$\Lambda$ piecewise constant with jumps at the observed events

$\displaystyle{l}_{n}(\beta,\Lambda,\gamma)-\hat{l}_{n}^{*}(\beta,\Lambda_{|\tau^{*}},\gamma)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{{\tau^{*}\leq
Y_{i}<\tau_{0}}\\}}\Delta_{i}\log\Lambda(\\{Y_{i}\\})$
$\displaystyle\quad+\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{{\tau^{*}\leq
Y_{i}<\tau_{0}}\\}}\Delta_{i}\beta^{\prime}Z_{i}$
$\displaystyle\quad-\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\}}e^{\beta^{\prime}Z_{i}}\\{\Delta_{i}\Lambda(Y_{i})-\mathds{1}_{\\{B_{i}=1\\}}\Lambda(\tau^{*}-)\\}$
$\displaystyle\quad+\frac{1}{n}\\!\sum_{i=1}^{n}\\!\mathds{1}_{\\{\tau_{0}\geq
Y_{i}\geq\tau^{*}\\!\\}}\\!\\!\left[\\!(1\\!-\\!\Delta_{i})\log\\!\left\\{\\!1\\!-\\!\phi_{i}(\gamma)\\!+\\!\phi_{i}(\gamma)\exp\left(\\!-\Lambda(Y_{i})e^{\beta^{\prime}Z_{i}}\\!\right)\\!\\!\right\\}\right.\\!$
$\displaystyle\quad\quad\left.-\\!\mathds{1}_{\\{B_{i}=0\\}}\\!\log\left\\{1\\!-\\!\phi_{i}(\gamma)\right\\}\\!\right]$
$\displaystyle=:r_{1n}(\Lambda;\tau^{*})+r_{2n}(\beta;\tau^{*})-r_{3n}(\Lambda,\beta;\tau^{*})+r_{4n}(\Lambda,\beta,\gamma;\tau^{*}),$

where $\phi_{i}(\gamma)$ is a short notation for $\phi(\gamma,X_{i})$. For
proving the Lemma, we have to suitably bound $r_{1n},\ldots,r_{4n}$. For this
purpose, let us notice that, by definition, all the cumulative hazard
functions we have to consider ($\hat{\Lambda}_{n}$, $\tilde{\Lambda}_{0,n}$,…)
have bounded jumps at the event times. More precisely, because the parameter
space $\mathcal{B}$ and $Z$ are supposed bounded, there exist constants
$0<c_{l}<c_{u}$ such that

$c_{l}\leq\exp(\beta^{\prime}Z)\leq c_{u}.$

Then the largest jump of any of the cumulative hazard functions we need to
consider is bounded by $1/c_{l}$ (which is located at the last uncensored
observation), the second largest one (and is located at the before last
uncensored observation) is bounded by $1/2c_{l}$,…

To control $r_{1n}(\Lambda;\tau^{*})$, one would look for a suitable lower
bound forthe jumps of $\Lambda$. However, no meaningful lower bound could be
derived for these jumps. More precisely, such a bound is necessarily of order
$1/n$, so that the sequence of the logarithm of the jumps is unbounded.
Fortunately, for our purposes it suffices to find a bound for

$\left|r_{1n}(\hat{\Lambda}_{n};\tau^{*})-r_{1n}(\tilde{\Lambda}_{0,n};\tau^{*})\right|=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{{\tau^{*}\leq
Y_{i}<\tau_{0}}\\}}\Delta_{i}\left|\log\frac{\hat{\Lambda}_{n}(\\{Y_{i}\\})}{\tilde{\Lambda}_{0,n}(\\{Y_{i}\\})}\right|,$

where

$\frac{\hat{\Lambda}_{n}(\\{Y_{i}\\})}{\tilde{\Lambda}_{0,n}(\\{Y_{i}\\})}=\frac{\sum_{j=1}^{n}\mathds{1}_{\\{\tau_{0}\geq
Y_{j}\geq
Y_{i}\\}}\exp(\beta^{\prime}_{0}Z_{j})\left\\{\Delta_{j}+(1-\Delta_{j})g_{j}(Y_{j},\Lambda_{0},\beta_{0},\hat{\gamma}_{n}))\right\\}}{\sum_{j=1}^{n}\mathds{1}_{\\{{\tau_{0}\geq
Y_{j}\geq
Y_{i}}\\}}\exp(\beta^{\prime}Z_{j})\left\\{\Delta_{j}+(1-\Delta_{j})g_{j}(Y_{j},{\Lambda}_{n,\hat{\beta},\hat{\gamma}_{n}},\beta,\hat{\gamma}_{n})\right\\}}.$

Since all $g_{j}$’s are between 0 and 1, it is easy to see that for any
uncensored $Y_{i}\geq\tau^{*}$,

$\frac{1}{\rho_{n}(Y_{i})}\frac{c_{l}}{c_{u}}\leq\frac{\hat{\Lambda}_{n}(\\{Y_{i}\\})}{\tilde{\Lambda}_{0,n}(\\{Y_{i}\\})}\leq\frac{c_{u}}{c_{l}}\rho_{n}(Y_{i}),$

where

$\rho_{n}(t)=\frac{\sum_{j=1}^{n}\mathds{1}_{\\{\tau_{0}\geq Y_{j}\geq
t\geq\tau^{*}\\}}}{\sum_{j=1}^{n}\Delta_{j}\mathds{1}_{\\{\tau_{0}\geq
Y_{j}\geq t\geq\tau^{*}\\}}},\qquad t\in[\tau^{*},\tau_{0}].$

Thus, since all $\rho_{n}(Y_{i})$’s are larger than 1, it suffices to suitably
bound

$0\leq
A_{n}=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\}}\Delta_{i}\log(\rho_{n}(Y_{i})),$

which we decompose as

$A_{n}\\!=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\in[\tau^{*}\\!,\tau_{0}-a_{n}]\\}}\Delta_{i}\log(\rho_{n}(Y_{i}))+\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\in[\tau_{0}-a_{n},\tau_{0}]\\}}\Delta_{i}\log(\rho_{n}(Y_{i}))=:A_{1n}\\!+\\!A_{2n},$

for some sequence of real numbers $a_{n}$, $n\geq 1$, decreasing to zero. The
rate of $a_{n}$ should be taken such that, on one hand, for any constant
$C>0$,

$\mathbb{P}(\limsup_{n\rightarrow\infty}A_{2n}>C)=0,$ (A20)

and, on the other hand, the $\limsup$ of $A_{1n}$ could be controlled by a
function of $\tau^{*}$ almost surely. More precisely, since

$A_{2n}\leq\frac{\log
n}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\in[\tau_{0}-a_{n},\tau_{0}]\\}}\Delta_{i},$

we take $a_{n}$ such that $p_{n}\log n\rightarrow 0$ and
$p_{n}\log^{2}n\rightarrow\infty$, where

$p_{n}=\mathbb{P}(Y\in[\tau_{0}-a_{n},\tau_{0}],\Delta=1)=\mathbb{E}\left[\phi(\gamma_{0},X)\int_{[\tau_{0}-a_{n},\tau_{0}]}F_{C}([t,\infty)|X,Z)F_{u}(dt|Z)\right].$

Then, by Theorem 1(i) from [9], we have

$\lim_{n\rightarrow\infty}\frac{1}{p_{n}}\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\in[\tau_{0}-a_{n},\tau_{0}]\\}}\Delta_{i}=1,\quad
a.s.,$

which implies (A20). On the other hand, we have

$A_{1n}\leq\log\left(\sup_{t\in[\tau^{*},\tau_{0}-a_{n}]}\rho_{n}(t)\right)\times\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\in[\tau^{*},\tau_{0}]\\}}\Delta_{i}.$

By the same Theorem 1(i) from [9],

$\lim_{n\rightarrow\infty}\sup_{t\in[\tau^{*},\tau_{0}-a_{n}]}\left[\rho_{n}(t)\frac{\mathbb{P}(Y\in[t,\tau_{0}-a_{n}],\Delta=1)}{\mathbb{P}(Y\in[t,\tau_{0}-a_{n}])}\right]=1,\quad
a.s.$

By our assumptions, there exists a constant $C_{r}$, independent of
$\tau^{*}$, $\beta$, $\gamma$ and $\Lambda$, such that

$1<\inf_{t\in[\tau^{*},\tau_{0}{-a_{n}}]}\frac{\mathbb{P}(Y\in[t,\tau_{0}-a_{n}])}{\mathbb{P}(Y\in[t,\tau_{0}-a_{n}],\Delta=1)}<\sup_{t\in[\tau^{*},\tau_{0}{-a_{n}}]}\frac{\mathbb{P}(Y\in[t,\tau_{0}-a_{n}])}{\mathbb{P}(Y\in[t,\tau_{0}-a_{n}],\Delta=1)}\leq
C_{r}.$

Gathering facts, deduce with probability 1, for sufficiently large $n$,

$\left|r_{1n}(\hat{\Lambda}_{n};\tau^{*})-r_{1n}(\tilde{\Lambda}_{0,n};\tau^{*})\right|\leq
c\frac{N^{*}}{n},$

where $N^{*}$ be the number of uncensored observations in
$[\tau^{*},\tau_{0}]$ and $c$ is some constant (independent of $\tau^{*}$,
$\beta$, $\gamma$ and $\Lambda$). Here, $N^{*}$ is a binomial random variable
with $n$ trials and success probability

$p^{*}=\mathbb{P}(Y\geq\tau^{*},\Delta=1)=\mathbb{E}\left[\phi(\gamma_{0},X)\int_{[\tau^{*},\tau_{0}]}F_{C}([t,\infty)|X,Z)F_{u}(dt|Z)\right]\\\
\leq\sup_{x}\phi(\gamma_{0},x)\mathbb{P}(Y\geq\tau^{*}).$

To bound $r_{3n}=r_{3n}(\Lambda,\beta;\tau^{*})$, we note that
$\mathds{1}_{\\{B_{i}=1\\}}=\Delta_{i}+(1-\Delta_{i})\mathds{1}_{\\{B_{i}=1\\}}$
and rewrite

$r_{3n}=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\}}e^{\beta^{\prime}Z_{i}}\Delta_{i}\\{\Lambda(Y_{i})-\Lambda(\tau^{*}-)\\}\\\
-\frac{\Lambda(\tau^{*}-)}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\}}e^{\beta^{\prime}Z_{i}}\mathds{1}_{\\{B_{i}=1\\}}(1-\Delta_{i})=r_{3an}-r_{3bn}.$

On one hand,

$r_{3an}\leq\frac{c_{u}}{c_{l}}\times\frac{N^{*}}{n}.$

The last inequality is obtained by bounding the jumps of $\Lambda$ and using
the following identity: for any integer $M\geq 1$,

$\sum_{k=1}^{M}\sum_{j=k}^{M}\frac{1}{j}=\sum_{j,k=1}^{M}\frac{\mathds{1}_{\\{k\leq
j\\}}}{j}=M.$

To bound $r_{3bn}$, let us note that

$\Lambda(\tau^{*}-)\leq\frac{1}{c_{l}}\sum_{j=N^{*}+1}^{N}\frac{1}{j}\leq
c_{1}\log\frac{N}{N^{*}},$

with $c_{1}$ some constant depending only on $c_{l}$ and the maximal value of
the convergent sequence

$\sum_{j=1}^{m}\frac{1}{j}-\log m,\quad m\geq 1.$

Here, $N=\sum_{i=1}^{n}\Delta_{i}$ is a binomial random variable with $n$
trials and success probability

$p=\mathbb{P}(\Delta=1)=\mathbb{E}\left[\phi(\gamma_{0},X)\int_{[0,\tau_{0}]}F_{C}([t,\infty)|X,Z)F_{u}(dt|Z)\right].$

Thus

$r_{3bn}\leq
c\log\frac{N}{N^{*}}\times\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\}}\mathds{1}_{\\{B_{i}=1\\}}(1-\Delta_{i})=c\log\frac{N/n}{N^{*}/n}\times\frac{Q^{*}}{n},$

where $Q^{*}$ is a binomial variable with $n$ trials and success probability

$q^{*}=\mathbb{E}\left[\phi(\gamma_{0},X)\int_{[\tau^{*},\tau_{0}]}F_{u}([t,\tau_{0}]|X,Z)F_{C}(dt|X,Z)\right]\\\
\leq\mathbb{E}\left[\phi(\gamma_{0},X)F_{C}([\tau^{*},\tau_{0}]|X,Z)F_{u}([\tau^{*},\tau_{0}]|Z)\right]\\\
\leq\left[\sup_{x}\phi(\gamma_{0},x)\right]\left[\sup_{x,z}\frac{F_{C}([\tau^{*},\tau_{0}]|X=x,Z=z)}{\tau_{0}-\tau^{*}}\right]\times(\tau^{*}-\tau_{0})\times\mathbb{P}(T_{0}\geq\tau^{*})\\\
\leq c(\tau^{*}-\tau_{0})\times\mathbb{P}(T_{0}\geq\tau^{*}),$

and $c$ is some constant. By the strong Law of Large Numbers,

$\lim_{n\rightarrow\infty}\log\frac{N}{N^{*}}=\log\frac{p}{p^{*}},\,a.s.$

Next, to bound $r_{2n}=r_{2n}(\beta;\tau^{*})$, we write

$r_{2n}=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{{\tau^{*}\leq
T_{0}<\tau_{0}}\\}}e^{\beta^{\prime}Z_{i}}\Delta_{i}\leq
c_{u}\frac{N^{*}}{n}.$

Finally, to control $r_{4n}=r_{4n}(\Lambda,\beta,\gamma;\tau^{*})$, since
$\mathds{1}_{\\{B_{i}=0\\}}=(1-\Delta_{i})\mathds{1}_{\\{B_{i}=0\\}}$ and
$\log(1+u)\leq u,$ $\forall u\geq 0$, we have

$r_{4n}=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{\tau_{0}\geq
Y_{i}\geq\tau^{*}\\}}\mathds{1}_{\\{B_{i}=0\\}}\log\left\\{1+\frac{\phi_{i}(\gamma)\exp\left(-\Lambda(Y_{i})e^{\beta^{\prime}Z_{i}}\right)}{1-\phi_{i}(\gamma)}\right\\}\\\
+\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\!\\}}(1-\Delta_{i})\mathds{1}_{\\{B_{i}=1\\}}\log\left\\{1-\phi_{i}(\gamma)+\phi_{i}(\gamma)\exp\left(-\Lambda(Y_{i})e^{\beta^{\prime}Z_{i}}\right)\right\\}.$

Thus

$|r_{4n}|\leq\sup_{\gamma,x}\left|\frac{\phi(\gamma,x)}{1-\phi(\gamma,x)}\right|\exp\left(-c_{l}\Lambda(\tau^{*}-)\right)\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\}}\mathds{1}_{\\{B_{i}=0\\}}\\\
+\sup_{\gamma,x}\left|\log\left\\{1-\phi(\gamma,x)\right\\}\right|\times\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Y_{i}\geq\tau^{*}\\!\\}}(1-\Delta_{i})\mathds{1}_{\\{B_{i}=1\\}}\\\
=c_{1}\exp\left(-c_{l}\Lambda(\tau^{*}-)\right)\frac{R^{*}}{n}+c_{2}\frac{Q^{*}}{n},$

where $R^{*}$ is a binomial variable with $n$ trials and success probability

$r^{*}=\mathbb{E}\left[\\{1-\phi(\gamma_{0},X)\\}F_{C}([\tau^{*},\tau_{0}]|X,Z)\right],$

and $c_{1}$ and $c_{2}$ are some constants. Deduce that there exists a
constant $c_{3}$ such that

