# The Effect of Stroke on Dementia Onset: Left-Truncation and Right-Censoring

R Weißbach, A Dörre, D Wied, G Doblhammer, A Fink

###### Abstract

We observe a quarter million people over a period of nine years and are
interested in the effect of a stroke on the probability of dementia onset.
Randomly left-truncated has a person been that was already deceased before the
period. The ages at a stroke event or dementia onset are conditionally fixed
right-censored, when either event may still occur, but after the period. We
incorporate death and model the history of the three events by a homogeneous
Markov process. The compensator for the respective counting processes is
derived and Jacod’s formula yields the likelihood contribution, conditional on
observation. An Appendix is devoted to the role of filtrations in deriving the
likelihood, for the simplification of merged non-dead states. Asymptotic
normality of the estimated intensities is derived by martingale theory,
relative to the size of the sample including the truncated persons. The data
of a German health insurance reveals that after a stroke, the intensity of
dementia onset is increased from 0.02 to 0.07, for Germans born in the first
half on the 20th century. The intensity difference has a $95\%$-confidence
interval of $[0.048,0.051]$ and the difference halves when we extent to an
age-inhomogeneous model due to Simpson’s paradox.

###### keywords:

Censoring; Truncation; Filtering; Confidence interval; Markov process.

##  1 Introduction

We are interested in the consequences of a stroke (as e.g. Hbid et al., 2020).
Let $t$ count the years after a person’s 50th birthday, typically the earliest
age of a stroke, and we simply call $t$ ‘age’. In our multi-morbidity
analysis, we consider the four increasingly pour states of being healthy (H),
having had at least one stroke (S1), having dementia (D) and being dead (d).
Let $X_{t}$ indicate the worst thereof at the age of $t$. Our multi-state
model about the connection between stroke and dementia for Germany is similar
that for South Korea by Kim and Lee (2018), both using national Health Claims
Data (HCD). We collect the possible state transitions in
$\mathcal{I}:=\\{HS^{1},S^{1}D,HD,Hd,S^{1}d,Dd\\}$. We do not look any further
than $\tau$ years, and let, in continuous time, the history
$\mathbf{X}=\\{X_{t},t\in[0,\tau]\\}$ be defined on probability spaces
$(\Omega,\mathcal{F},P_{\boldsymbol{\lambda}})$. We assume throughout a
Markovian history, being is determined by the transition intensities
$\lambda_{hj}(t):=\lim_{u\searrow 0}P_{\boldsymbol{\lambda}}(X_{t+u}=j\mid
X_{t}=h)/u$. Our age-homogeneous population model for Germany (in the first
half of the 20th century) is throughout the paper, with the exception of
Section 3.3, that $\lambda_{hj}(t)\equiv\lambda_{hj}$. By parameter we mean
the vector $\boldsymbol{\lambda}:=(\lambda_{hj},hj\in\mathcal{I})^{\prime}$
and assume for the true parameter $\boldsymbol{\lambda}_{0}$:

  1. (B0)

It is $\lambda_{hj^{0}}\in\Lambda_{hj}:=[\varepsilon_{hj};1/\varepsilon_{hj}]$
for small $\varepsilon_{hj}\in]0,1[$.

The counting processes $N_{hj}(t):=\sum_{s\leq
t}\mathds{1}_{\\{X_{i}(s-)=h,X_{i}(s)=j\\}}$, indicating a possible transition
from state $h$ to $j$ until age $t$, and
$Y_{h}(t):=\mathds{1}_{\\{X_{i}(t-)=h\\}}$, denoting whether a person can
migrate out of state $h$ at age $t$, reformulate one history. The vector
$\mathbf{N}_{X}(t):=(N_{hj}(t),hj\in\mathcal{I})^{\prime}$ is adapted to the
filtration $\mathcal{N}_{t}:=\sigma\\{\mathbf{N}_{X}(s),0\leq s\leq t\\}$,
because we assume a fifty-years old to be healthy, i.e. $X_{0}=H$.

We are not in possession neither of the population data nor of a simple sample
of size $n_{all}$ therefore. All we observe are those $n<n_{all}$ history
parts that occur in a study period between 2003 ans 20013. What we also
observe, for each observed history, is the duration $U$ (if random independent
of $\mathbf{X}$, or $u$ if not), between the calendar dates of the 50th
birthday and the study begin on 01/01/2004, and we denote it as ‘age-at-study-
begin’.

The change in information must change the $\sigma$-algebra $\mathcal{F}$ and
filtration $\mathcal{N}_{t}\in\mathcal{F}$. We need to distinguish existing
information and information available to the researcher. The new existing
information, namely the age-at-study-begin as second measurement, must lead to
a finer $\sigma$-algebra and hence to a finer filtration. In contrast, less
information available, namely the loss of left-truncated persons deceased
before 2004 and right-censored events after 2013, must lead to a coarser
filtration.

We use the Doob-Meyer decomposition into compensator and martingale for the
processes counting age-specific strokes and dementia onsets in Germany from
2004 to 2013, to derive a conditional likelihood by Jacod’s formula. And by
verifying regularity conditions we arrive at an asymptotic confidence
intervals for the parameters by standard results on martingales.

All that is easiest explained methodologically with merging the not-dead
states to an ‘alive’ state and we therefore included a mortality model in A.
For the four states we repeat the main arguments in the next Section 2. Our
result for Germany, taking account of left-truncation, that the hazard of a
dementia onset approximately doubles after a stroke, is similar to that for
Korea by Kim and Lee (2018).

##  2 Methodology

As preliminary model, in A, we specialized to mortality, and if we further
simplify to a non-random age-at-study-begin, $u$, the measurement of one
sample unit is a univariate random variable. In the morbidity model here, with
the four states and six transitions (from $\mathcal{I}$), the measurement is
an age-continuous, discrete-state history. The compensator of $\mathbf{N}_{X}$
has an intensity with respect to $\mathcal{N}_{t}$ and
$P_{\boldsymbol{\lambda}}$ of
$\boldsymbol{\alpha}(t):=(Y_{h}(t)\lambda_{hj},hj\in\mathcal{I})^{\prime}$.111When
in an expression with $hj\in\mathcal{I}$ only $h$ appears, it means the first
coordinate of $hj$, and especially $h\neq d$ because death is absorbing.

A person is not left-truncated when surviving 2003, i.e. is observed in case
of $A:=\\{X_{u}\neq d\\}$. We assume $P_{\boldsymbol{\lambda}}(A)>0$, and the
observable filtration is

$\leftindex_{u}{\mathcal{G}}_{t}:=\sigma\\{\leftindex_{u}{\mathbf{N}}(s),u\leq
s\leq
t\\}\quad\text{with}\quad\leftindex_{u}{\mathbf{N}}(t):=\mathbf{N}(t)-\mathbf{N}(t\land
u),$

where $t\land u:=\min(t,u)$.

###### Lemma 1

With respect to the probability measure

$P^{A}_{\boldsymbol{\lambda}}(F):=\frac{P_{\boldsymbol{\lambda}}(F\cap
A)}{P_{\boldsymbol{\lambda}}(A)}\quad\text{for}\quad F\in\mathcal{F},$

the intensity process of $\leftindex_{u}{\mathbf{N}}(t)$ is
$\leftindex_{u}{\boldsymbol{\alpha}}(t):=\mathds{1}_{\\{u<t\\}}\boldsymbol{\alpha}(t)$.

The deterministic $u$ is a special case of the random $U$, so that the proof
of the latter can be found in Proposition 4.1 and Example 4.3 of Andersen et
al. (1988), for details see A.1.

Note that $P^{A}_{\boldsymbol{\lambda}}$ depends on the parameter
$\boldsymbol{\lambda}$ and on $u$. With
$\leftindex_{u}{Y}_{h}(t):=\mathds{1}_{\\{u<t\\}}Y_{h}(t)$, the coordinates of
$\leftindex_{u}{\boldsymbol{\alpha}}$ are

$\leftindex_{u}{\alpha}_{hj}(t):=\mathds{1}_{\\{u<t\\}}\alpha_{hj}(t)=\leftindex_{u}{Y}(t)\lambda_{hj}.$

The observed left-truncated and right-censored counting process is

$\leftindex_{u}{\mathbf{N}}^{c}(t):=\int_{0}^{t}C(s)d\leftindex_{u}{\mathbf{N}}(s),$

with $C(t):=\mathds{1}_{\\{t\leq u+9\\}}$ and
$\leftindex_{u}{\mathbf{Y}}^{c}(t):=C(t)\leftindex_{u}{\mathbf{Y}}(t)$
(compare Fleming and Harrington, 1991, Example 1.4.2). It has intensity

$\leftindex_{u}{\boldsymbol{\alpha}}^{c}(t):=\mathds{1}_{\\{t\leq
u+9\\}}\leftindex_{u}{\boldsymbol{\alpha}}(t)=\mathds{1}_{\\{u<t\leq
u+9\\}}\boldsymbol{\alpha}(t)$ (1)

with respect to $P^{A}_{\boldsymbol{\lambda}}$ and observed filtration
$\leftindex_{u}{\mathcal{F}}^{c}_{t}:=\sigma\\{\leftindex_{u}{\mathbf{N}}^{c}(s),u\leq
s\leq t\\}$.

