilde{\epsilon}_{n}^{2})=o(1),$

we have that, conditionally on the data,

$\sup_{\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}\leq\epsilon_{n_{i}}}\sqrt{n}\int\delta(\Lambda-\bar{\Lambda})(y)h_{1}(y)e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)=o^{*}_{P}(1).$

Finally, since $\|\bar{\Lambda}_{n_{1}+n_{2}}-\bar{\Lambda}\|_{\infty}\to 0$
a.s., by Proposition A.5.3 in [49] it follows that, conditionally on the data,

$\int\delta(\bar{\Lambda}-\bar{\Lambda}_{n_{1}+n_{2}})(y)h_{1}(y)e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})(\delta,y,x,z)=o^{*}_{P}(1).$

Combining all the results we obtain that

$\sup_{\begin{subarray}{c}\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}\leq\epsilon_{n_{i}},\\\
\,\|\beta-\bar{\beta}_{n_{1}+n_{2}}\|\leq\epsilon_{n_{i}},\\\
\|\gamma-\bar{\gamma}_{n_{1}+n_{2}}\|\leq\epsilon_{n_{i}}\end{subarray}}\frac{\left|(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})a_{3,h}\right|}{n_{i}^{-1/2}\vee\|\beta-\bar{\beta}_{n_{1}+n_{2}}\|\vee\|\gamma-\bar{\gamma}_{n_{1}+n_{2}}\|\vee\|\Lambda-\bar{\Lambda}_{n_{1}+n_{2}}\|_{\infty}}$

converges to zero in probability, given the data. The terms related to the
other $a_{j}$ can be treated similarly.

b) Next we check condition (II.28). From (II.19) it follows in particular
that, conditionally on the data,

$\sqrt{n_{i}}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{(\bar{\Lambda},\bar{\theta}),h}-\sqrt{n_{i}}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}}),h}=o_{P}^{*}(1)$

almost surely. Hence, it is sufficient to show that

$\sqrt{n_{i}}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{(\bar{\Lambda},\bar{\theta}),h}\rightsquigarrow
Z_{1}$ (II.22)

on $l^{\infty}(\mathcal{H}_{m})$ in outer probability, where $Z_{1}$ is a
tight random element (actually a Gaussian process). This follows from Theorem
3.7.1. in [49] since the class of functions
$\\{\psi_{(\bar{\Lambda},\bar{\theta}),h}:h\in\mathcal{H}_{m}\\}$ is Donsker
and bounded. This is already shown in step 1 of the proof of Theorem 3 in [29]
(the class of functions is the same, just evaluated at a different point
$(\bar{\Lambda},\bar{\theta})$).

c) Since the functions $\psi_{(\Lambda,\theta),h}$, $h\in\mathcal{H}_{m}$ are
the same as in [29], it can be proved in the same way that
$\bar{\mathbb{P}}{\psi_{(\bar{\Lambda},\bar{\theta}),h}}$ is Fréchet-
differentiable at $(\bar{\Lambda},\bar{\theta})$ and the derivative is given
by

$(\bar{\mathbb{P}}{\dot{\psi}_{(\bar{\Lambda},\bar{\theta})}})((\Lambda,\theta)-(\bar{\Lambda},\bar{\theta}))(h)=\int_{0}^{\bar{\tau}_{0}}\bar{\sigma}_{(1)}(h)\,\mathrm{d}(\Lambda-\bar{\Lambda})(t)+(\theta-\bar{\theta})^{T}\bar{\sigma}_{(2)}(h)$

where $\bar{\sigma}_{(1)}$, $\bar{\sigma}_{(2)}$ are defined as in
(II.12)-(II.13) respectively, with $\mathbb{E}_{\mathbb{P}_{i}}$ replaced by
$\mathbb{E}_{\bar{\mathbb{P}}}$ and evaluated at
$(\bar{\Lambda},\bar{\theta})$ instead of $(\Lambda_{i},\theta_{i})$. Also the
proof that the derivative is continuously invertible remains the same as in
[29].

d) For condition (II.29), consider a sequence
$(\Lambda_{n_{1}+n_{2}},\theta_{n_{1}+n_{2}})$ converging to
$(\bar{\Lambda},\bar{\theta})$ as $n_{1}+n_{2}\to\infty$. We have