$|r_{4n}|\leq c_{3}\left(\frac{R^{*}}{n}+\frac{Q^{*}}{n}\right).$

Gathering facts, there exists a constants $C^{*}$ and $c^{*}$, independent of
$\tau^{*}$, $\beta$, $\gamma$ and $\Lambda$, such that

$\left|\hat{l}_{n}(\beta,\Lambda,\gamma)-\hat{l}_{n}^{*}(\beta,\Lambda_{|\tau^{*}},\gamma)\right|\leq
C^{*}\left\\{\frac{N^{*}}{n}+\frac{Q^{*}}{n}\left[1+\log{\frac{n}{N^{*}}}\right]+\frac{R^{*}}{n}\right\\}+o_{a.s.}(1),$

where $N^{*}$, $Q^{*}$ and $R^{*}$ are binomial with $n$ trials and success
probabilities $p^{*}$, $q^{*}$ and $r^{*}$, respectively, and

$p^{*}+q^{*}\leq c^{*}\mathbb{P}(T_{0}\geq\tau^{*})\quad\text{and}\quad
r^{*}\leq c^{*}\mathbb{P}(C\in[\tau^{*},\tau_{0}]).$

∎

######  Lemma 6.

Assume that for any $x$ and $z$, the conditional distribution of the censoring
times given $X=x$ and $Z=z$ is such that there exists a constant $C>0$ such
that

$\inf_{[t_{1},t_{2}]\subset[0,\tau_{0}]}\inf_{x,z}\\{F_{C}(t_{2}|x,z)-F_{C}(t_{1}|x,z)\\}>C(t_{2}-t_{1}),\qquad\forall\delta>0.$

Let $0<\bar{\tau}<\tau_{0}$ and $\epsilon>0$. There exist $c_{1},c_{2}>0$,
$\delta>0$ such that $c_{1}+c_{2}=1$ and

$\begin{split}&\inf\Bigg{\\{}\mathbb{E}[\ell(Y^{*},\Delta^{*},X,Z;\beta_{0},\Lambda_{0|\tau^{*}},\gamma_{0})]-\mathbb{E}[\ell(Y^{*},\Delta^{*},X,Z;\beta,\Lambda_{|\tau^{*}},\gamma_{0})]:\\\
&\left.\qquad\quad\bar{\tau}+\delta\leq\tau^{*}<\tau_{0},\quad\|\beta-\beta_{0}\|\geq
c_{1}\epsilon\quad\text{
or}\quad\sup\limits_{t\in[0,\bar{\tau}]}|\Lambda(t)-\Lambda_{0}(t)|\geq
c_{2}\epsilon\right\\}>0\end{split}$

###### Proof.

Note that, for any $\tau^{*}\in(\bar{\tau},\tau_{0})$,

$\mathbb{E}[\ell(Y^{*},\Delta^{*},X,Z;\beta_{0},\Lambda_{0|\tau^{*}},\gamma_{0})]-\mathbb{E}[\ell(Y^{*},\Delta^{*},X,Z;\beta,\Lambda_{|\tau^{*}},\gamma_{0})]$

is the Kullback-Leibler divergence $KL(\mathbb{P}|Q)$, where $\mathbb{P}$ and
$Q$ are the probability measures of $(Y^{*},\Delta^{*},X,Z)$ when the true
parameters are $(\Lambda_{0},\beta_{0},\gamma_{0})$ and
$(\Lambda,\beta,\gamma_{0})$ respectively. By Pinsker’s inequality, we have

$KL(\mathbb{P}|Q)\geq 2\delta(\mathbb{P},Q)^{2},$

where $\delta(\mathbb{P},Q)$ is the total variation distance between the two
probability measures, defined as

$\delta(\mathbb{P},Q)=\sup_{A}|\mathbb{P}(A)-Q(A)|,$

where the supremum is taken over all measurable sets $A$. We want to find a
positive lower bound for $\delta(\mathbb{P},Q)$ independent of $\tau^{*}$ and
$Q$, for all $Q$ such that $\|\beta-\beta_{0}\|\geq c_{1}\epsilon$ or
$\sup_{t\in[0,\bar{\tau}]}|\Lambda(t)-\Lambda_{0}(t)|\geq c_{2}\epsilon$.
Hence, it is sufficient to find $k>0$ and for each such $Q$ an event $A$,
which could depend on $Q$, for which $|\mathbb{P}(A)-Q(A)|>k.$ Without loss of
generality we can assume that the covariate vector $Z$ has mean zero.

Case 1. If $\sup_{t\in[0,\bar{\tau}]}|\Lambda(t)-\Lambda_{0}(t)|\geq
c_{2}\epsilon$, there exists $\bar{t}\in[0,\bar{\tau}]$ such that either

$\Lambda(\bar{t})\geq\Lambda_{0}(\bar{t})+c_{2}\epsilon$ (A21)

or

$\Lambda(\bar{t})\leq\Lambda_{0}(\bar{t})-c_{2}\epsilon.$ (A22)

We first consider (A21) and define

$\delta=\min\left\\{\frac{\tau_{0}-\bar{\tau}}{2},\;\frac{c_{2}\epsilon}{2\sup_{t\in[0,(\bar{\tau}+\tau_{0})/2]}\lambda_{0}(t)}\right\\}.$

It follows that for all
$t\in[\bar{t},\bar{t}+\delta]\subset[0,(\bar{\tau}+\tau_{0})/2]\subset[0,{\tau_{0}})$
we have $\Lambda(t)\geq\Lambda_{0}(t)+\frac{1}{2}c_{2}\epsilon$. Indeed, we
can write

$\Lambda(t)\geq\Lambda(\bar{t})\geq\Lambda_{0}(\bar{t})+c_{2}\epsilon\geq\Lambda_{0}(t)-\delta\sup_{u\in[0,(\bar{\tau}+\tau_{0})/2]}\lambda_{0}(u)+c_{2}\epsilon\geq\Lambda_{0}(t)+\frac{1}{2}c_{2}\epsilon,\quad\forall
t\in[\bar{t},\bar{t}+\delta].$

Since $Z$ has mean zero, $(\beta-\beta_{0})^{\prime}Z$ also has zero mean.
Moreover, since $B$ is compact and $Z$ is bounded non degenerated variance, we
have

$\inf_{\beta\in B}\mathbb{P}((\beta-\beta_{0})^{\prime}Z\geq 0)>0\quad\text{
and }\quad\inf_{\beta\in B}\mathbb{P}((\beta-\beta_{0})^{\prime}Z\leq 0)>0$
(A23)

(see proof below). Let $A_{\beta}$ be the event
$\\{\Delta^{*}=0,Y^{*}\in[\bar{t},\bar{t}+\delta],{(\beta-\beta_{0})^{\prime}Z}\geq
0\\}$, which depends on $\beta$ and thus on $Q$. However, by (A23) and the
construction of the model, the event $A_{\beta}$ has positive probability
which stays bounded away from zero. Moreover, we have

$\begin{split}{\mathbb{P}(A_{\beta})-Q(A_{\beta})}&=\iint\limits_{{(\beta-\beta_{0})^{\prime}z\geq
0}}\int_{\bar{t}}^{\bar{t}+\delta}\phi(\gamma_{0},x)\\\
&\qquad\quad\times\left\\{\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\right\\}F_{C}(\mathrm{d}t|x,z)G(\mathrm{d}x,\mathrm{d}z).\end{split}$

Whenever $(\beta-\beta_{0})^{\prime}z\geq 0$, by the mean value theorem, we
obtain

$\begin{split}\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)&=\left\\{\Lambda(t)e^{\beta^{\prime}z}-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right\\}e^{-\xi}\\\
&=\left[\\{\Lambda(t)-\Lambda_{0}(t)\\}e^{\beta_{0}^{\prime}z}+\Lambda(t)\\{e^{\beta^{\prime}z}-e^{\beta_{0}^{\prime}z}\\}\right]e^{-\xi}\\\
&{\geq\\{\Lambda(t)-\Lambda_{0}(t)\\}e^{\beta_{0}^{\prime}z}e^{-\xi},}\end{split}$

for some $\xi>0$ such that
$|\xi-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}|\leq|\Lambda(t)e^{\beta^{\prime}z}-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}|$,
$t\in[\bar{t},\bar{t}+\delta]$. Now, let

$M(t)=\Lambda_{0}(t)\frac{\sup_{\beta,z}e^{\beta^{\prime}z}}{\inf_{\beta,z}e^{\beta^{\prime}z}}+\frac{\log
2}{\inf_{\beta,z}e^{\beta^{\prime}z}},\quad{t\in[\bar{t},\bar{t}+\delta].}$

Then, for $(\beta-\beta_{0})^{\prime}z\geq 0$ and
$t\in[\bar{t},\bar{t}+\delta],$ such that $\Lambda(t)\leq M(t)$ we simply use
(A21) and write

$\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\geq\frac{1}{2}c_{2}\epsilon
e^{\beta_{0}^{\prime}z}e^{-\xi}\geq k_{1}\epsilon,$

for some constant $k_{1}>0$ independent of $\Lambda$, $\beta$ and the event
$A_{\beta}$, because $M(t)$ is uniformly bounded on
$[0,(\bar{\tau}+\tau_{0})/2]$ and thus $e^{-\xi}$ is bounded away from zero.
On the other hand, for $t\in[\bar{t},\bar{t}+\delta]$ such that
$\Lambda(t)>M(t)$, we have

$\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\leq\exp\left(-M(t)\inf_{\beta,z}e^{\beta^{\prime}z}\right)\leq\frac{1}{2}\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right).$

Consequently,

$\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\geq\frac{1}{2}\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)\\\
\geq\frac{1}{2}\exp\left(-\Lambda_{0}((\bar{\tau}+\tau_{0})/2)\sup_{\beta,z}e^{\beta_{0}^{\prime}z}\right)=k_{2}(\bar{\tau})>0.$

We conclude that, for any $t\in[\bar{t},\bar{t}+\delta]$,

$\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\geq{\min\\{k_{1}\epsilon,k_{2}(\bar{\tau})\\}}>0.$

It follows that

$|\mathbb{P}(A_{\beta})\\!-Q(A_{\beta})|\geq{\min\\{k_{1}\epsilon,k_{2}(\bar{\tau})\\}}\inf_{x}\phi(\gamma_{0},x)\\\
\times\iint\limits_{{(\beta-\beta_{0})^{\prime}z\geq
0}}\left\\{F_{C}(\bar{t}+\delta|x,z)\\!-F_{C}(\bar{t}|x,z)\right\\}G(\mathrm{d}x,\mathrm{d}z).$

By assumption we have

$\inf_{x,z}F_{C}(\bar{t}+\delta|x,z)-F_{C}(\bar{t}|x,z)\geq C\delta,$

yielding that there exist another constant $k_{3}>0$ independent of $\Lambda$,
$\beta$ and the event $A_{\beta}$ (but depending on $\epsilon$ and
$\bar{\tau}$) such that

$\forall\beta\in B,\quad|\mathbb{P}(A_{\beta})-Q(A_{\beta})|\geq
k_{3}\inf_{\beta\in B}\mathbb{P}((\beta-\beta_{0})^{\prime}Z\geq 0)>0.$

Note that the uniform lower bound holds for any choice of the constants
$c_{1}$ and $c_{2}$ in the statement of the Lemma.

We next consider (A22). Let

${\bar{\delta}}=\min\left\\{\frac{\bar{\tau}}{2},\,\frac{c_{2}\epsilon}{2\sup_{t\in[0,\bar{\tau}]}\lambda_{0}(t)}\right\\}.$

It follows that for all $t\in[\,\bar{t}-{\bar{\delta}},\bar{t}\,]$ we have
$\Lambda(t)\leq\Lambda_{0}(t)-\frac{1}{2}c_{2}\epsilon$. Indeed, we can write

$\Lambda(t)\leq\Lambda(\bar{t})\leq\\{\Lambda_{0}(\bar{t})-\Lambda_{0}(t)\\}+\Lambda_{0}(t)-c_{2}\epsilon\\\
\leq{\bar{\delta}}\sup_{u\in[0,\bar{\tau}]}\lambda_{0}(u)+\Lambda_{0}(t)-c_{2}\epsilon\leq\Lambda_{0}(t)-\frac{1}{2}c_{2}\epsilon,\quad\forall
t\in[\,\bar{t}-{\bar{\delta}},\bar{t}\,].$

Next we redefine $A_{\beta}$ as the event
$\\{\Delta^{*}=0,Y^{*}\in[\bar{t}-{\bar{\delta}},\bar{t}\,],{(\beta-\beta_{0})^{\prime}Z}\leq
0\\}$, which depends on $\beta$ and thus on $Q$. However, by (A23) and the
construction of the model, the event $A_{\beta}$ has positive probability
which stays bounded away from zero. Moreover, we have

$\begin{split}\mathbb{P}(A_{\beta})-Q(A_{\beta})&=\iint\limits_{{(\beta-\beta_{0})^{\prime}z\leq
0}}\int_{\bar{t}-{\bar{\delta}}}^{\bar{t}}\phi(\gamma_{0},x)\\\
&\qquad\quad\times\left\\{\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\right\\}F_{C}(\mathrm{d}t|x,z)G(\mathrm{d}x,\mathrm{d}z).\end{split}$

Whenever $(\beta-\beta_{0})^{\prime}z\leq 0$, by the mean value theorem, we
obtain

$\begin{split}\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\leq\\{\Lambda(t)-\Lambda_{0}(t)\\}e^{\beta_{0}^{\prime}z}e^{-\xi}\leq-\frac{1}{2}c_{2}\epsilon
e^{\beta_{0}^{\prime}z}e^{-\xi},\end{split}$

for some $\xi>0$ such that
$|\xi-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}|\leq|\Lambda(t)e^{\beta^{\prime}z}-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}|\leq
2\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}$,
$t\in[\bar{t}-{\bar{\delta}},\bar{t}\,]$. Thus necessarily $0<\xi\leq
2\Lambda_{0}(\bar{\tau})e^{\beta_{0}^{\prime}z}$, and thus $e^{-\xi}$ stays
away from zero. Using arguments as we used for the case (A22), we deduce that
$\mathbb{P}(A_{\beta})-Q(A_{\beta})$ is negative and away from zero. Thus we
obtain the result with $\bar{\tau}\leq\tau^{*}<\tau_{0}$ instead of
$\bar{\tau}+\delta\leq\tau^{*}<\tau_{0}$. Finally, it remains to recall that
$\inf$ is a decreasing function of nested sets. Now the arguments for _Case 1_
are complete for any choice of the constants $c_{1}$ and $c_{2}$ in the
statement of the Lemma.