As $\leftindex_{u}{\mathcal{F}}^{c}_{t}$ is a required self-exciting
filtration, by Jacod’s formula (see Andersen et al., 1988, Formula (4.3))
likelihood and log-likelihood are:

$\displaystyle dP$ $\displaystyle=$ $\displaystyle\Prodi_{u<t\leq
u+9}\\{(1-\leftindex_{u}{\alpha}_{\cdot}^{c}(t)dt)^{1-d\leftindex_{u}{N}_{\cdot}^{c}(t)}\prod_{h\in\\{H,S^{1},D\\}}\prod_{\stackrel{{\scriptstyle
j\in\\{S^{1},D,d\\}}}{{j\neq
h}}}(\leftindex_{u}{\alpha}_{hj}^{c}(t)dt)^{d\leftindex_{u}{N}_{hj}^{c}(t)}\\}$
$\displaystyle=$ $\displaystyle\left[\prod_{\stackrel{{\scriptstyle
t\in[u,u+9],\exists
hj\in\mathcal{I}}}{{\leftindex_{u}{N}^{c}_{hj}(t-)\not=\leftindex_{u}{N}^{c}_{hj}(t)}}}\prod_{hj\in\mathcal{I}}\left(\leftindex_{u}{Y}^{c}_{h}(t)\lambda_{hj}\right)^{\Delta\leftindex_{u}{N}^{c}_{hj}(t)}\right]$
$\displaystyle\exp\left(-\sum_{hj\in\mathcal{I}}\int_{u}^{u+9}\leftindex_{u}{Y}^{c}_{h}(t)\lambda_{hj}dt\right)$
$\displaystyle\log dP$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\sum_{hj\in\mathcal{I}}(\log\leftindex_{u}{Y}^{c}_{h}(t)+\log\lambda_{hj})d\leftindex_{u}{N}^{c}_{hj}(t)-\sum_{hj\in\mathcal{I}}\int_{0}^{\tau}\leftindex_{u}{Y}^{c}_{h}(t)\lambda_{hj}dt$
(2)

Note
$\leftindex_{u}{\alpha}_{\cdot}^{c}(t):=\sum_{hj\in\mathcal{I}}\leftindex_{u}{\alpha}_{hj}^{c}(t)$
together with
$\leftindex_{u}{N}_{\cdot}^{c}(t):=\sum_{hj\in\mathcal{I}}\leftindex_{u}{N}_{hj}^{c}(t)$.
The double-use of the integration symbol $dt$ in the first line is explained
in (14). For the second line see e.g. Formula (11) in Weißbach et al. (2009):
For whatever process $\mathbf{Z}$, it is $dZ(t):=Z(t)-Z(t-dt)$ for some
‘small’ $dt$ (see Fleming and Harrington, 1991, Sect. 1.4) and $\Delta
Z(t):=Z(t)-Z(t-)$ (see Andersen et al., 1993, Sect. II.2), hence
$\Delta\leftindex_{u}{N}^{c}_{hj}(t)=\leftindex_{u}{N}^{c}_{hj}(t)-\leftindex_{u}{N}^{c}_{hj}(t-)$
is only not zero if $\leftindex_{u}{N}^{c}_{hj}(t)$ jumps. These jumps are of
height one, since the transition times are continuous. Further note that
$\int_{u}^{u+9}$ can be replaced by $\int_{0}^{\tau}$ because
$\leftindex_{u}{Y}^{c}_{h}(t)$ already accounts for the limits, and alike, in
the product, $[u,u+9]$ is accounted for in
$\leftindex_{u}{\mathbf{N}}^{c}(t)$. Note that, because
$\leftindex_{u}{N}^{c}_{hj}(u)=0$,

$\frac{\partial}{\partial\lambda_{hj}}\log
dP=\frac{\leftindex_{u}{N}^{c}_{hj}(\tau)}{\lambda_{hj}}-\int_{0}^{\tau}\leftindex_{u}{Y}^{c}_{h}(t)dt.$
(3)

However, we cannot assume $u$ to be non-random. If we assume $\mathbf{X}$ to
be random, and with it the age-at-death, we should also assume the age-at-
study-begin to be random, i.e. $U$. Increasing the filtration to a finer has
not been necessary so far, for non-random (and universally known) $u$. We
impose as additional assumption, saying that not everybody is dead, prior to
2004:

  1. (B0)

$\beta_{\boldsymbol{\lambda}_{0}}:=P(X_{U}\neq d)>0$

The additional information by $U$, i.e. for $(\mathbf{X},U)$, and at the same
time the loss in information by truncation, is reflected by including
$\leftindex_{U}{\mathbf{Y}}^{c}$ into the filtration
$\leftindex_{u}{\mathcal{F}}^{c}_{t}$,
$\leftindex_{U}{\mathcal{F}}^{c}_{t}:=\sigma\\{\leftindex_{U}{\mathbf{N}}^{c}(s),\leftindex_{U}{\mathbf{Y}}^{c}(s),u\leq
s\leq t\\}$, for details see A.2. Consult A.3 to see that, similar to non-
random truncation (2), conditional on the last two coordinates, the
logarithmic density of $(\mathbf{X},U,\mathds{1}_{\\{X_{U}\neq d\\}})$ up to
$\tau$ is

$\begin{split}\log\leftindex_{U}{L}_{\tau}^{c}(\mathbf{X},U|\boldsymbol{\lambda})=&\int_{0}^{\tau}\sum_{hj\in\mathcal{I}}(\log\leftindex_{U}{Y}^{c}_{h}(t)+\log\lambda_{hj})d\leftindex_{U}{N}^{c}_{hj}(t)\\\
&-\sum_{hj\in\mathcal{I}}\int_{0}^{\tau}\leftindex_{U}{Y}^{c}_{h}(t)\lambda_{hj}dt,\end{split}$
(4)

where $U$ replaces $u$ in the definitions of $\leftindex_{u}{N}^{c}_{hj}(t)$,
$\leftindex_{u}{Y}^{c}_{h}(t)$ and
$\leftindex_{u}{\boldsymbol{\alpha}}^{c}(t)$ of (1).

So far we studied one person of the sample, truncated or not. Now we consider
all persons in a simple sample (of size $n_{all}$). The size of the observed
sample is now $n:=\sum_{i=1}^{n_{all}}\mathds{1}_{A_{i}}$, where $i$
enumerates the persons. Without loss of generality, we sort those truncated to
the end of the unobserved sample, all other summands or factors in the sequel
will be ineffective. Now, due to the simple sample assumption, the conditional
density of the sample is the product of the persons’ conditional densities
(see A.3),

$\log\leftindex_{U}{L}_{\tau}^{c}(data;\boldsymbol{\lambda})=\sum_{i=1}^{n}\log\leftindex_{U}{L}_{\tau}^{c}(\mathbf{X}_{i},U_{i}|\boldsymbol{\lambda}).$
(5)

This also requires $U_{i}$ to be random. Due to (3), and in spite of (4),
$\leftindex_{U}{N}^{c}_{hj\bullet}(t):=\sum_{i=1}^{n}\leftindex_{U}{N}^{c}_{hji}(t)$
and
$\leftindex_{U}{Y}^{c}_{h\bullet}(t):=\sum_{i=1}^{n}\leftindex_{U}{Y}^{c}_{hi}(t)$
suffice for representing the point estimator because the root of
$\frac{\partial}{\partial\lambda_{hj}}\log\leftindex_{U}{L}_{\tau}^{c}(data|\boldsymbol{\lambda})$
is uniquely

$\hat{\lambda}_{hj}=\frac{\leftindex_{U}{N}^{c}_{hj\bullet}(\tau)}{\int_{0}^{\tau}\leftindex_{U}{Y}^{c}_{h\bullet}(t)dt}.$
(6)

Note that by the simple sample assumption, among those who survive $U_{i}$
(i.e. 2003), the portion in the study period at the age $t$ in state $h$ is
asymptotically the same in the observed sample and in the population. By the
LLN, for fixed $t$,

$\frac{\leftindex_{U}{Y}^{c}_{h\bullet}(t)}{n}\stackrel{{\scriptstyle
P}}{{\longrightarrow}}m_{h}(t).$ (7)

The latter will typically be positive, however for our parametric model we
only need to assume:

  1. (B0)

$\int_{0}^{\tau}\sum_{h\in\\{H,S^{1},D\\}}m_{h}(t)dt>0$

###### Theorem 1

Under Conditions (B0)-(B0) it is $\hat{\boldsymbol{\lambda}}$, composed of
(6), consistent and

$\sqrt{n_{all}}(\hat{\boldsymbol{\lambda}}-\boldsymbol{\lambda}_{0})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\mathcal{N}(0,\boldsymbol{\Sigma}^{-1}(\boldsymbol{\lambda}_{0}))$

with diagonal matrix $\boldsymbol{\Sigma}(\boldsymbol{\lambda}_{0})$ of
diagonal elements

$\sigma_{hj,hj}(\boldsymbol{\lambda}_{0}):=\beta_{\boldsymbol{\lambda}_{0}}\int_{0}^{\tau}\sum_{hj\in\mathcal{I}}\frac{m_{h}(t)}{\lambda_{hj^{0}}}dt,\quad\text{for}\quad
hj\in\mathcal{I}.$

Roughly speaking, the arguments are similar to the case of the univariate
parameter space in A, the multivariate parameter space here results in a
diagonal matrix of asymptotic variances and positive definiteness follows from
positivity of the diagonal elements.

Proof: Due to the uniqueness of $\hat{\boldsymbol{\lambda}}$, for both,
consistency (see Andersen et al., 1993, Theorem VI.1.1) and asymptotic
normality (see Andersen et al., 1993, Theorem VI.1.2), we need to verify
Conditions (A)-(E) (see Andersen et al., 1993, Condition VI.1.1). In detail,
in order to map the notations, note that $n$ becomes $n_{all}$,
$a_{n}:=\sqrt{n_{all}}$, $\boldsymbol{\theta}$ becomes $\boldsymbol{\lambda}$,
$h$ becomes $hj$ and $\lambda_{h}(t;\boldsymbol{\theta})$ becomes

$\leftindex_{U}{\alpha}^{c}_{hj\bullet}(t;\boldsymbol{\lambda})=\lambda_{hj}\leftindex_{U}{Y}^{c}_{h\bullet}(t),\
hj\in\mathcal{I},$ (8)

bacause by interchanging conditional expectations, the multiplicative
intensity process of $\leftindex_{U}{N}^{c}_{hj\bullet}(t)$ is the sum of the
compensators for each person’s counting process.

The fulfilment of (A) is now as in Lemma 2 of A.4. For (B), note first that
for mixed derivatives, terms are only non-zero when
$h_{1}j_{1}=h_{2}j_{2}=h_{3}j_{3}$ due to (8). In that case,

$\frac{1}{n_{all}}\int_{0}^{\tau}\sum_{hj\in\mathcal{I}}\left(\frac{\partial}{\partial\lambda_{hj}}\log\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}_{0})\right)^{2}\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}_{0})dt\\\
=\frac{n}{n_{all}}\frac{1}{n}\int_{0}^{\tau}\sum_{hj}\frac{1}{\lambda_{hj^{0}}^{2}}\lambda_{hj^{0}}\leftindex_{U}{Y}^{c}_{h\bullet}(t)dt\rightarrow\sigma_{hj,hj}(\boldsymbol{\lambda}_{0}),$

because, by the LLN, for fixed $t\in[0,\tau]$, (including Slutzky’s Lemma, the
CMT, and (7) (as in Lemma 2)) it is $n/n_{all}\stackrel{{\scriptstyle
P}}{{\longrightarrow}}P(X_{U}\neq d)=\beta_{\boldsymbol{\lambda}_{0}}>0$ by
(B0). For (C),