$\begin{split}&(\mathbb{P}_{n_{1}+n_{2}}\psi_{\theta_{n_{1}+n_{2}},h}-\mathbb{P}_{n_{1}+n_{2}}\psi_{\bar{\theta},h})-(\bar{\mathbb{P}}\psi_{\theta_{n_{1}+n_{2}},h}-\bar{\mathbb{P}}\psi_{\bar{\theta},h})\\\
&=(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})\psi_{\theta_{n_{1}+n_{2}},h}-(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})\psi_{\bar{\theta},h}=\sum_{j=1}^{6}(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})\bar{a}_{j,h}\end{split}$

where $\bar{a}_{j,h}$ are defined as in (II.20) replacing ($\Lambda$,$\theta$)
and $(\bar{\Lambda}_{n_{1}+n_{2}},\bar{\theta}_{n_{1}+n_{2}})$ by
$(\Lambda_{n_{1}+n_{2}},\theta_{n_{1}+n_{2}})$ and
$(\bar{\Lambda},\bar{\theta})$ respectively. We can deal with this similarly
to what we did to show (II.19). The difference is that now
$(\Lambda_{n_{1}+n_{2}},\theta_{n_{1}+n_{2}})$ and
$(\bar{\Lambda},\bar{\theta})$ are fixed and we consider the class of
functions with respect to $h\in\mathcal{H}_{m}$. For example, for the term
corresponding to $\bar{a}_{3,h}$ we have

$\begin{split}&(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})\bar{a}_{3,h}\\\
&=(\beta_{n_{1}+n_{2}}-\bar{\beta})^{T}\int z\delta
e^{\bar{\beta}^{T}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\bar{\Lambda}(s)\mathrm{d}(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})(\delta,y,x,z)\\\
&\quad+\int\delta
e^{\bar{\beta}^{T}z}\int_{0}^{y}h_{1}(s)\mathrm{d}(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(s)\mathrm{d}(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})(\delta,y,x,z)\\\
&\quad+o^{*}_{P}\left(\int\delta
e^{\bar{\beta}^{T}z}\int_{0}^{y}h_{1}(s)\mathrm{d}(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(s)\mathrm{d}(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})(\delta,y,x,z)\right)\\\
&\quad+o^{*}_{P}(\|\beta_{n_{1}+n_{2}}-\bar{\beta}\|).\\\ \end{split}$ (II.23)

The integral in the first term converges to zero since the class of functions
that we are integrating is Donsker and uniformly bounded. The second term can
be rewritten as

$\int_{0}^{\bar{\tau}_{0}}\bar{D}_{n_{1}+n_{2}}(s)h_{1}(s)\mathrm{d}(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(s)$

where

$\bar{D}_{n_{1}+n_{2}}(s)=\int\delta\mathds{1}_{\\{y>s\\}}e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})(\delta,y,x,z).$

By integration by parts and the chain rule we again have

$\begin{split}&\int_{0}^{\bar{\tau}_{0}}\bar{D}_{n_{1}+n_{2}}(s)h_{1}(s)\mathrm{d}(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(s)\\\
&=\bar{D}_{n_{1}+n_{2}}(\bar{\tau}_{0})h_{1}(\bar{\tau}_{0})(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(\bar{\tau}_{0})-\int_{0}^{\bar{\tau}_{0}}(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(s)\bar{D}_{n_{1}+n_{2}}(s)\mathrm{d}h_{1}(s)\\\
&\quad+\int\delta(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(y)h_{1}(y)e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})(\delta,y,x,z).\end{split}$

By the Glivenko-Cantelli theorem
$\sup_{s\in[0,\bar{\tau}_{0}]}\bar{D}_{n_{1}+n_{2}}(s)=o_{P}(1)$. Since
$h_{1}\in\mathcal{H}_{m}$ are uniformly bounded, it follows that

$\sup_{h_{1}\in\mathcal{H}_{m}}\frac{\left|\bar{D}_{n_{1}+n_{2}}(\bar{\tau}_{0})h_{1}(\bar{\tau}_{0})(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(\bar{\tau}_{0})\right|}{\|\Lambda_{n_{1}+n_{2}}-\bar{\Lambda}\|_{\infty}}=o_{P}(1).$

In addition, since $h_{1}$ are functions of bounded variation and uniformly
bounded norm,