Case 2. If $\sup_{t\in[0,\bar{\tau}]}|\Lambda(t)-\Lambda_{0}(t)|\leq
c_{2}\epsilon$, then necessarily $\|\beta-\beta_{0}\|\geq c_{1}\epsilon.$ In
particular we also have that
$\Lambda(\bar{\tau})\leq\Lambda_{0}(\bar{\tau})+c_{2}\epsilon$, so all such
functions $\Lambda$ are uniformly bounded on $[0,\bar{\tau}]$. Without loss of
generality we can also assume that $\Lambda_{0}(\bar{\tau}/2)\geq 1$
(otherwise we can take a larger $\bar{\tau}$). Note that

$Var((\beta-\beta_{0})^{\prime}Z)=(\beta-\beta_{0})^{\prime}Var(Z)(\beta-\beta_{0})\geq(c_{1}\epsilon)^{2}\lambda_{min},$

with $\lambda_{min}$ the smallest eigenvalue of $Var(Z)$. From this lower
bound for the variance of $(\beta-\beta_{0})^{\prime}Z$, and since $Z$ is
centered and has a bounded support, we have

$\inf_{|\beta-\beta_{0}|\geq
c_{1}\epsilon}\left[\mathbb{P}((\beta-\beta_{0})^{\prime}Z\geq
z_{0})+\mathbb{P}((\beta-\beta_{0})^{\prime}Z\leq-
z_{0})\right]>\frac{(c_{1}\epsilon)^{2}\lambda_{min}}{2\sup\|Z\|^{2}},$ (A24)

for $z_{0}=c_{1}\epsilon\lambda^{1/2}_{min}/2$ (see proof below). If

$\inf_{|\beta-\beta_{0}|\geq
c_{1}\epsilon}\mathbb{P}\left((\beta-\beta_{0})^{\prime}Z\geq
z_{0}\right)>\frac{(c_{1}\epsilon)^{2}\lambda_{min}}{2\sup\|Z\|^{2}},$

let $A_{\beta}$ be the event
$\\{\Delta^{*}=0,Y^{*}\in[\bar{\tau}/2,\bar{\tau}],(\beta-\beta_{0})^{\prime}Z\geq
z_{0}\\}$. By (A24) and the construction of the model, the event $A_{\beta}$
has positive probability which stays bounded away from zero. Next, as in _Case
1_ , we write

$\begin{split}\mathbb{P}(A_{\beta})-Q(A_{\beta})&=\iint\limits_{(\beta-\beta_{0})^{\prime}z\geq
z_{0}}\int_{\frac{1}{2}\bar{\tau}}^{\bar{\tau}}\phi(\gamma_{0},x)\\\
&\qquad\qquad\times\left\\{\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\right\\}F_{C}(\mathrm{d}t|x,z)\,G(\mathrm{d}x,\mathrm{d}z),\end{split}$

and

$\begin{split}\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)=\left[\\{\Lambda(t)-\Lambda_{0}(t)\\}e^{\beta_{0}^{\prime}z}+\Lambda(t)\\{e^{\beta^{\prime}z}-e^{\beta_{0}^{\prime}z}\\}\right]e^{-\xi},\end{split}$

for some $\xi>0$ such that
$|\xi-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}|\leq|\Lambda(t)e^{\beta^{\prime}z}-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}|$.
From the boundedness of $\beta$, $z$, $\Lambda$ and $\Lambda_{0}$ on
$[0,\bar{\tau}]$, it follows that $e^{-\xi}\geq k_{4}>0$ for some $k_{4}$
independent of $\Lambda,$ $\beta$ and the event $A_{\beta}$ (but depending on
$\bar{\tau}$). Moreover, since for $t\in[\bar{\tau}/2,\bar{\tau}]$,
$(\beta-\beta_{0})^{\prime}z\geq z_{0}$,

$\left|\Lambda(t)-\Lambda_{0}(t)\right|e^{\beta_{0}^{\prime}z}\leq
c_{2}\epsilon e^{\beta_{0}^{\prime}z},$

and

$\begin{split}\Lambda(t)\\{e^{\beta^{\prime}z}-e^{\beta_{0}^{\prime}z}\\}\geq\Lambda_{0}(\bar{\tau}/2)e^{\beta_{0}^{\prime}z}\\{e^{(\beta-\beta_{0})^{\prime}z}-1\\}\geq
e^{\beta_{0}^{\prime}z}z_{0}=e^{\beta_{0}^{\prime}z}\lambda^{1/2}_{min}\,c_{1}\epsilon/2,\end{split}$

we obtain

$\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\geq\epsilon\left[\lambda^{1/2}_{min}\,c_{1}/2-c_{2}\right]e^{\beta_{0}^{\prime}z}e^{-\xi}.$

Define

$c_{1}=\frac{4}{\lambda_{min}^{1/2}+4}\qquad\text{and}\qquad
c_{2}=\frac{\lambda_{min}^{1/2}}{\lambda_{min}^{1/2}+4},$

such that $0<c_{1},c_{2}<1$ and $c_{1}+c_{2}=1$, and deduce that

$\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)\geq\epsilon\lambda^{1/2}_{min}e^{\beta_{0}^{\prime}z}k_{4}\left\\{\lambda_{min}^{1/2}+4\right\\}^{-1}.$

Deduce that, for any $\|\beta-\beta_{0}\|\geq c_{1}\epsilon$,

$\mathbb{P}(A_{\beta})-Q(A_{\beta})\geq\epsilon\left[\lambda^{1/2}_{min}e^{\beta_{0}^{\prime}z}k_{4}\left\\{\lambda_{min}^{1/2}+4\right\\}^{-1}\right]\inf_{z}e^{\beta_{0}^{\prime}z}\inf_{x}\phi(\gamma_{0},x)\\\
\times\inf_{|\beta-\beta_{0}|\geq
c_{1}\epsilon}\mathbb{P}(C\in[\bar{\tau}/2,\bar{\tau}],(\beta-\beta_{0})^{\prime}Z\geq
z_{0})>0.$

Note that this bound does not depend on $\delta$ in the statement of the
Lemma. Finally it is easy to see that similar arguments apply when
$\inf_{|\beta-\beta_{0}|\geq
c_{1}\epsilon}\mathbb{P}((\beta-\beta_{0})^{\prime}Z\leq-
z_{0})>(c_{1}\epsilon)^{2}\lambda_{min}/\\{4\sup\|Z\|^{2}\\}$, for the same
expression of $z_{0}$. In this case, we define
$A_{\beta}=\\{\Delta^{*}=0,Y^{*}\in[\bar{\tau}/2,\bar{\tau}],(\beta-\beta_{0})^{\prime}Z\leq-
z_{0}\\}$ and follow the same steps as above to show that

$\exp\left(-\Lambda_{0}(t)e^{\beta_{0}^{\prime}z}\right)-\exp\left(-\Lambda(t)e^{\beta^{\prime}z}\right)<0,$

and the difference of exponentials stays away from zero. Now, the proof of
Lemma 6 is complete. ∎

###### Proof of Equation (A23)..

For $\beta=\beta_{0}$ we have $\mathbb{P}((\beta-\beta_{0})^{\prime}Z\geq
0)=1$ and thus we only have to study $\beta\neq\beta_{0}$. Since $Var(Z)$ is
non degenerated, $\mathbb{P}((\beta-\beta_{0})^{\prime}Z=0)<1$ for any
$\beta\neq\beta_{0}$. Next, we could write

$0=\mathbb{E}\left(\frac{(\beta-\beta_{0})^{\prime}Z}{\|\beta-\beta_{0}\|}\right)=\mathbb{E}\left(\frac{(\beta-\beta_{0})^{\prime}Z}{\|\beta-\beta_{0}\|}\mathds{1}_{\\{(\beta-\beta_{0})^{\prime}Z\geq
0\\}}\right)+\mathbb{E}\left(\frac{(\beta-\beta_{0})^{\prime}Z}{\|\beta-\beta_{0}\|}\mathds{1}_{\\{(\beta-\beta_{0})^{\prime}Z<0\\}}\right)\\\
\leq\|Z\|\mathbb{P}((\beta-\beta_{0})^{\prime}Z\geq
0)+\mathbb{E}\left(\frac{(\beta-\beta_{0})^{\prime}Z}{\|\beta-\beta_{0}\|}\mathds{1}_{\\{(\beta-\beta_{0})^{\prime}Z<0\\}}\right).$

It remains to notice that

$\sup_{\|\tilde{\beta}\|=1}\mathbb{E}\left(\tilde{\beta}^{\prime}Z\mathds{1}_{\\{\tilde{\beta}^{\prime}Z<0\\}}\right)<0.$

Indeed, if
$\mathbb{E}\left(\tilde{\beta}^{\prime}Z\mathds{1}_{\\{\tilde{\beta}^{\prime}Z<0\\}}\right)$,
which is negative, could be arbitrarily close to zero, and since
$\mathbb{E}\left(\tilde{\beta}^{\prime}Z\mathds{1}_{\\{\tilde{\beta}^{\prime}Z<0\\}}\right)=-\mathbb{E}\left(\tilde{\beta}^{\prime}Z\mathds{1}_{\\{\tilde{\beta}^{\prime}Z\geq
0\\}}\right)$, we deduce that $\mathbb{E}(|\tilde{\beta}^{\prime}Z|)$ could be
arbitrarily close to zero, for suitable $\tilde{\beta}$ with unit norm. Since
the support of $Z$ is bounded and

$\lambda_{min}(Var(Z))\leq\mathbb{E}(|\tilde{\beta}^{\prime}Z|^{2})\leq\mathbb{E}(|\tilde{\beta}^{\prime}Z|)\sup\|Z\|,\quad\forall\|\tilde{\beta}\|=1,$

we thus get a contradiction with the assumption that $\lambda_{min}$, the
smallest eigenvalue of $Var(Z)$, is positive. Deduce that (A23) holds true. ∎

###### Proof of Equation (A24)..

It suffices to prove the following property. Let $U$ be a centered variable
such that $|U|\leq M$ for some constant $M$ and $Var(U)$ is bounded from below
by some constant $0<C<M^{2}$. Then there exists $z_{0}>0$ such that
$\mathbb{P}(|U|\geq z_{0})>C/2M^{2}$ with $z_{0}$ depending on $M$ and $C$ but
independent of the law of $U$.

For any $0<z_{0}<M$ we can write

$C\leq\mathbb{E}(U^{2})=\mathbb{E}\left(U^{2}\mathds{1}_{\\{|U|\geq
z_{0}\\}}\right)+\mathbb{E}\left(U^{2}\mathds{1}_{\\{|U|<z_{0}\\}}\right)\\\
\leq M^{2}\mathbb{P}(|U|\geq z_{0})+z_{0}^{2}\\{1-\mathbb{P}(|U|\geq
z_{0})\\}.$

Deduce that

$\mathbb{P}(|U|\geq z_{0})\geq\frac{C-z_{0}^{2}}{M^{2}-z_{0}^{2}}.$

Finally, it suffices to take $z_{0}^{2}\leq CM^{2}/(2M^{2}-C)$ in order to
obtain

$\mathbb{P}(|U|\geq z_{0})\geq\frac{C}{2M^{2}}.$

Since $C$ could be arbitrarily small, we could take $z_{0}^{2}=C/4$. ∎

###  A.3 Asymptotic normality

###### Proof of Theorem 3..

Let $m$ be as in (21) and define
$M_{n}(\gamma,\pi)=\frac{1}{n}\sum_{i=1}^{n}m(X_{i};\gamma,\pi)$ and
$M(\gamma,\pi)=\mathbb{E}[m(X;\gamma,\pi)]$. Note that
$m(x;\gamma_{0},\pi_{0})=0$ and $M(\gamma_{0},\pi_{0})=0$. If we take partial
derivatives with respect to $\gamma$ of the vector $M(\gamma,\pi)$ evaluated
at $(\gamma_{0},\pi_{0})$ we obtain a matrix
$\nabla_{\gamma}M(\gamma_{0},\pi_{0})$ with elements

$\nabla_{\gamma}M(\gamma_{0},\pi_{0})_{kl}=\mathbb{E}\left[\frac{\partial}{\partial\gamma_{l}}\left\\{\frac{1-\pi(X)}{\phi(\gamma,X)}-\frac{\pi(X)}{1-\phi(\gamma,X)}\right\\}\frac{\partial}{\partial\gamma_{k}}\phi(\gamma,X)\right]\Bigg{|}_{\gamma_{0},\pi_{0}},$

for $k,l\in\\{1,\dots,p\\}$, because

$\frac{1-\pi_{0}(x)}{\phi(\gamma_{0},x)}-\frac{\pi_{0}(x)}{1-\phi(\gamma_{0},x)}=0\quad\forall
x\in\mathcal{X}.$

Hence,

$\Gamma_{1}:=\nabla_{\gamma}M(\gamma_{0},\pi_{0})=-\mathbb{E}\left[W(X)\nabla_{\gamma}\phi(\gamma_{0},X)\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right]$
(A25)

where

$W(X)=\frac{1-\pi_{0}(X)}{\phi(\gamma_{0},X)^{2}}+\frac{\pi_{0}(X)}{\left(1-\phi(\gamma_{0},X)\right)^{2}}=\frac{1}{\phi(\gamma_{0},X)}+\frac{1}{1-\phi(\gamma_{0},X)}>0.$

We will also use the Gateaux derivative of $M(\gamma,\pi_{0})$ in a direction
$[\pi-\pi_{0}]$ given by

$\begin{split}\Gamma_{2}(\gamma,\pi_{0})[\pi-\pi_{0}]&:=\nabla_{\pi}M(\gamma,\pi_{0})[\pi-\pi_{0}]\\\
&=\lim_{h\to
0}\frac{1}{h}\left[M(\gamma,\pi_{0}+h(\pi-\pi_{0}))-M(\gamma,\pi_{0})\right]\\\
&=-\lim_{h\to
0}\frac{1}{h}\mathbb{E}\left[\left\\{\frac{h\left\\{\pi(X)-\pi_{0}(X)\right\\}}{\phi(\gamma,X)}+\frac{h\left\\{\pi(X)-\pi_{0}(X)\right\\}}{1-\phi(\gamma,X)}\right\\}\nabla_{\gamma}\phi(\gamma,X)\right]\\\
&=-\mathbb{E}\left[\left\\{\pi(X)-\pi_{0}(X)\right\\}\left\\{\frac{1}{\phi(\gamma,X)}+\frac{1}{1-\phi(\gamma,X)}\right\\}\nabla_{\gamma}\phi(\gamma,X)\right].\end{split}$
(A26)

We apply Theorem 2 in [3] so we need to verify its conditions. Consistency of
$\hat{\gamma}_{n}$ is shown in Theorem 1, while condition (2.1) in [3] is
satisfied by construction since

$\|M_{n}(\hat{\gamma}_{n},\hat{\pi}_{n})\|=0=\inf_{\gamma\in
G}\|M_{n}(\gamma,\hat{\pi}_{n})\|.$

Note that assumption (AC1) was needed in Theorem 1 in order to obtain almost
sure convergence. However, here we only need convergence in probability for
which (AN4)-(ii) suffices. For condition (2.2) in [3], the derivative of $M$
with respect to $\gamma$ is computed in (A25) and the matrix is negative
definite (as a result also full rank) because of our assumption (AN3).
Moreover the directional derivative was computed in (A26) and for
$(\gamma,\pi)\in G_{\delta_{n}}\times\Pi_{\delta_{n}}$ with
$G_{\delta_{n}}=\\{\gamma\in G:\|\gamma-\gamma_{0}\|\leq\delta_{n}\\}$,
$\Pi_{\delta_{n}}=\\{\pi\in\Pi:\|\pi-\pi_{0}\|_{\infty}\leq\delta_{n}\\}$,
$\delta_{n}=o(1)$, we have

$\begin{split}&\left\|M(\gamma,\pi)-M(\gamma,\pi_{0})-\Gamma_{2}(\gamma,\pi_{0})[\pi-\pi_{0}]\right\|\\\
&=\left\|\mathbb{E}\left[\left\\{\frac{\pi_{0}(X)-\pi(X)}{\phi(\gamma,X)}+\frac{\pi_{0}(X)-\pi(X)}{1-\phi(\gamma,X)}\right\\}\nabla_{\gamma}\phi(\gamma,X)\right]\right.\\\
&\quad+\left.\mathbb{E}\left[\left\\{\pi(X)-\pi_{0}(X)\right\\}\left\\{\frac{1}{\phi(\gamma,X)}+\frac{1}{1-\phi(\gamma,X)}\right\\}\nabla_{\gamma}\phi(\gamma,X)\right]\right\|=0,\end{split}$

which means that condition (2.3i) is satisfied. For condition (2.3ii), we have

$\begin{split}&\Gamma_{2}(\gamma,\pi_{0})[\pi-\pi_{0}]-\Gamma_{2}(\gamma_{0},\pi_{0})[\pi-\pi_{0}]\\\
&=-\mathbb{E}\left[\left\\{\pi(X)-\pi_{0}(X)\right\\}\left\\{\left(\frac{1}{\phi(\gamma,X)}+\frac{1}{1-\phi(\gamma,X)}\right)\nabla_{\gamma}\phi(\gamma,X)\right.\right.\\\
&\quad\qquad-\left.\left.\left(\frac{1}{\phi(\gamma_{0},X)}+\frac{1}{1-\phi(\gamma_{0},X)}\right)\nabla_{\gamma}\phi(\gamma_{0},X)\right\\}\right].\end{split}$