$\frac{1}{n_{all}}\int_{0}^{\tau}\sum_{hj\in\mathcal{I}}\left(\frac{\partial}{\partial\lambda_{hj}}\log\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}_{0})\right)^{2}\mathds{1}_{\left\\{\left|\frac{1}{\sqrt{n_{all}}}\frac{\partial}{\partial\lambda_{hj}}\log\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}_{0})\right|>\varepsilon\right\\}}\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}_{0})dt\\\
=\frac{1}{n_{all}}\int_{0}^{\tau}\sum_{hj\in\mathcal{I}}\frac{1}{\lambda_{hj^{0}}^{2}}\mathds{1}_{\left\\{\left|\frac{1}{\sqrt{n_{all}}}\frac{1}{\lambda_{hj^{0}}}\right|>\varepsilon\right\\}}\lambda_{hj^{0}}\leftindex_{U}{Y}^{c}_{h\bullet}(t)dt\rightarrow
0,$

because $1/\sqrt{n_{all}}\to 0$ and $\lambda_{hj^{0}}>0$ due to (B0). For (D),
$\boldsymbol{\Sigma}(\boldsymbol{\lambda}_{0})$ is diagonal with
$\sigma_{hj,hj}(\boldsymbol{\lambda}_{0})>0$ due to (B0) and (B0). Now in (E),
generally (first line) and if $h_{1}j_{1}=h_{2}j_{2}=h_{3}j_{3}$ (second
line), it is:

$\displaystyle\sup_{\boldsymbol{\lambda}\in\boldsymbol{\Lambda}}\left|\frac{\partial^{3}}{\partial\lambda_{h_{1}j_{1}}\partial\lambda_{h_{2}j_{2}}\partial\lambda_{h_{3}j_{3}}}\lambda_{hj}\leftindex_{U}{Y}^{c}_{h\bullet}(t)\right|$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\sup_{\boldsymbol{\lambda}\in\boldsymbol{\Lambda}}\left|\frac{\partial^{3}}{\partial\lambda_{h_{1}j_{1}}\partial\lambda_{h_{2}j_{2}}\partial\lambda_{h_{3}j_{3}}}\log(\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}))\right|$
$\displaystyle=$
$\displaystyle\sup_{\lambda_{hj}\in\Lambda_{hj}}\frac{1}{2\lambda_{hj}^{3}}$

Now $1/(2\lambda_{hj}^{3})\leq 1/(2\varepsilon_{hj^{0}}^{3})<\infty$ by (B0).
(This explains the third and forth requirement.) For the fifth,

$\frac{1}{n_{all}}\int_{0}^{\tau}\sum_{hj\in\mathcal{I}}\left(\frac{\partial^{2}}{\partial\lambda_{hj}^{2}}\log\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}_{0})\right)^{2}\leftindex_{U}{\alpha}_{hj\bullet}^{c}(t;\boldsymbol{\lambda}_{0})dt\\\
=\frac{n}{n_{all}}\frac{1}{n}\int_{0}^{\tau}\sum_{(h,j)}\frac{1}{\lambda_{hj^{0}}^{4}}\lambda_{hj^{0}}\leftindex_{U}{Y}^{c}_{h\bullet}(t)dt\rightarrow\frac{\sigma_{hj,hj}(\boldsymbol{\lambda}_{0})}{\lambda_{hj^{0}}^{2}},$

being finite due to (B0) and (B0). The sixth is fulfilled with the same
argument as in (C).

It remains to consistently estimate $\boldsymbol{\Sigma}(\lambda_{0})$, in
order to construct a confidence interval with the standard errors. By Theorem
1,

$Var(\hat{\boldsymbol{\lambda}})=Var(\sqrt{n_{all}}\hat{\boldsymbol{\lambda}})/n_{all}\stackrel{{\scriptstyle\cdot}}{{=}}\boldsymbol{\Sigma}^{-1}(\boldsymbol{\lambda}_{0})/n_{all},$

so that for estimating the asymptotic variance in Theorem 1, define
$-\mathcal{J}_{\tau}(\boldsymbol{\lambda}_{0})$ as

$diag\left(\frac{\partial^{2}}{\partial\lambda_{hj}^{2}}\log\leftindex_{U}{L}_{\tau}^{c}(data|\boldsymbol{\lambda})|_{\boldsymbol{\lambda}=\boldsymbol{\lambda}_{0}};hj\in\mathcal{I}\right)=-diag\left(\frac{\leftindex_{U}{N}^{c}_{hj\bullet}(\tau)}{\lambda_{hj^{0}}^{2}};hj\in\mathcal{I}\right).$

Now, as
$\mathcal{J}_{\tau}(\boldsymbol{\lambda}_{0})/n_{all}\stackrel{{\scriptstyle
n_{all}\to\infty,P}}{{\longrightarrow}}\boldsymbol{\Sigma}(\lambda_{0})$ (see
Andersen et al., 1993, Formula (6.1.11)), it is

$\widehat{Var(\hat{\lambda}_{hj})}=\mathcal{J}_{\tau}^{-1}(\hat{\boldsymbol{\lambda}})_{hj,hj}=\frac{\leftindex_{U}{N}^{c}_{hj\bullet}(\tau)}{\left(\int_{0}^{\tau}\leftindex_{U}{Y}^{c}_{h\bullet}(t)dt\right)^{2}},$
(9)

due to
$\mathcal{J}_{\tau}^{-1}(\boldsymbol{\lambda}_{0})_{hj,hj}=\lambda_{hj^{0}}^{2}/\leftindex_{U}{N}^{c}_{hj\bullet}(\tau)$,
(6) and the CMT. The estimated standard error of $\hat{\lambda}_{hj}$ is the
square root thereof. Note that even though $n_{all}$ is not observed, the
standard error estimate is.

##  3 Application: Stroke and Dementia

As population we consider Germans born between January 1st, 1900, and December
31st, 1954. On 01/01/2004 we drew a simple sample of size $n_{all}$ from that
population. Truncated (and not observed) is that part of the sample, that did
not survive $31/12/2003$ (see Figure 1). Of those $250.000$ persons made
accessible to us, after data cleaning $n=245{,}879$ contribute to the
conditional likelihood (5). The persons are all insurants of the Allgemeine
Ortskrankenkasse (AOK), with 25 million insurants at the time of sampling the
largest public health insurance company in Germany. We ignore any possible
selection bias in this respect. Simplifying we assume that a person cannot die
before, and stays healthy in our sense, until age 50. We follow-up histories
for transitions up to 31/12/2013, i.e. up to $\tau=54+9=63$. By this, the $n$
persons are at most followed until the age of 113 (see Figures 1 and 5).

Population: Germans born 1900–1954 (Size $76{,}239{,}006$)Left-Truncated:
Persons not surviving 31/12/2003Right-censored: Events $S^{1}$,$D$ or $d$ for
persons surviving 31/12/2013 Figure 1: Data: Trajectory from population to the
left-truncated mass and right-censored persons (Population size: German
Statistical Office (2004), without stillborn)

###  3.1 Mortality

Table 1: Statistics, point estimate (17) and standard error (SE) by (20) for mortality $\\#$ ‘Survivors’ |  $\\#$ ‘Deaths’ | ‘Time at Risk’ (in years) | Point | SE
---|---|---|---|---
$n_{cens}$ | $n_{uncens}$ | $\sum_{i=1}^{n}\mathds{1}_{\\{u_{i}\leq t_{i}<u_{i}+9\\}}(t_{i}-u_{i})$ | $\hat{\lambda}$ | $\sqrt{\mathcal{J}_{\tau}^{-1}(\hat{\lambda})}$
202,407 | 43,472 | 180,163 | 0.0217 | 0.000104

The AOK Health Claims Data (HCD) data include information about age, year of
birth and date of exit (death or switch to another insurance company). For the
preliminary mortality model, analysed in A, $n_{uncens}=43{,}457$ persons out
of $n$ died within the observation period between 2004 and 2013. For the
Exponentially distributed age-at-death, $T$, the point estimate (17) is rather
precisely $\hat{\lambda}=0.022$ (see Table 1) and results in an expected
lifetime of $1/0.022\approx 45$ (plus 50) years. That is apparently too large,
primarily due to the assumption of a constant hazard, and to some extent
because death is impossible before age 50 in this model. We account for the
well-documented age-inhomogeneous increase in the hazard, in demography
usually modelled as a Gompertz distribution, by piecewise constant hazards
shortly. Now we first disentangle the three non-dead states ‘Healthy’,
‘Stroke’ and ‘Dementia’, still age-homogeneously.

###  3.2 Age-Homogeneous Transition of Stroke to Dementia

The model is very similar to ‘Mortality of Diabetics in the County of Fyn’
(see Andersen et al., 1993). However, the age-at-study-begin, $U$, and the
age-at-study-end, $U+9$, share a simpler relation than in (Andersen et al.,
1993, Examples III.3.6, IV.1.7 and VI.1.4). Other than the life-threatening
effect of Diabetics, we investigate the well-known association of stroke with
all-cause dementia. Substantially, on the one hand, cerebrovascular lesions
may affect cognition through several mechanisms such as changes in blood flow
and oxygen supply or chronic inflammation (Hu and Chen, 2017). On the other
hand, dementia and stroke share common risk factors such as hypertension,
diabetes or atrial fibrillations (Pendlebury and Rothwell, 2009). Desmond et
al. (2002) revealed an increased relative risk (RR) of dementia of 3.8 after
stroke, adjusted for several demographic factors and cognitive status, and
within the Framingham Study (Ivan et al., 2002) the adjusted RR of dementia
was 2.0. The Rotterdam Study (Reitz et al., 2008) showed that stroke doubled
the risk of dementia (Hazard ratio: HR=2.1). A systematic review and meta-
analysis revealed a pooled HR of between 1.7 and 2.2 (Kuźma et al., 2018).
Another result, yet without multi-states, is that of Savva and Blossom (2010)
who report a hazard ratio of 2. Based on South Korean health claims data, and
also using multi-state methods, Kim and Lee (2018) found a 2.4-fold risk of
subsequent dementia after stroke.

Here, additional to mortality information, the AOK HCD data contain
information about outpatient and inpatient diagnoses, on a quarterly basis,
for each insured person, with at least one day of insurance coverage,
regardless of whether they sought medical treatment or not. All inpatient and
outpatient diagnoses are coded in the International Statistical Classification
of Diseases and Related Health Problems (ICD), revision 10, issued by the
World Health Organization. Dementia was defined as having at least one of
following diagnoses coded by ICD-10: G30, G31.0, G31.82, G23.1, F00, F01, F02,
F03, and F05.1. Stroke was defined as having at least one diagnosis of I63 or
I64. The data is sketched in Figure 2 for $n_{all}=4$.