$\begin{split}&\sup_{h_{1}\in\mathcal{H}_{m}}\frac{\left|\int_{0}^{\bar{\tau}_{0}}(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(s)\bar{D}_{n_{1}+n_{2}}(s)\mathrm{d}h_{1}(s)\right|}{\|\Lambda_{n_{1}+n_{2}}-\bar{\Lambda}\|_{\infty}}\\\
&\leq\sup_{s\in[0,\bar{\tau}_{0}]}|\bar{D}_{n_{1}+n_{2}}(s)|\int_{0}^{\bar{\tau}_{0}}\,\left|\mathrm{d}h_{1}(s)\right|=o_{P}(1).\end{split}$

Finally, from Theorem 2.11.23 in [49] it follows that

$\sup_{h_{1}\in\mathcal{H}_{m}}\frac{\sqrt{n_{1}+n_{2}}\int\delta(\Lambda_{n_{1}+n_{2}}-\bar{\Lambda})(y)h_{1}(y)e^{\bar{\beta}^{T}z}\mathrm{d}(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})(\delta,y,x,z)}{\|\Lambda_{n_{1}+n_{2}}-\bar{\Lambda}\|_{\infty}}$

is bounded in probability. Combining all the results we obtain

$\sup_{h_{1}\in\mathcal{H}_{m}}{\left\|(\mathbb{P}_{n_{1}+n_{2}}-\bar{\mathbb{P}})a_{3,h}\right\|}=o_{P}\left(\|\theta_{n_{1}+n_{2}}-\bar{\theta}\|\vee\|\Lambda_{n_{1}+n_{2}}-\bar{\Lambda}\|_{\infty}\right).$

The other terms can be handled similarly obtaining

$\begin{split}&\|(\mathbb{P}_{n_{1}+n_{2}}\psi_{\theta_{n_{1}+n_{2}},h}-\mathbb{P}_{n_{1}+n_{2}}\psi_{\bar{\theta},h})-(\bar{\mathbb{P}}\psi_{\theta_{n_{1}+n_{2}},h}-\bar{\mathbb{P}}\psi_{\bar{\theta},h})\|\\\
&=o_{P}\left({\|\theta_{n_{1}+n_{2}}-\bar{\theta}\|\vee\|\Lambda_{n_{1}+n_{2}}-\bar{\Lambda}\|_{\infty}}\right).\end{split}$

∎

###### Proof of Theorem 4.

We proceed similarly to the proof of Theorem 3. From Lemma 4, it follows that
the process

$\langle
n_{i}^{1/2}(\hat{\Lambda}^{\pi}_{n_{i},i}-\bar{\Lambda}_{n_{1}+n_{2}}),n_{i}^{1/2}(\hat{\theta}^{\pi}_{n_{i},i}-\bar{\theta}_{n_{1}+n_{2}})\rangle_{i=1,2}(h)$

converges to a Gaussian process $G^{*}=(G_{1}^{*},G_{2}^{*})$ with
$G_{2}^{*}=-\sqrt{\kappa/(1-\kappa)}G_{1}^{*}$. We start by deriving the weak
convergence of the process
$\sqrt{a_{n}}(\hat{S}^{\pi}_{n_{1},1}-\bar{S}_{n_{1}+n_{2}},\hat{S}^{\pi}_{n_{2},2}-\bar{S}_{n_{1}+n_{2}})$,
where $a_{n}=\sqrt{n_{1}n_{2}/(n_{1}+n_{2})}$. By a series of Taylor
expansions we can write