Then, from $\sup_{x}|\pi(x)-\pi_{0}(x)|\leq\delta_{n}$,
$|\gamma-\gamma_{0}|\leq\delta_{n}\to 0$ and (AN1), it follows that

$\|\Gamma_{2}(\gamma,\pi_{0})[\pi-\pi_{0}]-\Gamma_{2}(\gamma_{0},\pi_{0})[\pi-\pi_{0}]\|\leq
o(1)\delta_{n}.$

Conditions (2.4) and (2.6) in [3] are satisfied thanks to our assumption (AN4)
because

$\begin{split}&M_{n}(\gamma_{0},\pi_{0})+\Gamma_{2}(\gamma_{0},\pi_{0})[\hat{\pi}-\pi_{0}]\\\
&=\frac{1}{n}\sum_{i=1}^{n}\left\\{\frac{1-\pi_{0}(X_{i})}{\phi(\gamma_{0},X_{i})}-\frac{\pi_{0}(X_{i})}{1-\phi(\gamma_{0},X_{i})}\right\\}\nabla_{\gamma}\phi(\gamma_{0},X_{i})\\\
&\quad+\mathbb{E}^{*}\left[\left\\{\hat{\pi}(X)-\pi_{0}(X)\right\\}\left(\frac{1}{\phi(\gamma_{0},X)}+\frac{1}{1-\phi(\gamma_{0},X)}\right)\nabla_{\gamma}\phi(\gamma_{0},X)\right]\\\
&=\mathbb{E}^{*}\left[\left\\{\hat{\pi}(X)-\pi_{0}(X)\right\\}\left(\frac{1}{\phi(\gamma_{0},X)}+\frac{1}{1-\phi(\gamma_{0},X)}\right)\nabla_{\gamma}\phi(\gamma_{0},X)\right].\end{split}$
(A27)

Then we conclude by central limit theorem that

$\sqrt{n}\left(M_{n}(\gamma_{0},\pi_{0})+\Gamma_{2}(\gamma_{0},\pi_{0})[\hat{\pi}-\pi_{0}]\right)\xrightarrow{d}N(0,V)$

where $V=Var(\Psi(Y,\Delta,X,Z))$. It remains to deal with condition (2.5),
which is a consequence of Theorem 3 in [3] and assumption (AN2) because from
(AN1) we have

$\begin{split}\|m(x;\gamma_{1},\pi_{1})-m(x;\gamma_{2},\pi_{2})\|&\leq\left\|\left(\frac{1-\pi_{1}(x)}{\phi(\gamma_{1},x)}+\frac{\pi_{1}(x)}{1-\phi(\gamma_{1},X)}\right)\nabla_{\gamma}\phi(\gamma_{1},X)\right.\\\
&\left.\qquad\qquad-\left(\frac{1-\pi_{2}(x)}{\phi(\gamma_{2},x)}+\frac{\pi_{2}(x)}{1-\phi(\gamma_{2},X)}\right)\nabla_{\gamma}\phi(\gamma_{2},X)\right\|\\\
&\leq
C_{1}\|\gamma_{1}-\gamma_{2}\|+C_{2}\|\pi_{1}-\pi_{2}\|_{\infty}.\end{split}$

Finally, the asymptotic normality follows from Theorem 2 in [3] and the
asymptotic covariance matrix is given by

$\Sigma_{\gamma}=(\Gamma^{\prime}_{1}\Gamma_{1})^{-1}\Gamma^{\prime}_{1}V\Gamma_{1}(\Gamma^{\prime}_{1}\Gamma_{1})^{-1}=\Gamma_{1}^{-1}V\Gamma_{1}^{-1}$
(A28)

∎

###### Proof of Theorem 4..

We show that conditions 1 and 4 of Theorem 4 in [5] are satisfied. Define
$S_{n}$ as the version of $\hat{S}_{n}$ where $\hat{\gamma}_{n}$ is replaced
by $\gamma_{0}$

$\begin{split}{S}_{n}(\hat{\Lambda}_{n},\hat{\beta}_{n})(h_{1},h_{2})&=\frac{1}{n}\sum_{i=1}^{n}\Delta_{i}\mathds{1}_{\\{Y_{i}<\tau_{0}\\}}\left[h_{1}(Y_{i})+h^{\prime}_{2}Z_{i}\right]\\\
&\quad-\frac{1}{n}\sum_{i=1}^{n}\left\\{\Delta_{i}+(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}g_{i}(Y_{i},\hat{\Lambda}_{n},\hat{\beta}_{n},\gamma_{0})\right\\}\\\
&\qquad\qquad\quad\times\left\\{e^{\hat{\beta}^{\prime}_{n}Z_{i}}\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\hat{\Lambda}_{n}(s)+e^{\hat{\beta}^{\prime}_{n}Z_{i}}\hat{\Lambda}_{n}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}\end{split}$

Condition 1. We start by writing

$\begin{split}\hat{S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})-{S}(\Lambda_{0},\beta_{0})(h_{1},h_{2})&=\left[\hat{S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})-{S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})\right]\\\
&\quad+\left[{S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})-{S}(\Lambda_{0},\beta_{0})(h_{1},h_{2})\right].\end{split}$
(A29)

For the second term on the right hand side we have

${S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})-{S}(\Lambda_{0},\beta_{0})(h_{1},h_{2})=\int
f_{h}(y,\delta,x,z)\,\mathrm{d}(\mathbb{P}_{n}-\mathbb{P})(y,\delta,x,z)$
(A30)

where

$\begin{split}f_{h}(y,\delta,x,z)&=h_{2}^{\prime}z\left\\{\delta\mathds{1}_{\\{y<\tau_{0}\\}}-\left[\delta-(1-\delta)\mathds{1}_{\\{y\leq\tau_{0}\\}}g(y,\Lambda_{0},\beta_{0},\gamma_{0})\right]e^{\beta^{\prime}_{0}z}\Lambda_{0}(y)\right\\}\\\
&+\delta\mathds{1}_{\\{y<\tau_{0}\\}}h_{1}(y)-\left[\delta-(1-\delta)\mathds{1}_{\\{y\leq\tau_{0}\\}}g(y,\Lambda_{0},\beta_{0},\gamma_{0})\right]e^{\beta^{\prime}_{0}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda_{0}(s).\end{split}$

The classes $\\{h_{2}\in\mathbb{R}^{q},\|h_{2}\|\leq\mathfrak{m}\\}$,
$\\{h_{1}\in BV[0,\tau_{0}],\,\|h_{1}\|_{v}\leq\mathfrak{m}\\}$ and

$\left\\{\int_{0}^{y}h_{1}(t)\,\mathrm{d}\Lambda_{0}(t),\,h_{1}\in
BV[0,\tau_{0}],\|h_{1}\|_{v}\leq\mathfrak{m}\right\\}$

are Donsker classes (the last one because it consists of monotone bounded
functions). As in [5], because of the boundedness of the covariates and
$\Lambda_{0}$, it follows that
$\\{f_{h}(y,\delta,x,z),\,h\in\mathcal{H}_{\mathfrak{m}}\\}$ is also a Donsker
class since it is sum of products of Donsker classes with fixed uniformly
bounded functions.

On the other hand, for the first term on the right hand side of (A29), we have

$\begin{split}&\left[\hat{S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})-{S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})\right]\\\
&=-\frac{1}{n}\sum_{i=1}^{n}(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}\left\\{e^{\beta^{\prime}_{0}Z_{i}}\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+e^{\beta^{\prime}_{0}Z_{i}}\Lambda_{0}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left\\{g_{i}(Y_{i},\Lambda_{0},\beta_{0},\hat{\gamma}_{n})-g_{i}(Y_{i},\Lambda_{0},\beta_{0},\gamma_{0})\right\\}\\\
&=-\frac{1}{n}\sum_{i=1}^{n}(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z_{i}}\left\\{\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}\\\
&\quad\qquad\qquad\qquad\qquad\qquad\times\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\Lambda_{0},\beta_{0},\gamma_{0})\left\\{\phi(\hat{\gamma}_{n},X_{i})-\phi(\gamma_{0},X_{i})\right\\}+o_{P}(n^{-1/2}),\end{split}$
(A31)

where

$\begin{split}\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\Lambda_{0},\beta_{0},\gamma_{0})&=\frac{\exp\left(-\Lambda(Y_{i})e^{\beta^{\prime}Z_{j}}\right)}{1-\phi(\gamma,X_{j})+\phi(\gamma,X_{j})\exp\left(-\Lambda(Y_{i})e^{\beta^{\prime}Z_{j}}\right)}\\\
&\quad+\frac{\phi(\gamma,X_{j})\exp\left(-\Lambda(Y_{i})e^{\beta^{\prime}Z_{j}}\right)\left[\exp\left(-\Lambda(Y_{i})e^{\beta^{\prime}Z_{j}}\right)-1\right]}{\left[1-\phi(\gamma,X_{j})+\phi(\gamma,X_{j})\exp\left(-\Lambda(Y_{i})e^{\beta^{\prime}Z_{j}}\right)\right]^{2}}.\end{split}$

In order to conclude that the remainder term is of order $o_{P}(n^{-1/2})$ we
use

$\sup_{x}\left|\phi(\hat{\gamma}_{n},x)-\phi(\gamma_{0},x)\right|\leq\sup_{\gamma\in
G,x\in\mathcal{X}}\left\|\nabla_{\gamma}\phi(\gamma,x)\right\||\hat{\gamma}_{n}-\gamma_{0}|=O_{P}(n^{-1/2})$

and the fact that
$\frac{\partial^{2}g_{i}}{\partial\phi^{2}}(Y_{i},\Lambda_{0},\beta_{0},\gamma)$
and

$(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z_{i}}\left\\{\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}$

are uniformly bounded functions thanks to our assumptions on $Z,$ $\Lambda$,
$\Phi$ and $h$. From the same assumptions we also obtain

$\begin{split}&\frac{1}{n}\sum_{i=1}^{n}(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z_{i}}\left\\{\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}\\\
&\qquad\qquad\qquad\qquad\qquad\times\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\Lambda_{0},\beta_{0},\gamma_{0})\left\\{\phi(\hat{\gamma}_{n},X_{i})-\phi(\gamma_{0},X_{i})\right\\}\\\
&=\frac{1}{n}\sum_{i=1}^{n}(1-\Delta_{i})e^{\beta^{\prime}_{0}Z_{i}}\left\\{\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}\\\
&\qquad\qquad\qquad\qquad\qquad\times\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X_{i})^{\prime}\left\\{\hat{\gamma}_{n}-\gamma_{0}\right\\}+o_{P}(n^{-1/2})\\\
&=\mathbb{E}\left[(1-\Delta)\mathds{1}_{\\{Y\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z}\left\\{\int_{0}^{Y}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y)h^{\prime}_{2}Z\right\\}\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad\times\frac{\partial
g}{\partial\phi}(Y,\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right]\left(\hat{\gamma}_{n}-\gamma_{0}\right)+o_{P}(n^{-1/2}).\end{split}$
(A32)

and the expectation term is uniformly bounded. To prove the asymptotic
normality of $\hat{\gamma}_{n}-\gamma_{0}$ in Theorem 3 we used Theorem 2 in
[3]. Going through the proof of Theorem 2 in [3], we actually have

$(\hat{\gamma}_{n}-\gamma_{0})=-(\Gamma_{1}^{\prime}\Gamma_{1})^{-1}\Gamma^{\prime}_{1}\left\\{M_{n}(\gamma_{0},\pi_{0})+\Gamma_{2}(\gamma_{0},\pi_{0})[\hat{\pi}-\pi_{0}]\right\\}+o_{P}(n^{-1/2})$

where $\Gamma_{1}$ is defined in (A25) and $\Gamma_{2}$ in (A26). From
Assumption (AN4-iii) and (A27), it follows that

$(\hat{\gamma}_{n}-\gamma_{0})=-(\Gamma_{1}^{\prime}\Gamma_{1})^{-1}\Gamma^{\prime}_{1}\int\Psi(y,\delta,x)\,\d{(}\mathbb{P}_{n}-\mathbb{P})(y,\delta,x,z)+o_{P}(n^{-1/2}).$
(A33)

Putting together (A29)-(A33), we have

$\begin{split}&\hat{S}_{n}(\Lambda_{0},\beta_{0})(h_{1},h_{2})-{S}(\Lambda_{0},\beta_{0})(h_{1},h_{2})\\\
&=\int\left\\{f_{h}(y,\delta,x,z)-Q_{h}\Gamma_{1}^{-1}\Psi(y,\delta,x)\right\\}\,\mathrm{d}(\mathbb{P}_{n}-\mathbb{P})(y,\delta,x,z)+o_{P}(n^{-1/2})\end{split}$

where

$Q_{h}=\mathbb{E}\left[(1-\Delta)\mathds{1}_{\\{Y\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z}\left\\{\int_{0}^{Y}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y)h^{\prime}_{2}Z\right\\}\frac{\partial
g}{\partial\phi}(Y,\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right].$

In order to conclude the convergence of
$\sqrt{n}(\hat{S}_{n}(\Upsilon_{0})-S(\Upsilon_{0}))$ to a Gaussian process
$G^{*}$, we need to have that
$\\{Q_{h}\Gamma_{1}^{-1}\Psi(y,\delta,x),h\in\mathcal{H}_{\mathfrak{m}}\\}$ is
a bounded Dosker class of functions (since sum of bounded Donsker classes is
also Donsker). We can write

$\begin{split}&Q_{h}\Gamma_{1}^{-1}\Psi(y,\delta,x)\\\
&=h^{\prime}_{2}\mathbb{E}\left[(1-\Delta)\mathds{1}_{\\{Y\leq\tau_{0}\\}}Ze^{\beta^{\prime}_{0}Z}\Lambda_{0}(Y)\frac{\partial
g}{\partial\phi}(Y,\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right]\Gamma_{1}^{-1}\Psi(y,\delta,x)\\\
&\quad+\int_{0}^{\tau_{0}}\mathbb{E}\left[(1-\Delta)\mathds{1}_{\\{Y\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z}\frac{\partial
g}{\partial\phi}(Y,\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right]h_{1}(s)\mathrm{d}\Lambda_{0}(s)\Gamma_{1}^{-1}\Psi(y,\delta,x)\end{split}$

By assumption (AN1) and $\inf_{x}H((\tau_{0},\infty)|x)>0$,
$\Lambda_{0}(\tau_{0})<\infty$ and the boundedness of the covariates, we have
that

$\mathbb{E}\left[(1-\Delta)\mathds{1}_{\\{Y\leq\tau_{0}\\}}Ze^{\beta^{\prime}_{0}Z}\Lambda_{0}(Y)\frac{\partial
g}{\partial\phi}(Y,\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right]\Gamma_{1}^{-1}\Psi(y,\delta,x)$

is uniformly bounded. Hence

$\begin{split}&\left\\{h^{\prime}_{2}\mathbb{E}\left[(1-\Delta)\mathds{1}_{\\{Y\leq\tau_{0}\\}}Ze^{\beta^{\prime}_{0}Z}\Lambda_{0}(Y)\frac{\partial
g}{\partial\phi}(Y,\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right]\Gamma_{1}^{-1}\Psi(y,\delta,x):\right.\\\
&\,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
h_{2}\in\mathbb{R}^{q},\|h_{2}\|_{L_{1}}\leq\mathfrak{m}\bigg{\\}}\end{split}$

is a Donsker class (see Example 2.10.10 in [8]). It can also be shown that,
since $h_{1}$ belongs to the class of bounded functions with bounded variation
and all the other terms are uniformly bounded, that

$\begin{split}&\left\\{\int_{0}^{\tau_{0}}\mathbb{E}\left[(1-\Delta)\mathds{1}_{\\{Y\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z}\frac{\partial
g}{\partial\phi}(Y,\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X)^{\prime}\right]h_{1}(s)\mathrm{d}\Lambda_{0}(s)\Gamma_{1}^{-1}\Psi(y,\delta,x),\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
h_{1}\in BV[0,\tau_{0}],\,\|h_{1}\|_{v}\leq\mathfrak{m}\bigg{\\}}\end{split}$

is also a bounded Donsker class (covering numbers of order $\epsilon$ of
$\\{h_{1}\in BV[0,\tau_{0}],\|h_{1}\|_{v}\leq\mathfrak{m}\\}$ correspond to
covering numbers of order $c\epsilon$ for some constant $c>0$).