$0$$t$$S^{1}$$D$$H$$S^{1}$$D$$d$$H$$D$$H$$D$$X_{t}^{1}$$X_{t}^{2}$$X_{t}^{3}$$X_{t}^{4}$$N_{S^{1}D\bullet}(t)$$\leftindex_{U}{N}^{c}_{S^{1}D\bullet}(t)$$U_{1}$$U_{2}$$U_{3}$$U_{4}$
Figure 2: Four health histories, unobservable
$N_{S^{1}D\bullet}(t):=\sum_{i=1}^{n}N_{S^{1}Di}(t)$ and observed
$\leftindex_{U}{N}^{c}_{S^{1}D\bullet}(t)$ (off-set between
$N_{S^{1}D\bullet}$ and $\leftindex_{U}{N}^{c}_{S^{1}D\bullet}$ for clarity)

With interest solely in the inferential comparison of $\lambda_{HD}$ with
$\lambda_{S^{1}D}$, only their risk sets and counts are needed (see (6)). The
data contains $49{,}880$ persons with onset of dementia in the observation
period. Point estimates (6) and standard errors (roots of (9)) are given in
Table 2.

Table 2: Statistics, point estimates (6) and estimated standard errors (SE, (9)) for age-homogeneous model $\\#$ ‘Demented after Stroke’ | ‘Time after Stroke’ | Point | SE
---|---|---|---
$\leftindex_{U}{n}^{c}_{S^{1}D\bullet}(\tau)$ | $\int_{0}^{\tau}\leftindex_{U}{y}_{S^{1}\bullet}(t)dt$ | $\hat{\lambda}_{S^{1}D}$ | $\sqrt{\mathcal{J}_{\tau}^{-1}(\hat{\lambda}_{S^{1}D})}$
8,105 | 115,566 | 0.0701 | 0.00078
$\\#$ ‘Demented healthy’ | ‘Healthy Times’ | Point | SE
$\leftindex_{U}{n}^{c}_{HD\bullet}(\tau)$ | $\int_{0}^{\tau}\leftindex_{U}{y}_{H\bullet}(t)dt$ | $\hat{\lambda}_{HD}$ | $\sqrt{\mathcal{J}_{\tau}^{-1}(\hat{\lambda}_{HD})}$
41,775 | 1,997,092 | 0.0209 | 0.000102

We find that a stroke increases the intensity of suffering from dementia from
0.02 to 0.07. Due to the asymptotic independence of $\hat{\lambda}_{HD}$ and
$\hat{\lambda}_{S^{1}D}$ by Theorem 1 it is
$Var(\hat{\lambda}_{S^{1}D}-\hat{\lambda}_{HD})=Var(\hat{\lambda}_{S^{1}D})+Var(\hat{\lambda}_{HD})$
and (see Table 2) estimated to be $0.000000617$. Together with the asymptotic
normality of Theorem 1, it is approximately

$\hat{\lambda}_{S^{1}D}-\hat{\lambda}_{HD}\sim
N(\lambda_{S^{1}D0}-\lambda_{HD0},0.000785^{2}).$

So that a $95\%$-confidence interval (CI) of the intensity difference is
approximately $[0.492\pm 0.00154]=[4.8\%,5.1\%]$. The point estimates’ ratio
of 3.5 may exceed recent studies, because we do not adjust for further risk
factors of dementia. As one risk factor we now include age-inhomogeneous
intensities as piecewise constant. We will see that the effect of stroke on
dementia onset is markedly smaller because, simultaneously, intensities
increase with age and stroke is more likely at older ages.

###  3.3 Age-Inhomogeneous Transition from Stroke to Dementia

Figure 3: Age-inhomogeneous intensity from stroke ($S^{1}$) to dementia onset
($D$) $\hat{\lambda}_{S^{1}D}(t)$ (top, grey) and healthy ($H$) to dementia
onset ($D$) $\hat{\lambda}_{HD}(t)$ (bottom, black) on 5-year intervals: Point
estimate ((11), solid line) and 95% confidence interval ((12), dashed line)

In order to account for age-inhomogeneous intensities (also found for the data
at hand in Weißbach et al. (2021)), we here define (as in Weißbach et al.,
2009; Weißbach and Walter, 2010), with partition $0=t_{0},\ldots,t_{b}=\tau$,
$\mathbf{X}$ as Markov process with intensities
$\lambda_{hj}(t):=\sum_{l=1}^{b}\mathds{1}_{]t_{l-1},t_{l}]}(t)\lambda_{hjl}$.
Again, the two counting processes $N_{hj}(t)$ and $Y_{h}(t)$ reformulate a
history. When stacking $\lambda_{hj}(t)$ to $\boldsymbol{\lambda}(t)$ in the
same way as $N_{hj}$ to $\mathbf{N}_{X}$, $\mathbf{N}_{X}$ has a compensator -
with respect to $\mathcal{N}_{t}$ \- with intensity
$\boldsymbol{\alpha}(t):=(Y_{h}(t)\lambda_{hj}(t),hj\in\mathcal{I})^{\prime}$.
The compensator is with respect to the probability measure
$P_{\boldsymbol{\lambda}}$ where
$\boldsymbol{\lambda}:=(\lambda_{HS^{1}1},\ldots,\lambda_{Ddb})^{\prime}$
collects the $6b$ parameters. Hence with little change, as compared to (4):

$\begin{split}\log\leftindex_{U}{L}^{c}(\mathbf{X}|\boldsymbol{\lambda})=&\int_{U}^{U+9}\sum_{hj\in\mathcal{I}}(\log\leftindex_{U}{Y}_{h}(t)+\log\lambda_{hj}(t))d\leftindex_{U}{N}^{c}_{hj}(t)\\\
&-\sum_{hj\in\mathcal{I}}\int_{U}^{U+9}\leftindex_{U}{Y}_{h}(t)\lambda_{hj}(t)dt\end{split}$
(10)

$U$$U+9$$t_{l-1}$$t_{l}$$t_{l-1}$$t_{l}$$t_{l-1}$$t_{l}$$t_{l-1}$$t_{l}$$t_{l-1}$$t_{l}$
Figure 4: Possible intersections of $]t_{l},t_{l-1}]$ and $]U,U+9]$

Now note that there are five possibilities for the intersection of
$]t_{l},t_{l-1}]$ with $]U,U+9]$ (see Figure 4), so that (with $a\lor
b:=\max(a,b)$):

$\int_{U}^{U+9}\frac{\partial}{\partial\lambda_{hjl}}\log\left(\sum_{\tilde{l}=1}^{b}\mathds{1}_{]t_{\tilde{l}-1},t_{\tilde{l}}]}(t)\lambda_{hj\tilde{l}}\right)d\leftindex_{U}{N}^{c}_{hj}(t)\\\
=\int_{U}^{U+9}\frac{\mathds{1}_{]t_{l-1},t_{l}]}(t)}{\sum_{\tilde{l}=1}^{b}\mathds{1}_{]t_{\tilde{l}-1},t_{\tilde{l}}]}(t)\lambda_{hj\tilde{l}}}d\leftindex_{U}{N}^{c}_{hj}(t)\\\
=\int_{U\lor t_{l-1}}^{U+9\land
t_{l}}\frac{1}{\lambda_{hjl}}d\leftindex_{U}{N}^{c}_{hj}(t)\quad\text{if}\quad]t_{l},t_{l-1}]\cap]U,U+9]\neq\emptyset\quad\text{(0
elsewise)}\\\ =\frac{1}{\lambda_{hjl}}[\leftindex_{U}{N}^{c}_{hj}(U+9\land
t_{l})-\leftindex_{U}{N}^{c}_{hj}(U\lor t_{l-1})]$

Note further that

$\frac{\partial}{\partial\lambda_{hjl}}-\sum_{hj\in\mathcal{I}}\int_{U}^{U+9}\leftindex_{U}{Y}_{h}(t)\left(\sum_{\tilde{l}=1}^{b}\mathds{1}_{]t_{\tilde{l}-1},t_{\tilde{l}}]}(t)\lambda_{hj\tilde{l}}\right)dt=-\int_{U}^{U+9}\leftindex_{U}{Y}_{h}(t)\mathds{1}_{]t_{l-1},t_{l}]}(t)dt\\\
=-\int_{U\lor t_{l-1}}^{U+9\land
t_{l}}\leftindex_{U}{Y}_{h}(t)dt\quad\text{if}\quad]t_{l},t_{l-1}]\cap]U,U+9]\neq\emptyset\quad\text{(0
elsewise)}.$

Again by (4) with $\boldsymbol{\lambda}$, and comparable to (5), it is
$\log\leftindex_{U}{L}^{c}(data;\boldsymbol{\lambda})$ the sum of the (10), so
that
$(\partial/\partial\lambda_{hjl})\log\leftindex_{U}{L}^{c}(data|\boldsymbol{\lambda})=A_{l}/\lambda_{hjl}-B_{l}$
by interchanging differentiation and summation, with transitions $A_{l}$ and
time-at-risk $B_{l}$, per age-group, namely:

$\displaystyle A_{l}$ $\displaystyle:=$
$\displaystyle\sum_{\stackrel{{\scriptstyle
i=1}}{{]t_{l},t_{l-1}]\cap]U_{i},U_{i}+9]\neq\emptyset}}}^{n}[\leftindex_{U}{N}^{c}_{hji}(U_{i}+9\land
t_{l})-\leftindex_{U}{N}^{c}_{hji}(U_{i}\lor t_{l-1})]$ $\displaystyle B_{l}$
$\displaystyle:=$ $\displaystyle\sum_{\stackrel{{\scriptstyle
i=1}}{{]t_{l},t_{l-1}]\cap]U_{i},U_{i}+9]\neq\emptyset}}}^{n}\int_{U_{i}\lor
t_{l-1}}^{U_{i}+9\land t_{l}}\leftindex_{U}{Y}_{hi}(t)dt$

Similar to (6), restricted on the time interval $[t_{l},t_{l-1}]$ it is:

$\hat{\lambda}_{hjl}=A_{l}/B_{l}$ (11)

Again the Hessian of the log conditional likelihood is a diagonal matrix with
diagonal elements

$\displaystyle-\mathcal{J}_{\tau}(\boldsymbol{\lambda}_{0})_{hjl,hjl}$
$\displaystyle=$
$\displaystyle\frac{\partial^{2}}{\partial\lambda_{hjl}^{2}}\log\leftindex_{U}{L}^{c}(data|\boldsymbol{\lambda})|_{\boldsymbol{\lambda}=\boldsymbol{\lambda}_{0}}=-\frac{A_{l}}{\lambda_{hjl^{0}}^{2}},$
and hence $\displaystyle\widehat{Var(\hat{\lambda}_{hjl})}=A_{l}/B_{l}^{2}.$
(12)