$\begin{split}&\sqrt{a_{n}}\\{\hat{S}^{\pi}_{n_{1},1}(t|z)-\bar{S}_{n_{1}+n_{2}}(t|z)\\}\\\
&=\sqrt{a_{n}}\left\\{\exp\left(-\hat{\Lambda}^{\pi}_{n_{1},1}(t)e^{\hat{\beta}^{\pi^{T}}_{n_{1},1}z}\right)-\exp\left(-\bar{\Lambda}_{n_{1}+n_{2}}(t)e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\right)\right\\}\\\
&=-\sqrt{a_{n}}\left\\{\hat{\Lambda}^{\pi}_{n_{1},1}(t)e^{\hat{\beta}^{\pi^{T}}_{n_{1},1}z}-\bar{\Lambda}_{n_{1}+n_{2}}(t)e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\right\\}\exp\left(-\bar{\Lambda}_{n_{1}+n_{2}}(t)e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\right)+R_{1}\\\
&=-\sqrt{a_{n}}\left\\{\hat{\Lambda}^{\pi}_{n_{1},1}(t)-\bar{\Lambda}_{n_{1}+n_{2}}(t)\right\\}e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\exp\left(-\bar{\Lambda}_{n_{1}+n_{2}}(t)e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\right)\\\
&\quad-\sqrt{a_{n}}\left\\{e^{\hat{\beta}^{\pi^{T}}_{n_{1},1}z}-e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\right\\}\bar{\Lambda}_{n_{1}+n_{2}}(t)\exp\left(-\bar{\Lambda}_{n_{1}+n_{2}}(t)e^{\bar{\beta}^{T}_{n_{1}+n_{2}}z}\right)+R_{1}+R_{2}\\\
&=-\sqrt{a_{n}}\left\\{\hat{\Lambda}^{\pi}_{n_{1},1}(t)-\bar{\Lambda}_{n_{1}+n_{2}}(t)\right\\}e^{\bar{\beta}^{T}z}\exp\left(-\bar{\Lambda}(t)e^{\bar{\beta}^{T}z}\right)\\\
&\quad-\sqrt{a_{n}}\left\\{{\hat{\beta}^{\pi}_{n_{1},1}}-{\bar{\beta}_{n_{1}+n_{2}}}\right\\}^{T}ze^{\bar{\beta}^{T}z}\bar{\Lambda}(t)\exp\left(-{\bar{\Lambda}}(t)e^{\bar{\beta}^{T}z}\right)+R_{1}+R_{2}+R_{3}\\\
\end{split}$

where the remainder terms converge to zero in probability. Considering
functions $h\in\mathcal{H}_{m}$ of the form

$\begin{split}h_{t,z}&=(h_{1;t,z},h_{2;t,z})\\\
&=\left(\mathds{1}_{[0,t]}(\cdot)e^{\bar{\beta}^{T}z}\exp\left(-\bar{\Lambda}(t)e^{\bar{\beta}^{T}z}\right),\left(\mathbf{0}_{p},ze^{\bar{\beta}^{T}z}\bar{\Lambda}(t)\exp\left(-\bar{\Lambda}(t)e^{\bar{\beta}^{T}z}\right)\right)\right),\end{split}$

where $\mathbf{0}_{p}$ denotes a zero vector in $\mathbb{R}^{p}$, we have

$\begin{split}&\sqrt{a_{n}}\\{\hat{S}^{\pi}_{n_{1},1}(t|z)-\bar{S}_{n_{1}+n_{2}}(t|z)\\}\\\
&=\sqrt{1-\kappa}\langle
n_{1}^{1/2}(\hat{\Lambda}^{\pi}_{n_{1},1}-\bar{\Lambda}_{n_{1}+n_{2}}),n_{1}^{1/2}(\hat{\theta}^{\pi}_{n_{1},1}-\bar{\theta}_{n_{1}+n_{2}})\rangle(h_{t,z})+o_{P^{*}}(1)\end{split}$

from which we conclude that, given the data, the process
$\sqrt{a_{n}}\\{\hat{S}^{\pi}_{n_{1},1}(t|z)-\bar{S}_{n_{1}+n_{2}}(t|z)\\}$
converges weakly in $D^{\infty}([0,\bar{\tau}_{0}])$ to a mean zero Gaussian
process $\bar{G}_{1,z}$ with covariance

$\begin{split}\rho_{1,z}(s,t)&=Cov(\bar{G}_{1,z}(s),\bar{G}_{1,z}(t))\\\
&=(1-\kappa)\left\\{\int_{0}^{\tau_{0,1}}h_{1;s,z}(u)\bar{\sigma}_{(1)}^{-1}(h_{t,z})(u)\mathrm{d}\bar{\Lambda}(u)+h^{T}_{2;s,z}\bar{\sigma}_{(2)}^{-1}(h_{t,z})\right\\},\end{split}$
(II.24)

where $\bar{\sigma}_{(1)}$, $\bar{\sigma}_{(2)}$ are defined as in
(II.12)-(II.13) respectively, with $\mathbb{E}_{\mathbb{P}_{i}}$ replaced by
$\mathbb{E}_{\bar{\mathbb{P}}}$ and evaluated at
$(\bar{\Lambda},\bar{\theta})$ instead of $(\Lambda_{i},\theta_{i})$ Defining
$\overline{MST}_{u,z}$ as the conditional expected lifetime of the uncured in
the pooled sample, we have