The limit process $G^{*}$ has mean zero because

$\mathbb{E}[f_{h}(y,\delta,x,z)]=S(\Upsilon_{0})(h)=0\qquad\text{and}\qquad\mathbb{E}[\Psi(Y,\Delta,X)]=0.$

The covariance process of $G^{*}$ is

$\begin{split}&Cov\left(G^{*}(h),G^{*}(\tilde{h})\right)\\\
&=\mathbb{E}\left[\left\\{f_{h}(Y,\Delta,X,Z)-Q_{h}\Gamma_{1}^{-1}\Psi(Y,\Delta,X)\right\\}\left\\{f_{\tilde{h}}(Y,\Delta,X,Z)-Q_{\tilde{h}}\Gamma_{1}^{-1}\Psi(Y,\Delta,X)\right\\}\right]\\\
&=\mathbb{E}[f_{h}(Y,\Delta,X,Z)f_{\tilde{h}}(Y,\Delta,X,Z)]-Q_{h}\Gamma_{1}^{-1}\mathbb{E}[f_{\tilde{h}}(Y,\Delta,X,Z)\Psi(Y,\Delta,X)]\\\
&\quad-
Q_{\tilde{h}}\Gamma_{1}^{-1}\mathbb{E}[f_{h}(Y,\Delta,X,Z)\Psi(Y,\Delta,X)]+Q_{\tilde{h}}\Gamma_{1}^{-1}\mathbb{E}[\Psi(Y,\Delta,X)\Psi(Y,\Delta,X)^{\prime}]\Gamma_{1}^{-1}Q^{\prime}_{h}.\end{split}$
(A34)

Condition 4 of Theorem 4 in [5].  As for condition 1, we write

$\begin{split}\sqrt{n}\left\\{(\hat{S}_{n}-S)(\Upsilon_{n})-(\hat{S}_{n}-S)(\Upsilon_{0})\right\\}&=\sqrt{n}\left\\{({S}_{n}-S)(\Upsilon_{n})-({S}_{n}-S)(\Upsilon_{0})\right\\}\\\
&\quad+\sqrt{n}\left\\{(\hat{S}_{n}-S_{n})(\Upsilon_{n})-(\hat{S}_{n}-S_{n})(\Upsilon_{0})\right\\}\end{split}$
(A35)

For the second term in the right hand side of (A35), similarly to (A31)-(A32),
we have

$\begin{split}&(\hat{S}_{n}-S_{n})(\Upsilon_{n})-(\hat{S}_{n}-S_{n})(\Upsilon_{0})\\\
&=\frac{1}{n}\sum_{i=1}^{n}(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}e^{\hat{\beta}^{\prime}_{n}Z_{i}}\left\\{\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\hat{\Lambda}_{n}(s)+\hat{\Lambda}_{n}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}\\\
&\quad\qquad\qquad\qquad\qquad\qquad\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\hat{\Lambda}_{n},\hat{\beta}_{n},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X_{i})^{\prime}\left\\{\hat{\gamma}_{n}-\gamma_{0}\right\\}\\\
&\quad-\frac{1}{n}\sum_{i=1}^{n}(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}e^{\beta^{\prime}_{0}Z_{i}}\left\\{\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y_{i})h^{\prime}_{2}Z_{i}\right\\}\\\
&\quad\qquad\qquad\qquad\qquad\qquad\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\Lambda_{0},\beta_{0},\gamma_{0})\nabla_{\gamma}\phi(\gamma_{0},X_{i})^{\prime}\left\\{\hat{\gamma}_{n}-\gamma_{0}\right\\}+o_{P}(n^{-1/2})\\\
\end{split}$

Using the boundedness in probability of $\hat{\beta}_{n}$ and
$\hat{\Lambda}_{n}(\tau_{0})$, the boundedness of the covariates, $\beta_{0}$,
$\Lambda_{0}(\tau)$, $\nabla_{\gamma}\phi(\gamma,x)$ and the consistency
results in Theorem 2, it follows that

$\begin{split}&\left|\frac{1}{n}\sum_{i=1}^{n}(1-\Delta_{i})\mathds{1}_{\\{Y_{i}\leq\tau_{0}\\}}\left\\{e^{\hat{\beta}^{\prime}_{n}Z_{i}}\left(\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\hat{\Lambda}_{n}(s)+\hat{\Lambda}_{n}(Y_{i})h^{\prime}_{2}Z_{i}\right)\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\hat{\Lambda}_{n},\hat{\beta}_{n},\gamma_{0})\right.\right.\\\
&\left.\left.\quad-e^{\beta^{\prime}_{0}Z_{i}}\left(\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+\Lambda_{0}(Y_{i})h^{\prime}_{2}Z_{i}\right)\frac{\partial
g_{i}}{\partial\phi}(Y_{i},\Lambda_{0},\beta_{0},\gamma_{0})\right\\}\nabla_{\gamma}\phi(\gamma_{0},X_{i})\right|=o_{P}(1)\end{split}$

As a consequence, since $\hat{\gamma}_{n}-\gamma_{0}=O_{P}(n^{-1/2})$, we
obtain

$\sqrt{n}\left\\{(\hat{S}_{n}-S_{n})(\Upsilon_{n})-(\hat{S}_{n}-S_{n})(\Upsilon_{0})\right\\}=o_{P}(1).$

It remains to deal with the first term on the right hand side of (A35). It
suffices to show that, for any sequence $\epsilon_{n}\to 0$,

$\sup_{|\Lambda-\Lambda_{0}|_{\infty}\leq\epsilon_{n},\,\|\beta-\beta_{0}\|\leq\epsilon_{n}}\frac{\left|({S}_{n}-S)(\Upsilon)-({S}_{n}-S)(\Upsilon_{0})\right|}{n^{-1/2}\vee\|\beta-\beta_{0}\|\vee|\Lambda-\Lambda_{0}|_{\infty}}=o_{P}(1).$

Let

$a_{1}(y,\delta,z,x)=\delta
e^{\beta^{\prime}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)-\delta
e^{\beta^{\prime}_{0}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda_{0}(s),$
$a_{2}(y,\delta,z,x)=\delta
e^{\beta^{\prime}z}\Lambda(y)h^{\prime}_{2}z-\delta
e^{\beta^{\prime}_{0}z}\Lambda_{0}(y)h^{\prime}_{2}z,$

and

$\begin{split}a_{3}(y,\delta,z,x)&=(1-\delta)\mathds{1}_{\\{y\leq\tau_{0}\\}}g(y,\Lambda,\beta,\gamma_{0})\left\\{e^{\beta^{\prime}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)+e^{\beta^{\prime}z}\Lambda(y)h^{\prime}_{2}z\right\\}\\\
&\quad-(1-\delta)\mathds{1}_{\\{y\leq\tau_{0}\\}}g(y,\Lambda_{0},\beta_{0},\gamma_{0})\left\\{e^{\beta^{\prime}_{0}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+e^{\beta^{\prime}_{0}z}\Lambda_{0}(y)h^{\prime}_{2}z\right\\}\end{split}$

Then, we have

$\begin{split}({S}_{n}-S)(\Upsilon)-({S}_{n}-S)(\Upsilon_{0})&=-\frac{1}{n}\sum_{i=1}^{n}\left\\{a_{1}(Y_{i},\Delta_{i},Z_{i},X_{i})-\mathbb{E}\left[a_{1}(Y,\Delta,Z,X)\right]\right\\}\\\
&\quad-\frac{1}{n}\sum_{i=1}^{n}\left\\{a_{2}(Y_{i},\Delta_{i},Z_{i},X_{i})-\mathbb{E}\left[a_{2}(Y,\Delta,Z,X)\right]\right\\}\\\
&\quad-\frac{1}{n}\sum_{i=1}^{n}\left\\{a_{3}(Y_{i},\Delta_{i},Z_{i},X_{i})-\mathbb{E}\left[a_{3}(Y,\Delta,Z,X)\right]\right\\}\end{split}$

Next we consider the first term. The other two can be handled similarly. From
a Taylor expansion we have

$\begin{split}&\delta
e^{\beta^{\prime}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)-\delta
e^{\beta^{\prime}_{0}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda_{0}(s)\\\
&=(\beta-\beta_{0})^{\prime}z\delta
e^{\beta^{\prime}_{0}z}\int_{0}^{y}h_{1}(s)\mathrm{d}\Lambda(s)+\delta
e^{\beta^{\prime}_{0}z}\int_{0}^{y}h_{1}(s)\mathrm{d}(\Lambda-\Lambda_{0})(s)+o(\|\beta-\beta_{0}\|).\end{split}$

Hence

$\begin{split}&\frac{1}{n}\sum_{i=1}^{n}\left\\{a_{1}(Y_{i},\Delta_{i},Z_{i},X_{i})-\mathbb{E}\left[a_{1}(Y,\Delta,Z,X)\right]\right\\}\\\
&\leq(\beta-\beta_{0})\left\\{\frac{1}{n}\sum_{i=1}^{n}Z_{i}\Delta_{i}e^{\beta^{\prime}_{0}Z_{i}}\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda(s)-\mathbb{E}\left[Z\Delta
e^{\beta^{\prime}_{0}Z}\int_{0}^{Y}h_{1}(s)\mathrm{d}\Lambda(s)\right]+o(1)\right\\}\\\
&\quad+\frac{1}{n}\sum_{i=1}^{n}\left\\{\Delta_{i}e^{\beta^{\prime}_{0}Z_{i}}\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}(\Lambda-\Lambda_{0})(s)-\mathbb{E}\left[\Delta
e^{\beta^{\prime}_{0}Z}\int_{0}^{Y}h_{1}(s)\mathrm{d}(\Lambda-\Lambda_{0})(s)\right]\right\\}.\end{split}$
(A36)

By the law of large numbers

$\frac{1}{n}\sum_{i=1}^{n}\left\\{Z_{i}\Delta_{i}e^{\beta^{\prime}_{0}Z_{i}}\int_{0}^{Y_{i}}h_{1}(s)\mathrm{d}\Lambda(s)-\mathbb{E}\left[Z\Delta
e^{\beta^{\prime}_{0}Z}\int_{0}^{Y}h_{1}(s)\mathrm{d}\Lambda(s)\right]\right\\}=o_{P}(1)$

and as a result, the first term in the right hand side of (A36) is
$o_{P}(\|\beta-\beta_{0}\|)$. The second term can be rewritten as

$\int_{0}^{\tau_{0}}D_{n}(s)h_{1}(s)\mathrm{d}(\Lambda-\Lambda_{0})(s)$

where

$D_{n}(s)=\frac{1}{n}\sum_{i=1}^{n}\left\\{\Delta_{i}\mathds{1}_{\\{Y_{i}\geq
s\\}}e^{\beta^{\prime}_{0}Z_{i}}-\mathbb{E}\left[\Delta\mathds{1}_{\\{Y\geq
s\\}}e^{\beta^{\prime}_{0}Z}\right]\right\\}.$

By integration by parts and the chain rule we have

$\begin{split}&\int_{0}^{\tau_{0}}D_{n}(s)h_{1}(s)\mathrm{d}(\Lambda-\Lambda_{0})(s)\\\
&=D_{n}(\tau_{0})h_{1}(\tau_{0})(\Lambda-\Lambda_{0})(\tau_{0})-\int_{0}^{\tau_{0}}(\Lambda-\Lambda_{0})(s)\mathrm{d}\left[D_{n}(s)h_{1}(s)\right]\\\
&=D_{n}(\tau_{0})h_{1}(\tau_{0})(\Lambda-\Lambda_{0})(\tau_{0})-\int_{0}^{\tau_{0}}(\Lambda-\Lambda_{0})(s)D_{n}(s)\mathrm{d}h_{1}(s)\\\
&\quad+\frac{1}{n}\sum_{i=1}^{n}\left\\{\Delta_{i}(\Lambda-\Lambda_{0})(Y_{i})h_{1}(Y_{i})e^{\beta^{\prime}_{0}Z_{i}}-\mathbb{E}\left[(\Lambda-\Lambda_{0})(Y)h_{1}(Y)\Delta
e^{\beta^{\prime}_{0}Z}\right]\right\\}\end{split}$

Note that

$\begin{split}\mathbb{E}\left[(\Lambda-\Lambda_{0})(Y)h_{1}(Y)\Delta
e^{\beta^{\prime}_{0}Z}\right]&=\mathbb{E}\left[e^{\beta^{\prime}_{0}Z}\int_{0}^{\tau_{0}}(\Lambda-\Lambda_{0})(s)h_{1}(s)\mathrm{d}H_{1}(s|X,Z)\bigg{|}X,Z\right]\\\
&=-\int_{0}^{\tau_{0}}(\Lambda-\Lambda_{0})(s)h_{1}(s)\mathrm{d}\mathbb{E}\left[\Delta\mathds{1}_{\\{Y\geq
s\\}}e^{\beta^{\prime}_{0}Z}\right].\end{split}$

It can be shown that $\sqrt{n}D_{n}$ converges weakly to a tight, mean zero
Gaussian process $D$ in $l^{\infty}([0,\tau_{0}])$. Since $h_{1}$ is bounded,
it follows

$\frac{\left|D_{n}(\tau_{0})h_{1}(\tau_{0})(\Lambda-\Lambda_{0})(\tau_{0})\right|}{\|\Lambda-\Lambda_{0}\|_{\infty}}=o_{P}(1)$

Moreover, since $D_{n}\to 0$ and $h_{1}$ is of bounded variation

$\frac{\left|\int_{0}^{\tau_{0}}(\Lambda-\Lambda_{0})(s)D_{n}(s)\mathrm{d}h_{1}(s)\right|}{\|\Lambda-\Lambda_{0}\|_{\infty}}\leq\sup_{t\in[0,\tau_{0}]}|D_{n}(s)|\int_{0}^{\tau_{0}}\,\left|\mathrm{d}h(s)\right|\to
0.$

Finally, since
$\left\\{g_{\Lambda}(y,\delta,z)=\delta(\Lambda-\Lambda_{0})(y)h_{1}(y)e^{\beta^{\prime}_{0}Z}:\,\|\Lambda-\Lambda_{0}\|_{\infty}\leq\epsilon_{n}\right\\}$
is a Donsker class (product of bounded variation functions, uniformly bounded)
and

$\mathbb{E}\left[(\Lambda-\Lambda_{0})(Y)^{2}h_{1}(Y)^{2}\Delta
e^{2\beta^{\prime}_{0}Z}\right]=O(\epsilon_{n}^{2})=o(1),$

we have that

$\sqrt{n}\frac{1}{n}\sum_{i=1}^{n}\left\\{\Delta_{i}(\Lambda-\Lambda_{0})(Y_{i})h_{1}(Y_{i})e^{\beta^{\prime}_{0}Z_{i}}-\mathbb{E}\left[(\Lambda-\Lambda_{0})(Y)h_{1}(Y)\Delta
e^{\beta^{\prime}_{0}Z}\right]\right\\}$

converges to zero in probability. So we obtain

$\frac{\int_{0}^{\tau_{0}}D_{n}(s)h_{1}(s)\mathrm{d}(\Lambda-\Lambda_{0})(s)}{n^{-1/2}\vee\|\Lambda-\Lambda_{0}\|_{\infty}}=o_{P}(1)$

The other two terms related to $a_{2}$ and $a_{3}$ can be treated similarly.