Table 3: Statistics, point estimates (11) and standard errors (SE, root of (12)) for age-inhomogeneous model Age interval |  $\\#$ ‘Demented after stroke | ‘Time after stroke | Point |  $\\#$ ‘Demented healthy | ‘Healthy times | Point
---|---|---|---|---|---|---
$]t_{l-1}+50,t_{l}+50]$ | in age interval’ | in age interval’ | $\hat{\lambda}_{S^{1}Dl}^{\pm SE}$ | in age interval’ | in age interval’ | $\hat{\lambda}_{HDl}^{\pm SE}$
$]50,55]$ | 13 | 1199 |  0,0108±0.0030 | 157 | 116077 |  0,0014±0.0001
$]55,60]$ | 66 | 5156 |  0,0128±0.0016 | 369 | 245986 |  0,0015±0.0001
$]60,65]$ | 174 | 10369 |  0,0168±0.0013 | 800 | 318543 |  0,0025±0.0001
$]65,70]$ | 387 | 14839 |  0,0261±0.0013 | 1856 | 350043 |  0,0053±0.0001
$]70,75]$ | 815 | 21737 |  0,0375±0.0013 | 4112 | 360343 |  0,0114±0.0002
$]75,80]$ | 1300 | 23318 |  0,0558±0.0015 | 7212 | 291776 |  0,0247±0.0003
$]80,85]$ | 1584 | 18496 | 0,0856±0.0022 | 9151 | 188161 | 0,0486±0.0005
$]85,90]$ | 1138 | 10101 |  0,1127±0.0033 | 7087 | 87106 |  0,0814±0.0010
$]90,95]$ | 489 | 3544 |  0,1380±0.0062 | 3745 | 28899 |  0,1296±0.0021
$]95,100]$ | 122 | 716 |  0,1705±0.0154 | 963 | 5816 |  0,1656±0.0053
$]100,105]$ | 8 | 97 |  0,0827±0.0292 | 92 | 657 |  0,1400±0.0146
$]105,110]$ | 0 | 2 |  0±0 | 4 | 27 |  0,1509±0.0755
$]110,113]$ | 0 | 0 | NA | 0 | 0 | NA

Figure 3 and Table 3 exhibit point estimates, standard errors and confidence
intvervals, using five-year age-groups, i.e. $b=11$. For instance, in the age-
group with the most events, namely from 80 to 85 years, the intensity after
stroke of $\hat{\lambda}_{S^{1}D7}=0.086$ exceeds that without stroke of
$\hat{\lambda}_{HD7}=0.049$ (see framed numbers in Table 3). The ratio of 1.8
is now half of the ratio $0.07/0.02=3.5$ in Table 2, and more in line with the
recent literature. The reason is Simpson’s paradox: In the case for persons
with stroke, the age-homogeneous $\hat{\lambda}_{S^{1}D}=0.07$ of Section 3.2
is implicitly an average over a later part of the time span $[0,\tau]$,
because stoke occurs at higher ages. The age-homogeneous intensity is hence
larger than an average of the age-group intensities over the entire time span
due to the increasing estimates of dementia onset in age. This is not the case
for healthy persons’ intensities.

One may also see a decreasing stroke-effect in age here (Table 3 or Figure 3).
Respective Wald-type tests for pairwise differences (suppressed here) show
that there is no significant difference in the risk of dementia among between
persons with and without stroke for the highest age groups (90 years and
older). This coincides with the Framingham Study (Ivan et al., 2002) where the
adjusted RR was higher for persons aged younger than 80 years old (RR=2.6)
compared to persons at age 80 or older (RR=1.6). Similarly, a systematic
review by Savva and Blossom (2010) also does not find an excess risk of
dementia after stroke in those at ages 85 years or older.

##  4 Conclusion

It is encouraging to see that our model, applied to Germany, has similar point
estimates as the akin study of Kim and Lee (2018), for Korea. That the later
study takes more covariates into account clouds the enjoyment over the
similarity, because those will typically decrease the effect size. Note that
more exogeneous covariates could be integrated algorithmically in our event
history analysis as well using e.g. Kim et al. (2012). However, for reducing
bias, one should probably commence with including left-censored persons, i.e.
with prevalent dementia before 2004 (see Weißbach et al., 2021). However, our
thrust is more about precision and therefore takes truncation into account.
Our precision claim is holstered by assuming three sorts of independences.
First of all, within pairs $(T_{i},U_{i})$, and tests already exist whether
censoring (and with it $U$) is independent of the primary measure (see e.g.
Sun and Lee, 2011). Second, we assume independence between pairs
$(T_{i},U_{i})$, and generalizing to the conditions (7) and (8) would be an
ineffective improvement. In more detail, on the one hand we assume
independence between persons in the latent sample (of size $n_{all}$) what can
be defended in survival analysis with the argument of unforeseeable, staggered
entry. And if it holds, selection of a person into the observed sample (of
size $n$) relies only on a person’s information alone, so that - in contrast
to, say, selecting the obviously dependent largest and smallest value - we are
confident that the observations are also independent. On the other hand, it
would still sufficient to assume pairs $(T_{i},U_{i})$ to be independent
conditionally on $A_{i}$, with yet no reason. Third, close to longitudinal
independence, probably the most critical assumption seems to be the Markovian.
Especially as for our application Pendlebury and Rothwell (2009) and Corraini
et al. (2017) claim that the time elapsed since stroke is a risk factor to the
intensity of dementia onset. Such duration-dependence especially violates the
assumption of multiplicative intensities (8) and needs a different approach
(see e.g. Weißbach and Schmal, 2019).

Acknowledgment: The financial support from the Deutsche Forschungsgemeinschaft
(DFG) of R. Weißbach and G. Doblhammer is gratefully acknowledged (Grant
386913674 ‘Multi-state, multi-time, multi-level analysis of health-related
demographic events: Statistical aspects and applications’). For discussions at
an earlier stage of the study we are grateful to O. Gefeller. For the support
with data we thank the AOK Research Institute (WIdO). The linguistic and
idiomatic advice of Brian Bloch is also gratefully acknowledged.

## References

  * Andersen et al. (1988) P. Andersen, Ø. Borgan, R. Gill, and N. Keiding.  Censoring, truncation and filtering in statistical models based on counting processes.  In N. U. Prabhu, editor, _Statistical inference from stochastic processes_ , volume 80, pages 19–60. 1988.
  * Andersen et al. (1993) P. Andersen, Ø. Borgan, R. Gill, and N. Keiding.  _Statistical Models Based on Counting Processes_.  Springer, New York, 1993.
  * Corraini et al. (2017) P. Corraini, V.W. Henderson, A.G. Ording, L. Pedersen, E. Horváth-Puhó, and H.T. Sørensen.  Long-term risk of dementia among survivors of ischemic or hemorrhagic stroke.  _Stroke_ , 48:180–186, 2017.
  * Desmond et al. (2002) D.W. Desmond, J.T. Moroney, M. Sano, and Y. Stern.  Incidence of dementia after ischemic stroke.  _Stroke_ , 33:2254–2262, 2002.
  * Fleming and Harrington (1991) T. R. Fleming and D.P. Harrington.  _Counting Processes and Survival Analysis_.  Wiley, Hoboken, 1991.
  * Hbid et al. (2020) Y. Hbid, K. Mohamed, Ch.D.A. Wolfe, and A. Douiri.  Inverse problem approach to regularized regression models with application to predicting recovery after stroke.  _Biometrical Journal_ , 62:1926–1938, 2020.
  * Hu and Chen (2017) G.C. Hu and Y.M. Chen.  Post-stroke dementia: epidemiology, mechanisms and management.  _International Journal of Gerontology_ , 11:210–214, 2017\.
  * Ivan et al. (2002) C.S. Ivan, S. Seshadri, A. Beiser, R. Au, C.S. Kase, M. Kelly-Hayes, and P.A. Wolf.  Dementia after stroke: The Framingham study.  _Stroke_ , 35:1264–1268, 2002.
  * Kim and Lee (2018) J.H. Kim and Y. Lee.  Dementia and death after stroke in older adults during a 10-year follow-up: Results from a competing risk model.  _Journal of Nutrition, Health and Aging_ , 22:297–301, 2018\.
  * Kim et al. (2012) Y. Kim, L. James, and R. Weißbach.  Bayesian analysis of multi-state event history data: Beta-dirichlet process prior.  _Biometrika_ , 99:127–140, 2012.
  * Kremer et al. (2014) A. Kremer, R. Weißbach, and F. Liese.  Maximum likelihood estimation for left- and right-censored survival times with time-dependent covariates.  _Journal of Statistical Planning and Inference_ , 149:33–45, 2014.
  * Kuźma et al. (2018) E. Kuźma, I. Lourida, S.F. Moore, D.A. Levine abd O.C. Ukoumunne, and D.J. Llewellyn.  Stroke and dementia risk: A systematic review and meta-analysis.  _Alzheimer’s & Dementia_, 14:1416–1426, 2018.
  * Pendlebury and Rothwell (2009) S.T. Pendlebury and P.M. Rothwell.  Prevalence, incidence, and factors associated with pre-stroke and post-stroke dementia: A systematic review and meta-analysis.  _The Lancet Neurology_ , 8:1006–1018, 2009.
  * Reitz et al. (2008) C. Reitz, M.J. Bos, A. Hofman, P.J. Koudstaal, and M.M. Breteler.  Prestroke cognitive performance, incident stroke, and risk of dementia: the Rotterdam study.  _Stroke_ , 39:36–41, 2008.
  * Savva and Blossom (2010) G.M. Savva and S.C.M. Blossom.  Epidemiological studies of the effect of stroke on incident dementia: A systematic review.  _Stroke_ , 41:e41–e46, 2010.
  * Sun and Lee (2011) Y. Sun and J. Lee.  Testing independent censoring for longitudinal data.  _Statistica Sinica_ , 21:1315–1339, 2011.
  * Weißbach and Radloff (2020) R. Weißbach and L. Radloff.  Consistency for the negative binomial regression with fixed covariate.  _Metrika_ , 83:627–641, 2020.
  * Weißbach and Schmal (2019) R. Weißbach and F. Schmal.  Estimating the probability of a non-markovian rating transition from partially unobserved histories.  _Journal of Risk Management in Financial Institutions_ , 12:256–267, 2019.
  * Weißbach and Walter (2010) R. Weißbach and R. Walter.  A likelihood ratio test for stationarity of rating transitions.  _Journal of Econometrics_ , 155:188–194, 2010.
  * Weißbach et al. (2009) R. Weißbach, P. Tschiersch, and C. Lawrenz.  Testing time-homogeneity of rating transitions after origination of debt.  _Empirical Economics_ , 36:575–596, 2009.
  * Weißbach et al. (2021) R. Weißbach, Y. Kim, A. Dörre, A. Fink, and G. Doblhammer.  Left-censored dementia incidences in estimating cohort effects.  _Lifetime Data Analysis_ , 27:38–63, 2021.

##  Appendix A Filtrations in Mortality Model

###  A.1 Filtration in population model

The population are Germans born between 1900 and 1954, and we assume insurance
by Germany’s largest health insurer AOK to be representative. We consider a
simple sample thereof of size $n_{all}$. The lifetime $T$ after the 50th
birthday, called ‘age-at-death’ in the following, has hazard rate
$\lambda(\cdot)\equiv\lambda$ and CDF $F(\cdot)$. For a person randomly drawn
from the population, the age-at-death $T$ is defined on the probability space
$(\Omega,\mathcal{F},P_{\lambda})$.