$\begin{split}&\sqrt{a_{n}}(\widehat{MST}^{\pi}_{u,1,z}-\overline{MST}_{u,z})\\\
&=\sqrt{a_{n}}\int_{0}^{\bar{\tau}_{0}}\\{\hat{S}^{\pi}_{n_{1},1}(t|z)-\bar{S}_{n_{1}+n_{2}}(t|z)\\}\mathrm{d}t\\\
&\quad-\sqrt{a_{n}}\\{\bar{\tau}_{0}-Y^{\pi}_{1,(m_{1})}\\}\hat{S}^{\pi}_{n_{1},1}(\bar{\tau}_{0})-\sqrt{a_{n}}\\{\bar{\tau}_{0}-{Y}_{(m)}\\}\bar{S}_{n_{1}+n_{2}}(\bar{\tau}_{0}),\end{split}$
(II.25)

where $Y^{\pi}_{1,(m_{1})}$ and ${Y}_{(m)}$ denote the largest uncensored
observation in the first permuted sample and the pooled sample respectively.
As in the proof of Theorem 3, the second term and third term in the right hand
side of the previous equation can be shown to converge to zero in probability.
Considering the map

$\psi:{D}[0,\bar{\tau}_{0}]\to\mathbb{R}\qquad\psi(\xi)=\int_{0}^{\bar{\tau}_{0}}\xi(u)\,\mathrm{d}u$

which is Hadamard-differentiable, it follows that, given the data,
$\sqrt{a_{n}}(\widehat{MST}^{\pi}_{u,1,z}-\overline{MST}_{u,z},\widehat{MST}^{\pi}_{u,2,z}-\overline{MST}_{u,z})$
converges weakly to a two dimensional Gaussian random vector

$(N_{1},N_{2})=\left(\int_{0}^{\bar{\tau}_{0}}\bar{G}_{1,z}(u)\,\mathrm{d}u,\int_{0}^{\bar{\tau}_{0}}\bar{G}_{2,z}(u)\,\mathrm{d}u\right)=(N_{1},-\frac{\kappa}{1-\kappa}N_{1}).$

Taking the difference of both entries of the pair, we conclude that, given the
data, $\sqrt{a_{n}}(\widehat{MST}^{\pi}_{u,1,z}-\widehat{MST}^{\pi}_{u,2,z})$
converges weakly (in probability) to a mean-zero Gaussian random variable with
variance

$\sigma^{\pi
2}_{z}=\frac{1}{(1-\kappa)^{2}}\int_{0}^{\bar{\tau}_{0}}\int_{0}^{\bar{\tau}_{0}}\rho_{1,z}(s,t)\,\mathrm{d}s\,\mathrm{d}t,$
(II.26)

where $\rho_{1,z}$ is defined in (II.24). This concludes the proof. ∎

##  Appendix 2 Permutation of $Z$-estimators in a two-sample set-up

In this appendix, we discuss the asymptotic properties of randomly permuted
$Z$-estimators. For this, we consider a two independent samples set-up with
$n_{1}$ and $n_{2}$ i.i.d. random vectors
$W_{11},\dots,W_{1n_{1}}\sim\mathbb{P}_{1}$ and
$W_{21},\dots,W_{2n_{2}}\sim\mathbb{P}_{2}$, respectively. Let $\Theta$ be a
subset of a Banach space, $\Psi_{n_{1},1},\Psi_{n_{2},2}:\Theta\to\mathbb{L}$
be random maps, and $\Psi:\Theta\to\mathbb{L}$ be a deterministic map.
Solutions (or approximate solutions) $\hat{\theta}_{n_{i},i}$ to the equations
$\Psi_{n_{i},i}(\theta)\stackrel{{\scriptstyle!}}{{=}}0$ will be called _Z-
estimators_. Due to the i.i.d. set-up, we assume the structure
$\Psi_{n_{i},i}(\theta)h=\mathbb{P}_{n_{i},i}\psi_{\theta,h}$, for given
measurable functions $\psi_{\theta,h}$ indexed by $\Theta$ and
$h\in\mathcal{H}$ for some index set $\mathcal{H}$, where
$\mathbb{P}_{n_{i},i}$ denotes the $i$-th empirical process. Thus, we
understand the equation system in the space
$\mathbb{L}=\ell^{\infty}(\mathcal{H})$.