This concludes the verification of conditions of Theorem 4 in [5] (or Theorem
3.3.1 in [8]). Hence, the weak convergence of
$\sqrt{n}(\Upsilon_{n}-\Upsilon_{0})$ to a tight, mean zero Gaussian process
$G$ follows. Next we compute the covariance process of $G$. From Theorem 3.3.1
in [8] we have

$-\sqrt{n}\dot{S}(\Upsilon_{0})(\Upsilon_{n}-\Upsilon_{0})(h)=\sqrt{n}(\hat{S}_{n}(\Upsilon_{0})-S(\Upsilon_{0}))(h)+o_{P}(1).$
(A37)

Moreover, in [5] it is computed that

$\dot{S}(\Upsilon_{0})(\Upsilon_{n}-\Upsilon_{0})(h)=\int_{0}^{\tau_{0}}\sigma_{1}(h)(t)\mathrm{d}(\hat{\Lambda}_{n}(t)-\Lambda_{0}(t))+(\hat{\beta}_{n}-\beta_{0})^{\prime}\sigma_{2}(h)$
(A38)

where $\sigma=(\sigma_{1},\sigma_{2})$ is a continuous linear operator from
$\mathcal{H}_{\mathfrak{m}}$ to $\mathcal{H}_{\mathfrak{m}}$ of the form

$\begin{split}\sigma_{1}(h)(t)&=\mathbb{E}\left[\mathds{1}_{\\{Y\geq
t\\}}V(t,\Upsilon_{0})(h)g(t,\Upsilon_{0})e^{\beta^{\prime}_{0}Z}\right]\\\
&\quad-\mathbb{E}\left[\int_{t}^{\tau_{0}}\mathds{1}_{\\{Y\geq
s\\}}V(t,\Upsilon_{0})(h)g(s,\Upsilon_{0})\\{1-g(s,\Upsilon_{0})\\}e^{2\beta^{\prime}_{0}Z}\mathrm{d}\Lambda_{0}(s)\right]\end{split}$

and

$\sigma_{2}(h)(t)=\mathbb{E}\left[\int_{0}^{\tau_{0}}\mathds{1}_{\\{Y\geq
t\\}}W(t,\Upsilon_{0})V(t,\Upsilon_{0})(h)g(t,\Upsilon_{0})e^{\beta^{\prime}_{0}Z}\mathrm{d}\Lambda_{0}(t)\right]$

where

$V(t,\Upsilon_{0})(h)=h_{1}(t)-\left\\{1-g(t,\Upsilon_{0})\right\\}e^{\beta^{\prime}_{0}Z}\int_{0}^{t}h_{1}(s)\mathrm{d}\Lambda_{0}(s)+h^{\prime}_{2}W(t,\Upsilon_{0})$

and

$W(t,\Upsilon_{0})=\left[1-\left\\{1-g(t,\Upsilon_{0})\right\\}e^{\beta^{\prime}_{0}Z}\Lambda_{0}(t)\right]Z$

In [5], it is also shown that $\sigma$ is invertible with inverse
$\sigma^{-1}=(\sigma_{1}^{-1},\sigma_{2}^{-1})$. Hence, for all
$g\in\mathcal{H}_{\mathfrak{m}}$, let $h=\sigma^{-1}(g)$. If in (A38) we
replace $h$ by $\sigma^{-1}(g)$ and use (A37), we obtain

$\begin{split}&\int_{0}^{\tau_{0}}g_{1}(t)\mathrm{d}\sqrt{n}(\Lambda_{n}(t)-\Lambda_{0}(t))+\sqrt{n}(\hat{\beta}_{n}-\beta_{0})^{\prime}g_{2}\\\
&=-\sqrt{n}(\hat{S}_{n}(\Upsilon_{0})-S(\Upsilon_{0}))(\sigma^{-1}(g))+o_{P}(1)\xrightarrow{d}-G^{*}(\sigma^{-1}(g)).\end{split}$

Since the previous results holds for all $g\in\mathcal{H}_{\mathfrak{m}}$, it
follows that
$(\sqrt{n}(\hat{\Lambda}_{n}-\Lambda_{0}),\sqrt{n}(\hat{\beta}_{n}-\beta_{0}))$
converges to a tight mean zero Gaussian process $G$ with covariance

$Cov(G(g),G(\tilde{g}))=Cov\left(G^{*}(\sigma^{-1}(g)),G^{*}(\sigma^{-1}(\tilde{g}))\right)$
(A39)

and the covariance of $G^{*}$ is given in (A34). ∎

##  Appendix B Additional simulation results

In this section we report the simulation results for scenario 2 of the models
1-4, $n=200,400$, that were omitted from the main paper and the results for
sample size $n=1000$ (all models and scenarios). In addition, Table A2
complements Tables 4 and 5 containing results for $\hat{\beta}$.

## References

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  * [4] Murphy, S. A et. al. (1994) Consistency in a proportional hazards model incorporating a random effect. Ann. Stat., 22 (2), 712–731.
  * [5] Lu, W. (2008) Maximum likelihood estimation in the proportional hazards cure model. Ann. I. Stat. Math., 60 (3), 545–574.
  * [6] Patilea, V. and Van Keilegom, I. (2020) A general approach for cure models in survival analysis. To apperar in: Ann. Stat.
  * [7] Scharfstein, D. O., Tsiatis, A. A. and Gilbert, P. B. (1998) Semiparametric efficient estimation in the generalized odds-rate class of regression models for right-censored time-to-event data. Lifetime Data Anal., 4 (4), 355–391.
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  * [9] Wellner, J. A. (1978) Limit theorems for the ratio of the empirical distribution function to the true distribution function.  Z. Wahrsch. Verw. Gebiete, 45 (1), 73–88.

Table A1: Bias, variance and MSE of $\hat{\gamma}$ and $\hat{\beta}$ for
smcure (second rows) and our approach (first rows) in Model 1 and 2 (scenario
2).

Cens. level 1 Cens. level 2 Cens. level 3 Mod. n scen. Par. Bias Var. MSE Bias
Var. MSE Bias Var. MSE 1 $200$ $2$ $\gamma_{1}$ $0.005$ $0.032$ $0.032$
$-0.005$ $0.037$ $0.037$ $-0.005$ $0.045$ $0.045$ $0.013$ $0.032$ $0.032$
$0.009$ $0.038$ $0.038$ $0.015$ $0.046$ $0.046$ $\gamma_{2}$ $-0.024$ $0.093$
$0.094$ $-0.035$ $0.116$ $0.118$ $-0.018$ $0.137$ $0.137$ $0.017$ $0.096$
$0.096$ $0.016$ $0.123$ $0.123$ $0.046$ $0.144$ $0.146$ $\beta$ $0.019$
$0.033$ $0.033$ $0.025$ $0.035$ $0.036$ $0.011$ $0.041$ $0.041$ $0.018$
$0.033$ $0.033$ $0.023$ $0.036$ $0.036$ $0.007$ $0.042$ $0.042$ $400$ $2$
$\gamma_{1}$ $0.002$ $0.016$ $0.016$ $0.003$ $0.018$ $0.018$ $0.009$ $0.021$
$0.021$ $0.008$ $0.016$ $0.016$ $0.011$ $0.019$ $0.019$ $0.020$ $0.021$
$0.022$ $\gamma_{2}$ $-0.017$ $0.046$ $0.047$ $-0.016$ $0.054$ $0.054$
$-0.013$ $0.070$ $0.070$ $0.012$ $0.046$ $0.047$ $0.018$ $0.055$ $0.055$
$0.025$ $0.068$ $0.068$ $\beta$ $0.001$ $0.014$ $0.014$ $0.019$ $0.019$
$0.019$ $0.001$ $0.020$ $0.020$ $0.000$ $0.014$ $0.014$ $0.017$ $0.019$
$0.019$ $-0.001$ $0.020$ $0.020$ 2 $200$ $2$ $\gamma_{1}$ $-0.004$ $0.033$
$0.033$ $-0.016$ $0.039$ $0.039$ $-0.032$ $0.050$ $0.051$ $0.020$ $0.034$
$0.034$ $0.037$ $0.042$ $0.044$ $0.079$ $0.064$ $0.070$ $\gamma_{2}$ $-0.016$
$0.041$ $0.041$ $-0.037$ $0.044$ $0.045$ $-0.054$ $0.057$ $0.060$ $0.029$
$0.044$ $0.045$ $0.031$ $0.052$ $0.053$ $0.064$ $0.075$ $0.079$ $\beta$
$0.004$ $0.016$ $0.016$ $0.008$ $0.017$ $0.017$ $0.008$ $0.018$ $0.018$
$0.002$ $0.016$ $0.016$ $0.003$ $0.018$ $0.018$ $-0.005$ $0.019$ $0.019$ $400$
$2$ $\gamma_{1}$ $0.000$ $0.018$ $0.018$ $-0.002$ $0.022$ $0.022$ $-0.015$
$0.028$ $0.028$ $0.012$ $0.018$ $0.018$ $0.027$ $0.022$ $0.023$ $0.043$
$0.030$ $0.032$ $\gamma_{2}$ $-0.019$ $0.021$ $0.021$ $-0.022$ $0.025$ $0.026$
$-0.045$ $0.031$ $0.033$ $0.007$ $0.022$ $0.022$ $0.018$ $0.027$ $0.027$
$0.023$ $0.033$ $0.033$ $\beta$ $0.005$ $0.007$ $0.007$ $0.010$ $0.008$
$0.008$ $0.009$ $0.010$ $0.010$ $0.004$ $0.007$ $0.007$ $0.007$ $0.008$
$0.008$ $0.001$ $0.011$ $0.011$

Table A2:  Bias, variance and MSE of $\hat{\beta}$ for smcure (second rows)
and our approach (first rows) in Model 3 and 4.

Cens. level 1 Cens. level 2 Cens. level 3 Mod. n scen. Par. Bias Var. MSE Bias
Var. MSE Bias Var. MSE 3 $200$ 1 $\beta_{1}$ $-0.013$ $0.007$ $0.007$ $-0.007$
$0.006$ $0.006$ $-0.009$ $0.008$ $0.008$ $-0.014$ $0.007$ $0.007$ $-0.009$
$0.006$ $0.007$ $-0.014$ $0.008$ $0.008$ $\beta_{2}$ $0.007$ $0.003$ $0.003$
$0.006$ $0.003$ $0.003$ $0.005$ $0.004$ $0.004$ $0.007$ $0.003$ $0.003$
$0.008$ $0.003$ $0.003$ $0.007$ $0.004$ $0.004$ $\beta_{3}$ $0.016$ $0.043$
$0.043$ $0.004$ $0.043$ $0.043$ $0.007$ $0.053$ $0.053$ $0.017$ $0.043$
$0.043$ $0.006$ $0.043$ $0.043$ $0.015$ $0.053$ $0.054$ 3 $\beta_{1}$ $0.024$
$0.014$ $0.014$ $0.024$ $0.014$ $0.015$ $0.017$ $0.017$ $0.017$ $0.023$
$0.014$ $0.014$ $0.025$ $0.014$ $0.015$ $0.024$ $0.017$ $0.018$ $\beta_{2}$
$-0.005$ $0.004$ $0.004$ $-0.002$ $0.005$ $0.005$ $-0.003$ $0.006$ $0.006$
$-0.005$ $0.004$ $0.004$ $-0.003$ $0.005$ $0.005$ $-0.004$ $0.006$ $0.006$
$\beta_{3}$ $0.018$ $0.066$ $0.066$ $-0.002$ $0.081$ $0.081$ $0.021$ $0.099$
$0.100$ $0.018$ $0.066$ $0.066$ $0.001$ $0.083$ $0.083$ $0.033$ $0.104$
$0.105$ $400$ 1 $\beta_{1}$ $-0.007$ $0.003$ $0.003$ $-0.004$ $0.003$ $0.003$
$-0.002$ $0.003$ $0.003$ $-0.007$ $0.003$ $0.003$ $-0.005$ $0.003$ $0.003$
$-0.006$ $0.003$ $0.003$ $\beta_{2}$ $0.004$ $0.002$ $0.002$ $0.003$ $0.002$
$0.002$ $0.002$ $0.002$ $0.002$ $0.004$ $0.002$ $0.002$ $0.003$ $0.002$
$0.002$ $0.003$ $0.002$ $0.002$ $\beta_{3}$ $0.006$ $0.020$ $0.020$ $0.007$
$0.022$ $0.022$ $0.002$ $0.024$ $0.024$ $0.006$ $0.020$ $0.020$ $0.008$
$0.022$ $0.022$ $0.006$ $0.024$ $0.024$ 3 $\beta_{1}$ $0.013$ $0.006$ $0.006$
$0.009$ $0.007$ $0.007$ $0.003$ $0.008$ $0.008$ $0.012$ $0.006$ $0.006$
$0.009$ $0.007$ $0.007$ $0.007$ $0.008$ $0.008$ $\beta_{2}$ $-0.002$ $0.002$
$0.002$ $-0.001$ $0.003$ $0.003$ $0.000$ $0.003$ $0.003$ $-0.002$ $0.002$
$0.002$ $-0.001$ $0.003$ $0.003$ $-0.001$ $0.003$ $0.003$ $\beta_{3}$ $0.008$
$0.030$ $0.030$ $0.000$ $0.035$ $0.035$ $-0.007$ $0.045$ $0.045$ $0.007$
$0.030$ $0.030$ $0.001$ $0.035$ $0.035$ $0.001$ $0.047$ $0.047$ 4 $200$ 1
$\beta_{1}$ $-0.012$ $0.005$ $0.005$ $-0.010$ $0.005$ $0.005$ $-0.006$ $0.005$
$0.005$ $-0.013$ $0.005$ $0.005$ $-0.013$ $0.005$ $0.005$ $-0.013$ $0.005$
$0.006$ $\beta_{2}$ $0.003$ $0.003$ $0.003$ $0.002$ $0.003$ $0.003$ $0.001$
$0.003$ $0.003$ $0.003$ $0.003$ $0.003$ $0.002$ $0.003$ $0.003$ $0.003$
$0.003$ $0.003$ $\beta_{3}$ $0.002$ $0.032$ $0.032$ $0.000$ $0.036$ $0.036$
$0.000$ $0.040$ $0.040$ $0.002$ $0.032$ $0.032$ $0.002$ $0.037$ $0.037$
$0.005$ $0.041$ $0.041$ 3 $\beta_{1}$ $0.013$ $0.005$ $0.005$ $0.010$ $0.005$
$0.005$ $0.002$ $0.007$ $0.007$ $0.014$ $0.005$ $0.005$ $0.013$ $0.005$
$0.006$ $0.010$ $0.007$ $0.007$ $\beta_{2}$ $-0.003$ $0.004$ $0.004$ $-0.002$
$0.005$ $0.005$ $-0.002$ $0.006$ $0.006$ $-0.004$ $0.004$ $0.004$ $-0.003$
$0.005$ $0.005$ $-0.004$ $0.006$ $0.006$ $\beta_{3}$ $-0.001$ $0.057$ $0.057$
$-0.008$ $0.067$ $0.067$ $-0.023$ $0.085$ $0.085$ $0.001$ $0.057$ $0.057$
$-0.001$ $0.068$ $0.068$ $-0.005$ $0.089$ $0.089$ $400$ 1 $\beta_{1}$ $-0.005$
$0.002$ $0.002$ $-0.003$ $0.002$ $0.002$ $0.001$ $0.003$ $0.003$ $-0.005$
$0.002$ $0.002$ $-0.005$ $0.002$ $0.002$ $-0.003$ $0.003$ $0.003$ $\beta_{2}$
$0.000$ $0.001$ $0.001$ $0.000$ $0.001$ $0.001$ $-0.001$ $0.001$ $0.001$
$0.000$ $0.001$ $0.001$ $0.000$ $0.001$ $0.001$ $-0.001$ $0.001$ $0.001$
$\beta_{3}$ $0.002$ $0.015$ $0.015$ $0.000$ $0.017$ $0.017$ $-0.002$ $0.018$
$0.018$ $0.002$ $0.015$ $0.015$ $0.000$ $0.017$ $0.017$ $-0.001$ $0.018$
$0.018$ 3 $\beta_{1}$ $0.006$ $0.002$ $0.002$ $0.004$ $0.003$ $0.003$ $0.000$
$0.003$ $0.003$ $0.006$ $0.002$ $0.002$ $0.006$ $0.003$ $0.003$ $0.006$
$0.004$ $0.004$ $\beta_{2}$ $-0.004$ $0.002$ $0.002$ $-0.004$ $0.003$ $0.003$
$-0.004$ $0.003$ $0.003$ $-0.005$ $0.002$ $0.002$ $-0.005$ $0.003$ $0.003$
$-0.005$ $0.003$ $0.003$ $\beta_{3}$ $0.001$ $0.028$ $0.028$ $-0.004$ $0.033$
$0.033$ $-0.013$ $0.046$ $0.046$ $0.001$ $0.028$ $0.028$ $-0.001$ $0.034$
$0.034$ $0.000$ $0.048$ $0.048$