  1. (A0)

It is for the true parameter
$\lambda_{0}\in\Lambda:=[\varepsilon;1/\varepsilon]$ for some small
$\varepsilon\in]0;1[$.

In terms of a process, $N_{T}(t):=\mathds{1}_{\\{T\leq t\\}}$ is adapted to
$\mathcal{N}_{t}:=\sigma\\{\mathds{1}_{\\{T\leq s\\}},0\leq s\leq t\\}$.
$N_{T}$ has compensator with intensity

$\alpha(t):=\mathds{1}_{\\{N_{T}(t-)=0\\}}\lambda=Y(t)\lambda,$

with respect to $\mathcal{N}_{t}$ and $P_{\lambda}$ and with
$Y(t):=\mathds{1}_{\\{T\geq t\\}}$.

The population model is further described by acknowledging that $T$ is
accompanied by the time past since the age of 50 at study begin, $U$ (if
random independent of $T$, or $u$ if not, and denoted ‘age at study begin’).

###  A.2 Non-random left-truncation and right-censoring (Figure 5)

1/1/20041/1/20131/1/1950$\overbrace{{\color[rgb]{0.93,0.93,0.93}------------------------}}$$T=34$$\overbrace{{\color[rgb]{0.93,0.93,0.93}-----------------------------}}$$T=50$$\underbrace{{\color[rgb]{0.93,0.93,0.93}----------------------------}}$$u+9=48$\-
$\overbrace{{\color[rgb]{0.93,0.93,0.93}------------------------}}$$T=35$$\underbrace{{\color[rgb]{0.93,0.93,0.93}--------------------------}}$$u=39$
Figure 5:  (top: uncensored/untruncated) Path for person born 1/1/1925 with
death 1/1/2009, i.e. with $T=84-50$, $u\leq T\leq u+9$
(middle: left-truncated) Path for person born 1/1/1915 with death 1/1/2000,
i.e. with $T=85-50$, $T<u$, $u=89-50$
(bottom: right-censored) Path for person born 1/1/1915 with death 1/1/2015,
i.e. with $T=100-50$, $T\geq u+9$, $u+9=98-50$

In our study we begin to observe 01/01/2004 and by left-truncation,
information deteriorates even more than by left-censoring, where only the age
at an event gets lost for concerned persons (see e.g. Kremer et al., 2014;
Weißbach et al., 2021). Here also the age-at-censoring gets lost, and even the
existence of a truncated person,

As a preparation (and as in Andersen et al., 1993, Example III.3.6), we start
with deterministic $u$. Increasing the filtration (as will be necessary for
random truncation) is not necessary for deterministic (and hence universally
known) $u$. The (not) left-truncation event is $A:=\\{T>u\\}$. We assume
$P_{\lambda}(A)>0$. Now the observable filtration is

$\leftindex_{u}{\mathcal{G}}_{t}:=\sigma\\{\mathds{1}_{\\{u\leq T\leq
s\\}},u\leq s\leq t\\}=\sigma\\{\leftindex_{u}{N}_{T}(s),u\leq s\leq t\\}$

with $\leftindex_{u}{N}_{T}(t):=N_{T}(t)-N_{T}(t\land u)$ (where $t\land
u:=\min(t,u)$), i.e. $\leftindex_{u}{N}_{T}$ is $N_{T}$ if $u\leq T$ and zero
elsewise. The intensity process of $\leftindex_{u}{N}_{T}(t)$ is
$\leftindex_{u}{\alpha}(t):=\mathds{1}_{\\{u<t\leq T\\}}\lambda$, with respect
to the probability measure

$P^{A}_{\lambda}(F):=\frac{P_{\lambda}(F\cap
A)}{P_{\lambda}(A)}\quad\text{for}\quad F\in\mathcal{F}.$

The deterministic $u$ is a special case of the random $U$, the Example 4.1 in
Andersen et al. (1988).

Right-censored is the age-at-death if it occurs after 2012, or after having
left the AOK (see Figure 1). As in (Andersen et al., 1993, Examples III.3.6,
IV.1.7 and VI.1.4) we superimpose right-censoring on left-truncation (and not
vice versa). The observed left-truncated and right-censored counting process
is

$\leftindex_{u}{N}^{c}_{T}(t):=\int_{0}^{t}C(s)d\leftindex_{u}{N}_{T}(s)=\mathds{1}_{\\{u\leq
T\leq\min(t,u+9)\\}}$

with $C(t):=\mathds{1}_{\\{t\leq u+9\\}}$ (compare Fleming and Harrington,
1991, Example 1.4.2). It has intensity
$\leftindex_{u}{\alpha}_{T}^{c}(t):=\lambda\mathds{1}_{\\{u\leq
t\leq\min(T,u+9)\\}}$ with respect to $P^{A}_{\lambda}$ and observed
filtration

$\leftindex_{u}{\mathcal{F}}^{c}_{t}:=\sigma\\{\leftindex_{u}{N}^{c}_{T}(s),u\leq
s\leq t\\}=\sigma\\{\mathds{1}_{\\{u\leq T\leq\min(s,u+9)\\}},u\leq s\leq
t\\}.$

The $\leftindex_{u}{\mathcal{F}}^{c}_{t}$ is self-exciting (see Andersen et
al., 1988, p4), as required (see Andersen et al., 1988, p23), so that the
likelihood is by Jacod’s formula (see Andersen et al., 1988, Formula (4.3)),
(see also Andersen et al., 1993, Formula (2.7.2”) (and in extension 3.2.8)):

$\displaystyle dP$ $\displaystyle=$ $\displaystyle\Prodi_{u<t\leq
u+9}\\{(1-\leftindex_{u}{\alpha}_{T}^{c}(t)dt)^{1-d\leftindex_{u}{N}_{T}^{c}(t)}(\leftindex_{u}{\alpha}_{T}^{c}(t)dt)^{d\leftindex_{u}{N}_{T}^{c}(t)}\\}$
(13) $\displaystyle=$ $\displaystyle\begin{cases}1^{1}\cdot
0^{0}=1&\text{for}\;T<u\\\ \lambda(T)e^{-\int_{u}^{T}\lambda(t)dt}=\lambda
e^{-\lambda(T-u)}&\text{for}\;u\leq T\leq u+9\\\
e^{-\int_{u}^{u+9}\lambda(s)ds}=e^{-9\lambda}&\text{for}\;u+9<T\\\
\end{cases}$

For explanation of the first and third case see page 24 and, respectively,
Example 2.2 in Andersen et al. (1988). Note that in the second case
$d\leftindex_{u}{N}_{T}^{c}(t)\equiv 0$ and $\mathds{1}_{\\{u\leq
T\leq\min(t,u+9)\\}}\equiv 0$.

Independent persons are, due to different $u_{i}$, not identical distributed.
Each likelihood is a Radon-Nikodym derivative with respect to a different
measure, all of those are even dependent on the parameter,
$P_{\lambda}^{A_{i}}$. Even worse, measures are conditional on observation.
The usual formula of introductory statistics that the density for a collection
of independent persons is the product of the persons’ densities relies on the
equal (and parameter-independent) dominating measure (usually being
Lebesgues). Unconditionally, such fixed covariate regression could be analysed
as in Weißbach and Radloff (2020). To achieve equal dominating measures for
all persons, we follow Examples IV.1.7 and VI.1.4 of Andersen et al. (1993) in
using a random $U$. (In the end, we will draw a simple sample of size
$n_{all}$ from the population and begin our study on 1/1/2004.)

###  A.3 Random left-truncation and conditional right-censoring

The probability space for $(T,U)$ is
$(\Omega,\mathcal{F},\tilde{P}_{\lambda})$, where the distribution of $U$ will
not be important and we suppress its parameter (and indicate the difference by
tilde instead).

  1. (A0)

$U$ and $T$ are independent and
$\beta_{\lambda_{0}}:=\tilde{P}_{\lambda}(A)>0$ with redefined $A:=\\{T>U\\}$.

####  A.3.1 Left-truncation for the age-at-death (Figure 6)

Ignoring censoring for the moment, $T$ is recorded when being larger than $U$,
the age-at-study-begin (see Figure 6).

1/1/19101/1/19201/1/19301/1/19401/1/19501/1/19551/1/20041/1/1900$\overbrace{{\color[rgb]{0.93,0.93,0.93}------------------------}}$$T+50=84$$\overbrace{{\color[rgb]{0.93,0.93,0.93}------------------------}}$$T+50=85$$\underbrace{{\color[rgb]{0.93,0.93,0.93}--------------------------}}$$U+50=89$\-
Figure 6: (top: left-censored/truncated) Path for person born 1/1/1915 with
death 1/1/2000, i.e. with $T=85-50=35$, $T<U$, $U=89-50=39$
(bottom: uncensored/truncated) Path for person born 1/1/1925 with death
1/1/2009, i.e. with $T=84-50=34$, $T\geq U$

Consider the unobservable filtration

$\mathcal{G}_{t}:=\sigma\\{\mathds{1}_{\\{T\leq s\\}},\mathds{1}_{\\{U\leq
s\\}},0\leq s\leq t\\}.$

For left-truncation (see Andersen et al., 1993, Example III.3.2), similar to
non-random left-truncation, define
$\leftindex_{U}{N}_{T}(t):=N_{T}(t)-N_{T}(t\land U)$ with
$\mathcal{G}_{t}$-intensity, with respect to $\tilde{P}_{\lambda}$, again
being $\lambda\mathds{1}_{\\{T\geq t\\}}$ (because $U$ is independent of $T$).
We observe durations $U$ and $T$ in the case of $A$, and neither measurement
$U$ nor $T$ \- nor the person at all - when $T<U$. Now define
$\leftindex_{U}{Y}(t):=Y(t)\mathds{1}_{\\{t>U\\}}=\mathds{1}_{\\{T\geq
t>U\\}}$ and as $\mathcal{G}_{t}$ is unobservable, but due to $U$ being a
$\mathcal{G}_{t}$-stopping time, observable is
$\leftindex_{U}{\mathcal{F}}_{t}:=\\{\leftindex_{U}{N}_{T}(s),\leftindex_{U}{Y}(s),U\leq
s\leq t\\}$. The intensity process of $\leftindex_{U}{N}_{T}(t)$ is
$\leftindex_{U}{\alpha}_{T}(t)=\mathds{1}_{\\{U<t\leq T\\}}\lambda$, with
respect to $\leftindex_{U}{\mathcal{F}}_{t}$ and the probability measure