For the random permutation approach, we randomly re-assign the $n_{1}+n_{2}$
observations of the pooled sample
$(W_{11},\dots,W_{1n_{1}},W_{21},\dots,W_{2n_{2}})=:(W_{1},\dots,W_{n_{1}+n_{2}})$
to the groups 1 and 2 without changing the original sample sizes. For a random
permutation $\pi$ of the numbers $1,\dots,n_{1}+n_{2}$, the permuted samples
can be expressed as $W_{\pi(1)},\dots,W_{\pi(n_{1})}$ and
$W_{\pi(n_{1}+1)},\dots,W_{\pi(n_{1}+n_{2})}$. For notational convenience, we
denote the permuted samples by $W^{\pi}_{i1},\dots,W^{\pi}_{in_{i}}$, for
sample group $i=1,2$, and the corresponding $i$-th permutation empirical
process by $\mathbb{P}_{n_{i},i}^{\pi}$. Let
$\Psi_{n_{i},i}^{\pi}(\theta)h=\mathbb{P}_{n_{i},i}^{\pi}\psi_{\theta,h}\stackrel{{\scriptstyle!}}{{=}}0$
for all $h\in\mathcal{H}$ be the estimating equation corresponding to
$\Psi_{n_{i},i}(\theta)\stackrel{{\scriptstyle!}}{{=}}0$, just based on the
$i$-th permuted sample. We denote the (approximate) solution to the $i$-th
permuted estimating equation by $\theta^{\pi}_{n_{i},i}$. For future uses, let
$\mathbb{G}^{\pi}_{n_{i},i}=\sqrt{n_{i}}(\mathbb{P}_{n_{i},i}^{\pi}-\mathbb{P}_{n_{1}+n_{2}})$
be the $i$-th normalized permutation empirical process, where
$\mathbb{P}_{n_{1}+n_{2}}$ denotes the empirical process of the pooled sample.
The centering at $\mathbb{P}_{n_{1}+n_{2}}$ seems reasonable, as this has an
interpretation as a conditional expectation:

$E(\mathbb{P}_{n_{i},i}^{\pi}\psi_{\theta,h}\ |\
W_{ij}:i=1,2;j=1,\dots,n_{i})=\mathbb{P}_{n_{1}+n_{2}}\psi_{\theta,h}.$

Let $\bar{\theta}_{n_{1}+n_{2}}$ be the (approximate) solution to
$\mathbb{P}_{n_{1}+n_{2}}\psi_{\theta,h}\stackrel{{\scriptstyle!}}{{=}}0$ for
all $h\in\mathcal{H}$.

The following theorem represents a version of Theorem 3.3.1 of [49] for the
random permutation-based estimators.

######  Theorem 5.

Assume that $\frac{n_{1}}{n_{1}+n_{2}}\to\lambda\in(0,1)$ as
$n_{1}+n_{2}\to\infty$, define
$\bar{\mathbb{P}}=\lambda\mathbb{P}_{1}+(1-\lambda)\mathbb{P}_{2}$, and assume
that Theorem 3.3.1 holds for each sample-specific $Z$-estimator
$\hat{\theta}_{n_{1},1}$ and $\hat{\theta}_{n_{2},2}$. Let the criterion
functions $\psi_{\cdot,h}$ be such that

$\displaystyle\begin{split}\|\mathbb{G}_{n_{i},i}^{\pi}(\psi_{\theta_{n_{i},i}^{\pi},h}-\psi_{\bar{\theta}_{n_{1}+n_{2}},h})\|_{\mathcal{H}}=o_{P}^{*}(1+\sqrt{n_{i}}\|\theta^{\pi}_{n_{i},i}-\bar{\theta}_{n_{1}+n_{2}}\|).\end{split}$
(II.27)

Conditionally on $W_{11},W_{21},W_{12},W_{22},\dots$, assume that

$\displaystyle(\sqrt{n_{i}}(\mathbb{P}^{\pi}_{n_{i},i}-\mathbb{P}_{n_{1}+n_{2}})\psi_{\bar{\theta}_{n_{1}+n_{2}},h})_{i=1}^{2}\rightsquigarrow(Z_{1},Z_{2})$
(II.28)

on $(\ell^{\infty}(\mathcal{H}))^{2}$ in outer probability, where
$(Z_{1},Z_{2})$ is a tight random element.