Table A3:  Bias, variance and MSE of $\hat{\gamma}$ and $\hat{\beta}$ for
smcure (second rows) and our approach (first rows) in Model 3 (Scenario 2).

Cens. level 1 Cens. level 2 Cens. level 3 Mod. n scen. Par. Bias Var. MSE Bias
Var. MSE Bias Var. MSE 3 $200$ $2$ $\gamma_{1}$ $-0.056$ $0.261$ $0.264$
$-0.215$ $0.344$ $0.390$ $-0.379$ $0.422$ $0.566$ $0.091$ $0.299$ $0.308$
$0.090$ $0.454$ $0.462$ $0.127$ $0.755$ $0.771$ $\gamma_{2}$ $-0.055$ $0.124$
$0.127$ $-0.123$ $0.137$ $0.152$ $-0.198$ $0.165$ $0.204$ $0.183$ $0.178$
$0.212$ $0.275$ $0.268$ $0.344$ $0.390$ $0.485$ $0.638$ $\gamma_{3}$ $0.023$
$0.401$ $0.402$ $0.087$ $0.512$ $0.519$ $0.162$ $0.612$ $0.639$ $0.161$
$0.488$ $0.514$ $0.278$ $0.729$ $0.806$ $0.427$ $1.159$ $1.341$ $\gamma_{4}$
$-0.049$ $0.321$ $0.324$ $-0.093$ $0.421$ $0.429$ $-0.164$ $0.543$ $0.570$
$0.036$ $0.367$ $0.369$ $0.086$ $0.547$ $0.554$ $0.062$ $0.795$ $0.799$
$\beta_{1}$ $0.017$ $0.008$ $0.009$ $0.017$ $0.009$ $0.009$ $0.013$ $0.010$
$0.010$ $0.016$ $0.008$ $0.009$ $0.018$ $0.009$ $0.009$ $0.015$ $0.010$
$0.011$ $\beta_{2}$ $0.004$ $0.004$ $0.004$ $0.007$ $0.004$ $0.004$ $0.005$
$0.005$ $0.005$ $0.004$ $0.004$ $0.004$ $0.009$ $0.004$ $0.004$ $0.008$
$0.005$ $0.005$ $\beta_{3}$ $0.022$ $0.059$ $0.059$ $0.008$ $0.059$ $0.060$
$-0.003$ $0.074$ $0.074$ $0.024$ $0.059$ $0.060$ $0.015$ $0.060$ $0.060$
$0.013$ $0.074$ $0.075$ $400$ $2$ $\gamma_{1}$ $-0.037$ $0.120$ $0.121$
$-0.145$ $0.174$ $0.195$ $-0.255$ $0.240$ $0.305$ $0.047$ $0.122$ $0.124$
$0.029$ $0.179$ $0.179$ $0.042$ $0.265$ $0.267$ $\gamma_{2}$ $-0.059$ $0.056$
$0.060$ $-0.123$ $0.075$ $0.090$ $-0.177$ $0.077$ $0.0108$ $0.090$ $0.068$
$0.076$ $0.110$ $0.098$ $0.110$ $0.173$ $0.125$ $0.155$ $\gamma_{3}$ $-0.020$
$0.181$ $0.182$ $0.026$ $0.245$ $0.246$ $0.080$ $0.337$ $0.344$ $0.066$
$0.194$ $0.198$ $0.124$ $0.261$ $0.276$ $0.212$ $0.380$ $0.425$ $\gamma_{4}$
$-0.021$ $0.151$ $0.152$ $-0.069$ $0.220$ $0.225$ $-0.122$ $0.315$ $0.329$
$0.023$ $0.156$ $0.157$ $0.022$ $0.218$ $0.218$ $0.033$ $0.309$ $0.310$
$\beta_{1}$ $0.011$ $0.004$ $0.004$ $0.008$ $0.004$ $0.004$ $0.013$ $0.005$
$0.005$ $0.011$ $0.004$ $0.004$ $0.007$ $0.004$ $0.004$ $0.013$ $0.005$
$0.005$ $\beta_{2}$ $0.003$ $0.002$ $0.002$ $0.003$ $0.002$ $0.002$ $0.001$
$0.002$ $0.002$ $0.003$ $0.002$ $0.002$ $0.004$ $0.002$ $0.002$ $0.003$
$0.002$ $0.002$ $\beta_{3}$ $0.019$ $0.028$ $0.029$ $0.002$ $0.029$ $0.029$
$-0.001$ $0.033$ $0.033$ $0.020$ $0.028$ $0.029$ $0.006$ $0.029$ $0.029$
$0.009$ $0.034$ $0.034$

Table A4:  Bias, variance and MSE of $\hat{\gamma}$ and $\hat{\beta}$ for
smcure (second rows) and our approach (first rows) in Model 4 (Scenario 2).

Cens. level 1 Cens. level 2 Cens. level 3 Mod. n scen. Par. Bias Var. MSE Bias
Var. MSE Bias Var. MSE 4 $200$ $2$ $\gamma_{1}$ $-0.121$ $0.117$ $0.131$
$-0.240$ $0.140$ $0.198$ $-0.375$ $0.179$ $0.320$ $0.006$ $0.134$ $0.134$
$-0.011$ $0.170$ $0.170$ $-0.035$ $0.231$ $0.232$ $\gamma_{2}$ $0.053$ $0.014$
$0.017$ $0.091$ $0.017$ $0.026$ $0.122$ $0.018$ $0.033$ $0.041$ $0.016$
$0.018$ $0.056$ $0.020$ $0.024$ $0.077$ $0.027$ $0.033$ $\gamma_{3}$ $-0.195$
$0.137$ $0.175$ $-0.348$ $0.156$ $0.277$ $-0.511$ $0.165$ $0.426$ $0.119$
$0.169$ $0.183$ $0.122$ $0.216$ $0.231$ $0.139$ $0.273$ $0.293$ $\gamma_{4}$
$0.096$ $0.156$ $0.165$ $0.157$ $0.189$ $0.213$ $0.240$ $0.229$ $0.287$
$0.072$ $0.182$ $0.187$ $0.096$ $0.216$ $0.226$ $0.137$ $0.288$ $0.307$
$\gamma_{5}$ $-0.038$ $0.161$ $0.163$ $-0.078$ $0.185$ $0.191$ $-0.149$
$0.218$ $0.240$ $0.046$ $0.175$ $0.177$ $0.058$ $0.220$ $0.223$ $0.047$
$0.254$ $0.257$ $\beta_{1}$ $0.014$ $0.007$ $0.007$ $0.009$ $0.008$ $0.008$
$0.001$ $0.008$ $0.008$ $0.017$ $0.007$ $0.007$ $0.016$ $0.008$ $0.008$
$0.014$ $0.008$ $0.008$ $\beta_{2}$ $0.005$ $0.004$ $0.004$ $0.003$ $0.004$
$0.004$ $0.002$ $0.004$ $0.005$ $0.005$ $0.004$ $0.004$ $0.005$ $0.004$
$0.004$ $0.005$ $0.004$ $0.005$ $\beta_{3}$ $0.010$ $0.056$ $0.056$ $-0.003$
$0.063$ $0.063$ $-0.022$ $0.071$ $0.072$ $0.015$ $0.057$ $0.057$ $0.011$
$0.064$ $0.064$ $0.006$ $0.072$ $0.072$ $400$ $2$ $\gamma_{1}$ $-0.068$
$0.057$ $0.062$ $-0.169$ $0.073$ $0.102$ $-0.279$ $0.089$ $0.167$ $0.015$
$0.061$ $0.061$ $0.001$ $0.078$ $0.078$ $-0.022$ $0.097$ $0.098$ $\gamma_{2}$
$0.033$ $0.007$ $0.008$ $0.068$ $0.008$ $0.013$ $0.099$ $0.010$ $0.019$
$0.019$ $0.008$ $0.008$ $0.030$ $0.010$ $0.011$ $0.044$ $0.012$ $0.014$
$\gamma_{3}$ $-0.180$ $0.073$ $0.105$ $-0.311$ $0.084$ $0.181$ $-0.450$
$0.093$ $0.296$ $0.043$ $0.079$ $0.081$ $0.038$ $0.096$ $0.097$ $0.030$
$0.119$ $0.120$ $\gamma_{4}$ $0.050$ $0.078$ $0.080$ $0.119$ $0.096$ $0.110$
$0.177$ $0.118$ $0.149$ $0.028$ $0.082$ $0.083$ $0.049$ $0.101$ $0.104$
$0.066$ $0.124$ $0.129$ $\gamma_{5}$ $-0.060$ $0.077$ $0.081$ $-0.106$ $0.096$
$0.107$ $-0.150$ $0.101$ $0.124$ $0.003$ $0.082$ $0.082$ $0.001$ $0.099$
$0.099$ $0.003$ $0.116$ $0.116$ $\beta_{1}$ $0.006$ $0.003$ $0.003$ $0.002$
$0.004$ $0.004$ $-0.004$ $0.004$ $0.004$ $0.008$ $0.003$ $0.003$ $0.008$
$0.004$ $0.004$ $0.007$ $0.004$ $0.004$ $\beta_{2}$ $0.002$ $0.002$ $0.002$
$0.000$ $0.002$ $0.002$ $0.000$ $0.002$ $0.002$ $0.002$ $0.002$ $0.002$
$0.001$ $0.002$ $0.002$ $0.002$ $0.002$ $0.002$ $\beta_{3}$ $0.003$ $0.026$
$0.026$ $-0.010$ $0.030$ $0.030$ $-0.023$ $0.034$ $0.034$ $0.007$ $0.026$
$0.026$ $0.002$ $0.031$ $0.031$ $-0.001$ $0.034$ $0.034$

Table A5: Bias, variance and MSE of $\hat{\gamma}$ and $\hat{\beta}$ for
smcure (second rows) and our approach (first rows) in Model 1 and 2
($n=1000$).

Cens. level 1 Cens. level 2 Cens. level 3 Mod. n scen. Par. Bias Var. MSE Bias
Var. MSE Bias Var. MSE 1 $1000$ $1$ $\gamma_{1}$ $0.007$ $0.011$ $0.011$
$0.008$ $0.013$ $0.013$ $0.010$ $0.015$ $0.015$ $0.016$ $0.011$ $0.012$
$0.019$ $0.013$ $0.013$ $0.022$ $0.015$ $0.016$ $\gamma_{2}$ $-0.006$ $0.032$
$0.032$ $-0.006$ $0.037$ $0.037$ $-0.006$ $0.041$ $0.041$ $0.023$ $0.033$
$0.034$ $0.027$ $0.038$ $0.039$ $0.028$ $0.043$ $0.044$ $\beta$ $0.001$
$0.005$ $0.005$ $0.002$ $0.006$ $0.006$ $0.002$ $0.006$ $0.006$ $0.000$
$0.005$ $0.005$ $0.001$ $0.006$ $0.006$ $0.000$ $0.006$ $0.006$ $2$
$\gamma_{1}$ $0.005$ $0.007$ $0.007$ $0.006$ $0.008$ $0.008$ $0.007$ $0.010$
$0.010$ $0.008$ $0.007$ $0.007$ $0.011$ $0.008$ $0.008$ $0.014$ $0.010$
$0.010$ $\gamma_{2}$ $-0.008$ $0.019$ $0.019$ $-0.006$ $0.022$ $0.022$
$-0.007$ $0.026$ $0.026$ $0.012$ $0.019$ $0.020$ $0.016$ $0.023$ $0.023$
$0.018$ $0.027$ $0.027$ $\beta$ $0.000$ $0.006$ $0.006$ $-0.001$ $0.007$
$0.007$ $0.000$ $0.008$ $0.008$ $-0.001$ $0.006$ $0.006$ $-0.002$ $0.007$
$0.007$ $-0.002$ $0.008$ $0.008$ $3$ $\gamma_{1}$ $0.001$ $0.011$ $0.011$
$-0.008$ $0.012$ $0.012$ $-0.003$ $0.018$ $0.018$ $0.004$ $0.011$ $0.011$
$-0.002$ $0.013$ $0.013$ $0.009$ $0.019$ $0.019$ $\gamma_{2}$ $-0.064$ $0.093$
$0.097$ $-0.076$ $0.123$ $0.129$ $-0.107$ $0.163$ $0.175$ $0.046$ $0.099$
$0.102$ $0.058$ $0.130$ $0.134$ $0.072$ $0.167$ $0.173$ $\beta$ $0.011$
$0.018$ $0.018$ $0.005$ $0.022$ $0.022$ $0.004$ $0.026$ $0.026$ $0.009$
$0.018$ $0.018$ $0.001$ $0.022$ $0.022$ $-0.005$ $0.026$ $0.026$ 2 $1000$ $1$
$\gamma_{1}$ $0.006$ $0.008$ $0.008$ $0.000$ $0.009$ $0.009$ $0.007$ $0.012$
$0.012$ $0.009$ $0.008$ $0.008$ $0.007$ $0.009$ $0.009$ $0.022$ $0.013$
$0.013$ $\gamma_{2}$ $-0.010$ $0.007$ $0.007$ $-0.011$ $0.009$ $0.009$
$-0.018$ $0.011$ $0.011$ $-0.002$ $0.007$ $0.007$ $-0.002$ $0.009$ $0.009$
$-0.005$ $0.012$ $0.012$ $\beta$ $-0.001$ $0.002$ $0.002$ $0.000$ $0.003$
$0.003$ $0.001$ $0.003$ $0.003$ $-0.001$ $0.002$ $0.002$ $-0.001$ $0.003$
$0.003$ $-0.001$ $0.003$ $0.003$ $2$ $\gamma_{1}$ $-0.004$ $0.006$ $0.006$
$-0.010$ $0.007$ $0.007$ $-0.004$ $0.010$ $0.010$ $0.002$ $0.006$ $0.006$
$0.003$ $0.008$ $0.008$ $0.022$ $0.010$ $0.011$ $\gamma_{2}$ $-0.012$ $0.007$
$0.008$ $-0.023$ $0.009$ $0.010$ $-0.031$ $0.012$ $0.012$ $0.004$ $0.008$
$0.008$ $0.000$ $0.009$ $0.009$ $0.002$ $0.012$ $0.012$ $\beta$ $0.005$
$0.003$ $0.003$ $0.006$ $0.003$ $0.003$ $0.005$ $0.004$ $0.004$ $0.004$
$0.003$ $0.003$ $0.005$ $0.003$ $0.003$ $0.001$ $0.004$ $0.004$ $3$
$\gamma_{1}$ $-0.011$ $0.014$ $0.014$ $-0.029$ $0.014$ $0.015$ $-0.054$
$0.019$ $0.022$ $0.002$ $0.016$ $0.016$ $0.015$ $0.018$ $0.018$ $0.028$
$0.029$ $0.029$ $\gamma_{2}$ $-0.314$ $0.144$ $0.242$ $-0.583$ $0.189$ $0.529$
$-0.841$ $0.237$ $0.945$ $0.045$ $0.148$ $0.151$ $0.068$ $0.175$ $0.180$
$0.137$ $0.223$ $0.242$ $\beta$ $0.004$ $0.006$ $0.006$ $0.004$ $0.008$
$0.008$ $0.006$ $0.010$ $0.010$ $0.002$ $0.006$ $0.006$ $0.001$ $0.008$
$0.008$ $0.003$ $0.010$ $0.010$

Table A6:  Bias, variance and MSE of $\hat{\gamma}$ and $\hat{\beta}$ for
smcure and our approach in Model 3 ($n=1000$).