$\tilde{P}^{A}_{\lambda}(F):=\frac{\tilde{P}_{\lambda}(F\cap
A)}{\tilde{P}_{\lambda}(A)}\quad\text{for}\quad F\in\mathcal{F}.$

Intuitively, $\leftindex_{U}{N}_{T}$ will not yet jump prior to $U$, and not
anymore after $T$, and in between at the hazard rate of $T$. Now as
$\leftindex_{U}{\mathcal{F}}_{t}$ is self-exciting (see Andersen et al., 1988,
p4), we may apply Jacod’s formula (see Andersen et al., 1988, Formula (2.1))
in order to determine the conditional likelihood (see Andersen et al., 1988,
Formula (4.3)). And due to $U$ being independent of $T$ it is (see also
Andersen et al., 1993, Formula (3.3.3)), as the hazard rate is
$\lambda(\cdot)\equiv\lambda$, the conditional likelihood for $T\leq U$ one
and else:

$\displaystyle\leftindex_{U}{L}(\lambda)$ $\displaystyle=$
$\displaystyle\Prodi_{t>U}\\{\leftindex_{U}{\alpha}_{T}(t)^{\Delta\leftindex_{U}{N}_{T}(t)}(1-\leftindex_{U}{\alpha}_{T}(t)dt)^{1-\Delta\leftindex_{U}{N}_{T}(t)}\\}$
(14) $\displaystyle=$
$\displaystyle\lambda(T)e^{-\int_{U}^{T}\lambda(t)dt}=\frac{[1-F(T)]\lambda(T)}{1-F(U)}=\lambda
e^{-\lambda(T-U)}$

####  A.3.2 Acknowledging non-random censoring

First note that now $T>U+9$ is known, conditionally on $A$, because $U$ is
observable, rendering unnecessary a further increase of the filtration. Hence,
similar to A.2, we apply Definition III.2.1 in Andersen et al. (1993), with
$\leftindex_{U}{N}_{T}$ in the role of $N$,
$(\leftindex_{U}{\mathcal{F}}_{t})$ in the role of $(\mathcal{F}_{t})$,
$\leftindex_{U}{\alpha}_{T}$ in the role of $\lambda^{\theta}$ and
$\tilde{P}^{A}_{\lambda}$ in the role of $P_{\theta\psi}$. Being conditionally
deterministic, $C(t):=\mathds{1}_{\\{t\leq U+9\\}}$ is independent and
predictable. We define
$\leftindex_{U}{Y}^{c}(t):=C(t)\leftindex_{U}{Y}(t)=\mathds{1}_{\\{U\leq
t<\min(T,U+9)\\}}$ so that

$\leftindex_{U}{N}_{T}^{c}(t):=\int_{U}^{t}C(s)\leftindex_{U}{N}_{T}(s)=\mathds{1}_{\\{U\leq
T\leq\min(t,U+9)\\}}$

has intensity
$\leftindex_{U}{\alpha}_{T}^{c}(t)=\mathds{1}_{\\{U<t\leq\min(T,U+9)\\}}\lambda$,
with respect to the observable filtration

$\leftindex_{U}{\mathcal{F}}^{c}_{t}:=\\{\leftindex_{U}{N}^{c}_{T}(s),\leftindex_{U}{Y}^{c}(s),U\leq
s\leq t\\}$

and conditional distribution $\tilde{P}^{A}_{\lambda}$ (see Andersen et al.,
1993, Section III.2.2). By Formula (3.2.8) in Andersen et al. (1993), the
partial (and here only conditional) likelihood is for $T\leq U$ again one and
else:

$\displaystyle\leftindex_{U}{L}_{\tau}^{c}(\lambda)$ $\displaystyle=$
$\displaystyle\Prodi_{t>U}^{\tau}\\{(\leftindex_{U}{\alpha}^{c}_{T}(t))^{\Delta\leftindex_{U}{N}^{c}_{T}(t)}(1-\leftindex_{U}{\alpha}^{c}_{T}(t)dt)^{1-\Delta\leftindex_{U}{N}^{c}_{T}(t)}\\}$
(15) $\displaystyle=$
$\displaystyle\begin{cases}\lambda(T)e^{-\int_{U}^{T}\lambda(t)dt}=\lambda
e^{-\lambda(T-U)}&\text{for}\;U\leq T\leq U+9<\tau\\\
e^{-\int_{U}^{U+9}\lambda(s)ds}=e^{-9\lambda}&\text{for}\;U+9<T<\tau\\\
\end{cases}$

###  A.4 Sample result

We switch now from one person to the simple sample, of unobserved $n_{all}$.
Now the size of the observed sample persons surviving $31/12/2003$ is
$n:=\sum_{i=1}^{n_{all}}\mathds{1}_{A_{i}}$ where $i$ enumerates the persons
(see Figure 1). Again, without loss of generality, we sorted those not
truncated at the beginning of the sample, all other summands are zero, Due to
the assumption of a simple sample (and hence same dominating measure), the log
conditional likelihood for the data is the sum of the log conditional
likelihood contributions for each observed person. With
$N_{cens}:=\sum_{i=1}^{n}\mathds{1}_{\\{T_{i}>U_{i}+9\\}}$ denoting the number
of right-censored and
$N_{uncens}:=n-N_{cens}=\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}<T_{i}\leq
U_{i}+9\\}}$ the number of neither truncated nor censored,

$\begin{split}\log\leftindex_{U}{\mathbf{L}}_{\tau}^{c}(\lambda)&=\sum_{i=1}^{n}\log\leftindex_{U}{L}_{\tau}^{c}(\lambda)_{i}=N_{uncens}\log(\lambda)\\\
&-\lambda\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}\leq T_{i}\leq
U_{i}+9\\}}(T_{i}-U_{i})-9\lambda N_{cens},\end{split}$ (16)

so that the unique estimator becomes

$\hat{\lambda}=argmax_{\lambda\in\Lambda}\log\leftindex_{U}{\mathbf{L}}_{\tau}^{c}(\lambda)=\frac{N_{uncens}}{\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}\leq
T_{i}<U_{i}+9\\}}(T_{i}-U_{i})+9N_{cens}}.$ (17)

Note that due to the simple sample, among those who survive $U_{i}$ (i.e.
2004), for portion in the study period alive at the age $t$ in observed
sample, it is, with

$\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}<t\leq\min(T_{i},U_{i}+9)\\}}\stackrel{{\scriptstyle
P}}{{\longrightarrow}}\tilde{P}_{\lambda}(U<t\leq\min(T,U+9)|A),$ (18)

the latter is that portion the population, and analogue to $m_{h}(t)$. Again
for our parametric model we only assume:

  1. (A0)

$\int_{0}^{\tau}\tilde{P}_{\lambda}(U<t\leq\min(T,U+9)|A)dt>0$

###### Lemma 2

Under Conditions (A0)-(A0) it is $\hat{\lambda}$ of (17) consistent and

$\sqrt{n_{all}}(\hat{\lambda}-\lambda_{0})\stackrel{{\scriptstyle\mathcal{D}}}{{\longrightarrow}}\mathcal{N}(0,\sigma^{-1}(\lambda_{0}))$

with

$\sigma(\lambda_{0}):=\frac{\beta_{\lambda_{0}}}{\lambda_{0}}\int_{0}^{\tau}\tilde{P}_{\lambda}(U<t\leq\min(T,U+9)|T>U)dt.$

Proof: Due to the uniqueness of $\hat{\lambda}$, for both consistency and
asymptotic normality we need to verify Conditions (A)-(E) (see Andersen et
al., 1993, Theorems VI.1.1+2). In order to map the notations, note that $n$
becomes $n_{all}$, $a_{n}:=\sqrt{n_{all}}$, $\boldsymbol{\theta}$ becomes
$\lambda$ and $h$ is redundant. The $\lambda(t;\theta)$ becomes, by
interchanging conditional expectations with summation, the multiplicative
intensity process of $\leftindex_{U}{N}^{c}_{T\bullet}(t)$:

$\leftindex_{U}{\alpha}^{c}_{\bullet}(t;\lambda)=\lambda\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}<t\leq\min(T_{i},U_{i}+9)\\}}$

By (A0), (A) is fulfilled for intensity $\leftindex_{U}{\alpha}_{T}^{c}(t)$
and therefore for $\leftindex_{U}{\alpha}_{T\bullet}^{c}(t;\lambda)$ and the
logarithm of likelihood (15). For (B), because by the LLN, for fixed
$t\in[0,\tau]$, (18) and $n/n_{all}\stackrel{{\scriptstyle
P}}{{\longrightarrow}}\tilde{P}_{\lambda}(A)=\beta_{\lambda_{0}}$, so that
Slutzky’s Lemma and the CMT yield

$\frac{1}{n_{all}}\int_{0}^{\tau}\left(\frac{d}{d\lambda}(\log\lambda)\right)^{2}|_{\lambda=\lambda_{0}}\lambda_{0}\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}<t\leq\min(T_{i},U_{i}+9)\\}}dt\\\
=\frac{n}{n_{all}}\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\frac{1}{\lambda_{0}^{2}}\lambda_{0}\mathds{1}_{\\{U_{i}<t\leq\min(T_{i},U_{i}+9)\\}}dt\rightarrow\sigma(\lambda_{0}).$

For (C), because (i) $n\leq n_{all}$, (ii)
$1/(n\lambda_{0})\stackrel{{\scriptstyle P}}{{\rightarrow}}0$ by (A0) and
$n\stackrel{{\scriptstyle P}}{{\rightarrow}}\infty$ (due to $n$ being
distributed as Binomial with parameters $n_{all}$ and the selection
probability $\beta_{\lambda_{0}}$) and (iii) (A0), we have

$\frac{n}{n_{all}}\frac{1}{n}\int_{0}^{\tau}\left(\frac{d}{d\lambda}(\log\lambda)|_{\lambda=\lambda_{0}}\right)^{2}\mathds{1}_{\\{|\frac{1}{n}\frac{d}{d\lambda}(\log\lambda)|_{\lambda=\lambda_{0}}|>\varepsilon\\}}\lambda_{0}\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}<t\leq\min(T_{i},U_{i}+9)\\}}dt\\\
=\frac{n}{n_{all}}\frac{1}{\lambda_{0}^{2}}\mathds{1}_{\\{|1/(n\lambda_{0})|>\varepsilon\\}}\lambda_{0}\int_{0}^{\tau}\frac{1}{n}\mathds{1}_{\\{U_{i}<t\leq\min(T_{i},U_{i}+9)\\}}dt\stackrel{{\scriptstyle
P}}{{\rightarrow}}0.$ (19)

For (D), by (A0), it is $\sigma(\lambda_{0})$ positive. Condition (E) consists
of six conditions: For the first four note that

$\sup_{\lambda}\left|\frac{\partial^{3}}{d\lambda^{3}}\left(\lambda\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}<t\leq\min(T_{i},U_{i}+9)\\}}\right)\right|=0,\\\
sup_{\lambda}\left|\frac{\partial^{3}}{d\lambda^{3}}\log(\lambda)\right|=sup_{\lambda}|\lambda^{-3}/2|=:H<\infty\quad\text{and}\\\
\frac{1}{n_{all}}\int_{0}^{\tau}Hdt\rightarrow 0.$

For the fifth, note (19) (and also (A0)). For the sixth, note (19) and then
$H/\sqrt{n_{all}}\stackrel{{\scriptstyle P}}{{\rightarrow}}0$.