We assume that $\theta\mapsto\bar{\mathbb{P}}\psi_{\theta,h}$ is Fréchet-
differentiable in $\ell^{\infty}(\mathcal{H})$ at $\bar{\theta}$ with a
continuously invertible derivative
$\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h}$, and that, for any sequence
$(\theta_{n_{1}+n_{2}})_{n_{1},n_{2}}$ converging to $\bar{\theta}$,

$\displaystyle\|(\mathbb{P}_{n_{1}+n_{2}}\psi_{\theta_{n_{1}+n_{2}},h}-\mathbb{P}_{n_{1}+n_{2}}\psi_{\bar{\theta},h})-(\bar{\mathbb{P}}\psi_{\theta_{n_{1}+n_{2}},h}-\bar{\mathbb{P}}\psi_{\bar{\theta},h})\|_{\mathcal{H}}=o_{P}^{*}(\|\theta_{n_{1}+n_{2}}-\bar{\theta}\|)$
(II.29)

as $n_{1}+n_{2}\to\infty$.

If $\theta^{\pi}_{n_{i},i}$ and $\bar{\theta}_{n_{1}+n_{2}}$ satisfy
$\|\mathbb{P}^{\pi}_{n_{i},i}\psi_{\theta^{\pi}_{n_{i},i},h}\|_{\mathcal{H}}=o_{P}^{*}(n^{-1/2})$,
$i=1,2$, and
$\|\mathbb{P}_{n_{1}+n_{2}}\psi_{\bar{\theta}_{n_{1}+n_{2}},h}\|_{\mathcal{H}}=o_{P}^{*}(n^{-1/2})$,
respectively, and if all three estimators converge in outer probability to
$\bar{\theta}$, then

$\displaystyle\sqrt{n_{i}}(\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})(\theta_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}})=-\sqrt{n_{i}}(\mathbb{P}_{n_{i},i}^{\pi}-\mathbb{P}_{n_{1}+n_{2}})\psi_{\bar{\theta}_{n_{1}+n_{2}},h}+o_{P}^{*}(1)\rightsquigarrow-(Z_{1},Z_{2})$
(II.30)

as $n_{1}+n_{2}\to\infty$ conditionally on
$W_{11},W_{21},W_{12},W_{22},\dots$, in outer probability. Finally,
$(\sqrt{n_{1}}(\theta^{\pi}_{n_{1},1}-\bar{\theta}_{n_{1}+n_{2}}),\sqrt{n_{2}}(\theta^{\pi}_{n_{2},2}-\bar{\theta}_{n_{1}+n_{2}}))\rightsquigarrow-((\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})^{-1}Z_{1},(\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})^{-1}Z_{2})$
conditionally on $W_{11},W_{21},W_{12},W_{22},\dots$, in outer probability.

Note that, due to the equality
$\mathbb{P}^{\pi}_{n_{2},2}-\mathbb{P}_{n_{1}+n_{2}}=-\frac{n_{1}}{n_{2}}(\mathbb{P}^{\pi}_{n_{1},1}-\mathbb{P}_{n_{1}+n_{2}})$,
$Z_{1}$ and $Z_{2}$ are perfectly negatively linearly correlated:
$Z_{2}=-\sqrt{\frac{\lambda}{1-\lambda}}Z_{1}$.

###### Proof.

The essential steps of this proof are similar to those in the proof of Theorem
3.3.1 of [49]. But for the sake of completeness, we shall present the whole
proof.

The assumed consistencies of the pooled and the permuted estimators and then
assumption (II.27) entail that

$\displaystyle\begin{split}&\sqrt{n_{i}}(\mathbb{P}_{n_{1}+n_{2}}\psi_{\theta^{\pi}_{n_{i},i},h}-\mathbb{P}_{n_{1}+n_{2}}\psi_{\bar{\theta}_{n_{1}+n_{2}},h})\\\
&=\sqrt{n_{i}}(\mathbb{P}_{n_{1}+n_{2}}\psi_{\theta^{\pi}_{n_{i},i},h}-\mathbb{P}_{n_{i},i}^{\pi}\psi_{\theta_{n_{i},i}^{\pi},h})+o_{P}^{*}(1)\\\
&=-\mathbb{G}_{n_{i},i}^{\pi}(\psi_{\theta^{\pi}_{n_{i},i},h}-\psi_{\theta_{n_{i},i}^{\pi},h})+o_{P}^{*}(1)\\\
&=-\sqrt{n_{i}}(\mathbb{P}_{n_{i},i}^{\pi}-\mathbb{P}_{n_{1}+n_{2}})\psi_{\bar{\theta}_{n_{1}+n_{2}},h}+o_{P}^{*}(1+\sqrt{n_{i}}\|\theta_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}}\|).\end{split}$
(II.31)