Cens. level 1 Cens. level 2 Cens. level 3 n scen. Par. Bias Var. MSE Bias Var.
MSE Bias Var. MSE $1000$ $1$ $\gamma_{1}$ $0.000$ $0.026$ $0.026$ $0.010$
$0.035$ $0.035$ $0.013$ $0.050$ $0.050$ $0.001$ $0.026$ $0.026$ $0.003$
$0.031$ $0.031$ $0.006$ $0.040$ $0.040$ $\gamma_{2}$ $-0.009$ $0.008$ $0.008$
$-0.021$ $0.001$ $0.011$ $-0.051$ $0.014$ $0.017$ $-0.018$ $0.008$ $0.008$
$-0.022$ $0.010$ $0.010$ $-0.036$ $0.013$ $0.014$ $\gamma_{3}$ $0.019$ $0.071$
$0.072$ $0.022$ $0.094$ $0.094$ $0.022$ $0.128$ $0.128$ $0.041$ $0.069$
$0.070$ $0.060$ $0.078$ $0.082$ $0.062$ $0.105$ $0.109$ $\gamma_{4}$ $0.004$
$0.061$ $0.061$ $-0.030$ $0.087$ $0.088$ $-0.095$ $0.124$ $0.133$ $0.026$
$0.058$ $0.058$ $-0.002$ $0.077$ $0.078$ $0.033$ $0.098$ $0.099$ $\beta_{1}$
$-0.002$ $0.001$ $0.001$ $-0.001$ $0.001$ $0.001$ $0.001$ $0.001$ $0.001$
$-0.002$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$
$\beta_{2}$ $0.002$ $0.001$ $0.001$ $0.000$ $0.001$ $0.001$ $0.001$ $0.001$
$0.001$ $0.002$ $0.001$ $0.001$ $0.000$ $0.001$ $0.001$ $0.001$ $0.001$
$0.001$ $\beta_{3}$ $0.001$ $0.007$ $0.007$ $-0.006$ $0.008$ $0.008$ $-0.003$
$0.009$ $0.009$ $0.001$ $0.007$ $0.007$ $-0.006$ $0.0018$ $0.008$ $-0.001$
$0.009$ $0.009$ $2$ $\gamma_{1}$ $-0.025$ $0.046$ $0.047$ $-0.071$ $0.077$
$0.082$ $-0.142$ $0.117$ $0.137$ $0.020$ $0.046$ $0.047$ $0.007$ $0.066$
$0.066$ $0.005$ $0.090$ $0.090$ $\gamma_{2}$ $-0.057$ $0.022$ $0.025$ $-0.097$
$0.030$ $0.040$ $-0.108$ $0.038$ $0.050$ $0.032$ $0.023$ $0.024$ $0.037$
$0.031$ $0.032$ $0.056$ $0.038$ $0.041$ $\gamma_{3}$ $-0.041$ $0.068$ $0.070$
$-0.024$ $0.125$ $0.126$ $0.015$ $0.169$ $0.170$ $0.009$ $0.066$ $0.066$
$0.044$ $0.104$ $0.106$ $0.059$ $0.132$ $0.136$ $\gamma_{4}$ $-0.019$ $0.059$
$0.060$ $-0.025$ $0.093$ $0.093$ $-0.037$ $0.127$ $0.128$ $0.005$ $0.058$
$0.058$ $0.020$ $0.081$ $0.081$ $0.039$ $0.101$ $0.102$ $\beta_{1}$ $0.003$
$0.001$ $0.001$ $0.002$ $0.002$ $0.002$ $0.002$ $0.002$ $0.002$ $0.003$
$0.001$ $0.001$ $0.001$ $0.002$ $0.002$ $0.002$ $0.002$ $0.002$ $\beta_{2}$
$0.001$ $0.001$ $0.001$ $0.002$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$
$0.001$ $0.001$ $0.001$ $0.002$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$
$\beta_{3}$ $0.003$ $0.010$ $0.010$ $0.004$ $0.011$ $0.011$ $-0.010$ $0.013$
$0.013$ $0.003$ $0.010$ $0.010$ $0.005$ $0.011$ $0.011$ $-0.004$ $0.013$
$0.013$ $3$ $\gamma_{1}$ $0.004$ $0.030$ $0.031$ $-0.008$ $0.050$ $0.050$
$-0.058$ $0.093$ $0.096$ $-0.001$ $0.029$ $0.029$ $-0.008$ $0.043$ $0.043$
$-0.022$ $0.069$ $0.070$ $\gamma_{2}$ $-0.014$ $0.009$ $0.009$ $-0.011$
$0.014$ $0.014$ $0.031$ $0.022$ $0.023$ $0.015$ $0.009$ $0.009$ $0.018$
$0.012$ $0.012$ $0.045$ $0.018$ $0.020$ $\gamma_{3}$ $0.000$ $0.044$ $0.044$
$-0.004$ $0.072$ $0.072$ $0.049$ $0.123$ $0.125$ $0.017$ $0.043$ $0.043$
$0.016$ $0.058$ $0.058$ $0.043$ $0.089$ $0.091$ $\gamma_{4}$ $-0.002$ $0.037$
$0.037$ $0.002$ $0.060$ $0.060$ $-0.020$ $0.099$ $0.099$ $-0.007$ $0.036$
$0.036$ $0.003$ $0.051$ $0.051$ $-0.012$ $0.071$ $0.072$ $\beta_{1}$ $0.007$
$0.002$ $0.002$ $0.008$ $0.003$ $0.003$ $0.005$ $0.003$ $0.003$ $0.006$
$0.002$ $0.002$ $0.007$ $0.003$ $0.003$ $0.006$ $0.003$ $0.003$ $\beta_{2}$
$0.000$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$ $-0.001$ $0.001$ $0.001$
$0.000$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$ $-0.001$ $0.001$ $0.001$
$\beta_{3}$ $0.000$ $0.001$ $0.001$ $0.014$ $0.013$ $0.013$ $-0.004$ $0.017$
$0.017$ $0.000$ $0.001$ $0.001$ $0.014$ $0.012$ $0.013$ $-0.001$ $0.016$
$0.016$

Table A7:  Bias, variance and MSE of $\hat{\gamma}$ and $\hat{\beta}$ for
smcure and our approach in Model 4 ($n=1000$).

Cens. level 1 Cens. level 2 Cens. level 3 n scen. Par. Bias Var. MSE Bias Var.
MSE Bias Var. MSE $1000$ $1$ $\gamma_{1}$ $-0.002$ $0.027$ $0.027$ $0.007$
$0.033$ $0.033$ $0.010$ $0.038$ $0.038$ $0.004$ $0.027$ $0.027$ $0.008$
$0.032$ $0.032$ $0.012$ $0.038$ $0.038$ $\gamma_{2}$ $0.013$ $0.007$ $0.007$
$-0.001$ $0.008$ $0.008$ $-0.005$ $0.009$ $0.009$ $-0.015$ $0.007$ $0.007$
$-0.024$ $0.009$ $0.009$ $-0.029$ $0.011$ $0.012$ $\gamma_{3}$ $-0.192$
$0.041$ $0.078$ $-0.149$ $0.037$ $0.059$ $-0.196$ $0.046$ $0.084$ $0.020$
$0.037$ $0.037$ $0.029$ $0.047$ $0.048$ $0.035$ $0.058$ $0.059$ $\gamma_{4}$
$-0.054$ $0.063$ $0.066$ $-0.086$ $0.076$ $0.084$ $-0.155$ $0.090$ $0.114$
$0.030$ $0.067$ $0.068$ $0.047$ $0.083$ $0.085$ $0.053$ $0.104$ $0.107$
$\gamma_{5}$ $-0.043$ $0.060$ $0.062$ $-0.108$ $0.075$ $0.087$ $-0.188$
$0.083$ $0.118$ $0.025$ $0.061$ $0.062$ $0.035$ $0.075$ $0.076$ $0.039$
$0.092$ $0.093$ $\beta_{1}$ $-0.001$ $0.001$ $0.001$ $0.000$ $0.001$ $0.001$
$0.002$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$ $-0.002$ $0.001$ $0.001$
$-0.001$ $0.001$ $0.001$ $\beta_{2}$ $0.000$ $0.000$ $0.000$ $0.000$ $0.001$
$0.001$ $0.000$ $0.001$ $0.001$ $0.000$ $0.000$ $0.000$ $0.000$ $0.001$
$0.001$ $0.000$ $0.001$ $0.001$ $\beta_{3}$ $0.001$ $0.006$ $0.006$ $0.002$
$0.006$ $0.006$ $0.002$ $0.007$ $0.007$ $0.001$ $0.006$ $0.006$ $0.002$
$0.007$ $0.007$ $0.003$ $0.007$ $0.007$ $2$ $\gamma_{1}$ $-0.049$ $0.024$
$0.027$ $-0.116$ $0.031$ $0.044$ $-0.207$ $0.040$ $0.083$ $0.006$ $0.024$
$0.024$ $-0.004$ $0.031$ $0.031$ $-0.020$ $0.039$ $0.039$ $\gamma_{2}$ $0.019$
$0.003$ $0.003$ $0.049$ $0.003$ $0.006$ $0.079$ $0.004$ $0.011$ $0.006$
$0.003$ $0.003$ $0.011$ $0.004$ $0.004$ $0.018$ $0.005$ $0.005$ $\gamma_{3}$
$-0.142$ $0.031$ $0.051$ $-0.229$ $0.039$ $0.091$ $-0.330$ $0.045$ $0.154$
$0.017$ $0.032$ $0.033$ $0.021$ $0.040$ $0.040$ $0.017$ $0.046$ $0.046$
$\gamma_{4}$ $0.029$ $0.033$ $0.034$ $0.079$ $0.041$ $0.047$ $0.144$ $0.051$
$0.072$ $0.012$ $0.033$ $0.033$ $0.026$ $0.040$ $0.040$ $0.039$ $0.047$
$0.049$ $\gamma_{5}$ $-0.034$ $0.033$ $0.034$ $-0.068$ $0.039$ $0.044$
$-0.103$ $0.048$ $0.058$ $0.005$ $0.032$ $0.032$ $0.006$ $0.036$ $0.036$
$0.007$ $0.041$ $0.042$ $\beta_{1}$ $0.001$ $0.001$ $0.001$ $-0.003$ $0.001$
$0.001$ $-0.008$ $0.001$ $0.001$ $0.002$ $0.001$ $0.001$ $0.002$ $0.001$
$0.001$ $0.001$ $0.001$ $0.001$ $\beta_{2}$ $0.000$ $0.001$ $0.001$ $-0.001$
$0.001$ $0.001$ $-0.001$ $0.001$ $0.001$ $0.001$ $0.001$ $0.001$ $0.000$
$0.001$ $0.001$ $0.001$ $0.001$ $0.001$ $\beta_{3}$ $0.000$ $0.010$ $0.010$
$-0.009$ $0.011$ $0.011$ $-0.019$ $0.012$ $0.012$ $0.002$ $0.010$ $0.010$
$0.000$ $0.011$ $0.011$ $-0.001$ $0.012$ $0.012$ $3$ $\gamma_{1}$ $-0.002$
$0.015$ $0.015$ $-0.012$ $0.019$ $0.019$ $-0.029$ $0.026$ $0.027$ $0.004$
$0.015$ $0.015$ $0.001$ $0.018$ $0.018$ $0.001$ $0.023$ $0.023$ $\gamma_{2}$
$-0.005$ $0.001$ $0.001$ $-0.001$ $0.002$ $0.002$ $0.006$ $0.002$ $0.002$
$0.007$ $0.002$ $0.002$ $0.004$ $0.002$ $0.002$ $0.002$ $0.002$ $0.002$
$\gamma_{3}$ $0.051$ $0.016$ $0.019$ $0.034$ $0.015$ $0.017$ $0.041$ $0.021$
$0.022$ $-0.009$ $0.014$ $0.014$ $-0.006$ $0.016$ $0.016$ $-0.006$ $0.021$
$0.021$ $\gamma_{4}$ $0.000$ $0.020$ $0.020$ $0.009$ $0.024$ $0.024$ $0.018$
$0.031$ $0.031$ $-0.001$ $0.019$ $0.019$ $0.004$ $0.023$ $0.023$ $0.004$
$0.028$ $0.028$ $\gamma_{5}$ $-0.006$ $0.019$ $0.019$ $-0.011$ $0.024$ $0.024$
$-0.015$ $0.033$ $0.033$ $-0.005$ $0.019$ $0.019$ $-0.007$ $0.022$ $0.022$
$-0.011$ $0.028$ $0.028$ $\beta_{1}$ $0.003$ $0.001$ $0.001$ $0.001$ $0.001$
$0.001$ $-0.001$ $0.001$ $0.001$ $0.002$ $0.001$ $0.001$ $0.001$ $0.001$
$0.001$ $0.001$ $0.001$ $0.001$ $\beta_{2}$ $0.000$ $0.001$ $0.001$ $0.000$
$0.001$ $0.001$ $0.000$ $0.001$ $0.001$ $-0.001$ $0.001$ $0.001$ $0.000$
$0.001$ $0.001$ $0.000$ $0.001$ $0.001$ $\beta_{3}$ $0.003$ $0.010$ $0.010$
$0.001$ $0.012$ $0.012$ $-0.005$ $0.016$ $0.016$ $0.003$ $0.010$ $0.010$
$0.002$ $0.012$ $0.012$ $-0.001$ $0.016$ $0.016$