The consistencies of $\theta^{\pi}_{n_{i},i}$ and $\bar{\theta}_{n_{1}+n_{2}}$
for $\bar{\theta}$ in combination with the approximation (II.29) applied twice
imply that the norm of the left-hand side of (LABEL:eq:331_333_perm) equals

$\displaystyle\sqrt{n_{i}}(\|\bar{\mathbb{P}}\psi_{\theta^{\pi}_{n_{i},i},h}-\bar{\mathbb{P}}\psi_{\bar{\theta}_{n_{1}+n_{2}},h}\|_{\mathcal{H}}+o_{P}^{*}(\|\theta^{\pi}_{n_{i},i}-\bar{\theta}_{n_{1}+n_{2}}\|)).$
(II.32)

Additionally, the Fréchet-differentiability of
$\theta\mapsto\bar{\mathbb{P}}\psi_{\theta,h}$ at $\bar{\theta}$ and the
continuous invertibility of the derivative
$\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h}$ respectively imply that

$\displaystyle\|\bar{\mathbb{P}}\psi_{\theta,h}-\bar{\mathbb{P}}\psi_{\bar{\theta},h}\|_{\mathcal{H}}=\|(\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})(\theta-\bar{\theta})\|_{\mathcal{H}}+o(\|\theta-\bar{\theta}\|)$
(II.33)

and that the right-hand side in the previous display is bounded below by

$c\|\theta-\bar{\theta}\|+o(\|\theta-\bar{\theta}\|)$

for some positive constant $c$. Combine this with (II.32) and
(LABEL:eq:331_333_perm) to see that

$\displaystyle\sqrt{n_{i}}\|\theta^{\pi}_{n_{i},i}-\bar{\theta}_{n_{1}+n_{2}}\|(c+o_{P}(1))\leq
O_{P}(1)+o_{P}(1+\sqrt{n_{i}}\|\theta_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}}\|),$

conditionally on $W_{11},W_{21},W_{12},W_{22},\dots$ in probability. Thus, in
the same manner, $\theta^{\pi}_{n_{i},i}$ is $\sqrt{n_{i}}$-consistent for
$\bar{\theta}_{n_{1}+n_{2}}$ in norm.

Next, apply the approximation in (II.29) to the left-hand side of
(LABEL:eq:331_333_perm) and use the Fréchet-differentiability of
$\theta\mapsto\bar{\mathbb{P}}\psi_{\theta,h}$ at $\bar{\theta}$ to find that
(LABEL:eq:331_333_perm) equals

$\sqrt{n_{i}}(\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})(\theta_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}})+o_{P}^{*}(\sqrt{n_{i}}\|\theta_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}}\|).$

Since
$o_{P}^{*}(\sqrt{n_{i}}\|\theta_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}}\|)$
and also
$o_{P}^{*}(1+\sqrt{n_{i}}\|\theta_{n_{i},i}^{\pi}-\bar{\theta}_{n_{1}+n_{2}}\|)$
in the right-hand side of (LABEL:eq:331_333_perm) are both $o_{P}(1)$, the
assertion given in (II.30) follows from assumption (II.28).

The continuity of $(\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})^{-1}$
together with the continuous mapping theorem and the just established
conditional weak convergence (II.30) in outer probability imply the
corresponding conditional weak convergence

$(\sqrt{n_{1}}(\theta^{\pi}_{n_{1},1}-\bar{\theta}_{n_{1}+n_{2}}),\sqrt{n_{2}}(\theta^{\pi}_{n_{2},2}-\bar{\theta}_{n_{1}+n_{2}}))\rightsquigarrow-((\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})^{-1}Z_{1},(\bar{\mathbb{P}}\dot{\psi}_{\bar{\theta},h})^{-1}Z_{2})$

in outer probability. ∎

##  Appendix 3 R code for accessing the breast cancer data set

    if (!require("BiocManager", quietly = TRUE))
        install.packages("BiocManager")
    BiocManager::install("curatedBreastData")
    library(curatedBreastData)

    # read the data, study id 2034
    data(curatedBreastDataExprSetList)
    D=curatedBreastDataExprSetList$study_2034_GPL96_all

    # Relapse-free survival
    Data=data.frame(Y=D$RFS_months_or_MIN_months_of_RFS,status=1-D$RFS,
        age=D$age,ER=D$ER_preTrt,sizeTum=D$tumor_stage_preTrt)

[Acknowledgments] The authors would like to thank Sarah Friedrich for useful
comments regarding the data set.

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