It remains to consistently estimate $\sigma(\lambda_{0})$, in order to
construct a confidence interval:

$\displaystyle-\mathcal{J}_{\tau}(\lambda_{0})$ $\displaystyle:=$
$\displaystyle\frac{d^{2}}{d\lambda^{2}}\log\leftindex_{U}{\mathbf{L}}_{\tau}^{c}(\lambda)|_{\lambda=\lambda_{0}}\stackrel{{\scriptstyle\text{\eqref{loglikemodela}}}}{{=}}-\frac{N_{uncens}}{\lambda_{0}^{2}}$
$\displaystyle\frac{1}{n_{all}}\mathcal{J}_{\tau}(\lambda_{0})$
$\displaystyle\stackrel{{\scriptstyle n_{all}\to\infty,P}}{{\longrightarrow}}$
$\displaystyle\sigma(\lambda_{0})\quad\text{\cite[cite]{(\@@bibref{AuthorsPhrase1Year}{And}{\@@citephrase{,
}}{}, Formula (6.1.11))}}$
$\displaystyle\Rightarrow\mathcal{J}_{\tau}^{-1}(\lambda_{0})$
$\displaystyle=$
$\displaystyle\frac{\lambda_{0}^{2}}{N_{uncens}}\quad\text{and}\quad
n_{all}\mathcal{J}_{\tau}^{-1}(\lambda_{0})\stackrel{{\scriptstyle\cdot}}{{=}}\sigma^{-1}(\lambda_{0})$
$\displaystyle\Rightarrow Var(\hat{\lambda})$
$\displaystyle\stackrel{{\scriptstyle\cdot}}{{=}}$
$\displaystyle\mathcal{J}_{\tau}^{-1}(\hat{\lambda})\quad\text{(by
\eqref{expest} and CMT as $\hat{\lambda}$ is consistent)}$ (20)
$\displaystyle=$
$\displaystyle\frac{N_{uncens}}{\left(\sum_{i=1}^{n}\mathds{1}_{\\{U_{i}\leq
T_{i}<U_{i}+9\\}}(T_{i}-U_{i})+9N_{cens}\right)^{2}}$

Note that by Lemma 2,
$Var(\hat{\lambda})=Var(\sqrt{n_{all}}\hat{\lambda})/n_{all}\stackrel{{\scriptstyle\cdot}}{{=}}\sigma^{-1}(\lambda_{0})/n_{all}$
depends on the unobservable $n_{all}$. However, the estimated standard error
of $\hat{\lambda}$ (SE), as square root of (20), is observable.


# A refinement of the Šidák-Khatri inequality and a strong Gaussian
correlation conjecture

Rotem Assouline, Arnon Chor, and Shay Sadovsky

###### Abstract

We prove two special cases of a strengthened Gaussian correlation conjecture,
due to Tehranchi, and show that if the conjecture holds asymptotically, it
holds for any dimension. Additionally, we use these special cases to prove a
refined version of the Šidák-Khatri inequality.

##  1 Introduction

Over 50 years ago, Šidák [16, 17] and independently, Khatri [10], proved the
following inequality for $X_{1},\dots,X_{n}$ symmetric Gaussian random
variables:

$\Pr(|X_{i}|\leq 1\ \forall i\in[n])\geq\prod_{i=1}^{n}\Pr(|X_{i}|\leq 1).$

Our first theorem is a refinement of this result.

######  Theorem 1 (Refined Šidák-Khatri inequality).

Let $X_{1},\dots,X_{n}$ be zero-mean jointly Gaussian real random variables,
and let $a_{1}\in(0,\infty]$. Then,

$\displaystyle\begin{split}&\Pr(|X_{1}|\leq 1+a_{1})\Pr(|X_{i}|\leq 1\ \forall
i\in[n])\\\ &\quad\geq\Pr(|X_{1}|\leq 1)\Pr(|X_{1}|\leq 1+a_{1},|X_{2}|\leq
1,\dots,|X_{n}|\leq 1).\end{split}$ (1)

In particular, we find that

$\dfrac{\Pr(|X_{i}|\leq 1\ \forall i\in[n])}{\prod_{i=1}^{n}\Pr(|X_{i}|\leq
1)}\geq\sup_{a_{1},\dots,a_{n}\in(0,\infty)}\dfrac{\Pr(|X_{i}|\leq 1+a_{i}\
\forall i\in[n])}{\prod_{i=1}^{n}\Pr(|X_{i}|\leq 1+a_{i})}.$ (2)

Clearly, when $a_{i}=\infty$ for all $i\in[n]$, the inequality becomes the
Šidák-Kahtri inequality. In addition, renormalizing, we find that

$\dfrac{\Pr(|X_{i}|\leq c_{i}\ \forall
i\in[n])}{\prod_{i=1}^{n}\Pr(|X_{i}|\leq c_{i})}$

is increasing each of the $c_{i}$’s (see Remark 10).

The Gaussian Correlation inequality, which is a generalization of the Šidák-
Khatri inequality, was proved by Royen in [14] and states the following:

$\Pr(|X_{i}|\leq 1\ \forall i\in[n])\geq\Pr(|X_{i}|\leq 1\ \forall
i\in[k])\Pr(|X_{i}|\leq 1\ \forall i\in[n]\setminus[k]).$ (3)

In attempt to generalize Theorem 1, we present the following conjecture, which
is equivalent to a conjecture of Teheranchi [19] (stated below).

######  Conjecture 2.

Let $X_{1},\dots,X_{n}$ be zero-mean jointly Gaussian real random variables,
and let $s_{1},\dots,s_{n},t_{1},\dots,t_{n}\in[0,\infty)\cup\\{\infty\\}$.
Then,

$\displaystyle\Pr(|X_{i}|\leq s_{i}+t_{i}\ \forall i\in[n])$
$\displaystyle\Pr(|X_{i}|\leq\min\\{s_{i},t_{i}\\}\ \forall i\in[n])$ (4)
$\displaystyle\geq\Pr(|X_{i}|\leq s_{i}\ \forall i\in[n])$
$\displaystyle\Pr(|X_{i}|\leq t_{i}\ \forall i\in[n]).$

Our proof relies on the geometric interpretation of the inequality with
respect to the usual Gaussian probability measure in $\mathbb{R}^{d}$, denoted
henceforth as $\gamma_{d}$. The classical Šidák-Khatri inequality can be
interperted as the inequality $\gamma_{d}(K\cap
T)\geq\gamma_{d}(K)\gamma_{d}(T)$ when $K$ is an origin-symmetric convex set
and $T$ is a symmetric slab. In this geometric setting, this is due to
Giannopoulos [6] and Szarek and Werner [18].

Theorem 1 is restated and proved in the geometric setting as follows.

######  Theorem 3.

Let $K\subseteq\mathbb{R}^{d}$ be an origin-symmetric convex body and
$T=\\{x:\ |\langle x,u\rangle|\leq 1\\}\subset\mathbb{R}^{d}$ a symmetric slab
with $u\in\mathbb{R}^{d}$. Then,

$\gamma_{d}({\rm conv}\\{K\cup T\\})\gamma_{d}(K\cap
T)\geq\gamma_{d}(K)\gamma_{d}(T).$ (5)

Equation (5) does not hold for all origin symmetric and convex
$K,T\subseteq\mathbb{R}^{d}$, as will be seen in Example 14. However, the
following weaker inequality, which is a strengthening of Royen’s Gaussian
correlation inequality, was conjectured to be true by Tehranchi [19].

######  Conjecture 4.

Let $d>0$ and let $K,T\subseteq\mathbb{R}^{d}$ be origin-symmetric convex
sets, then

$\gamma_{d}(K+T)\gamma_{d}(K\cap T)\geq\gamma_{d}(K)\gamma_{d}(T)$ (6)

where $\gamma$ is the usual Gaussian probability measure.

In [19], Tehranchi showed that Conjecture 4 holds up to some constant,
improving on a result by Schechetman, Schlumprecht, and Zinn [15] (these
results and other related works are elaborated on in the Background section).

The proof of the equivalence of Conjectures 4 and 2 is a classical argument,
which we present in Proposition 7 in the Background section. Using a similar
argument we show that Theorem 3 implies Theorem 1.

We also present two partial results in the direction of the conjecture. First,
we prove it for unconditional convex sets.

######  Theorem 5.

Let $K,T\subseteq\mathbb{R}^{d}$ be unconditional convex sets, then (6) holds.

Second, we show that an asymptotic version of Conjecture 4 would imply the
conjecture itself.

######  Theorem 6.

Assume that there is a sequence of positive numbers $(c_{N})$ with
$\lim_{N\to\infty}c_{N}^{1/N}=1$, such that for any $N\in\mathbb{N}$ and $K,T$
origin-symmetric convex sets in $\mathbb{R}^{N}$,

$\gamma_{N}(K+T)\gamma_{N}(K\cap T)\geq c_{N}\gamma_{N}(K)\gamma_{N}(T).$

Then, Conjecture 4 holds for all $d>0$ and all $K,T\subseteq\mathbb{R}^{d}$.

###  1.1 Acknowledgements

The authors would like to thank Ruth Heller and Yosef Rinott for their useful
comments and discussion. The third named author would like to thank Matthieu
Fradelizi for introducing her to Tehranchi’s conjecture, and is thankful to
the Azrieli foundation for the award of an Azrieli fellowship. The first
author is supported by a grant from the Israel Science Foundation (ISF). The
second and third named authors were supported in part by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 770127).

##  2 Background

In this note, we switch between a probabilistic setting of $n$ random
variables and a geometric setting of functions and sets in $\mathbb{R}^{d}$.
We will use some capital letters (e.g. $X,Y,X_{i},Y_{j}$) to denote random
vectors and some to denote convex sets (e.g. $K,L,T$). The inner product
$\langle\cdot,\cdot\rangle$ will be the standard product in $\mathbb{R}^{d}$.
The Gaussian measure in $\mathbb{R}^{d}$, denoted by $\gamma_{d}$, is the
probability measure with density proportional to $e^{-\|x\|^{2}/2}$. Finally,
the Minkowski sum of two sets is the set $K+T:=\\{x+y:x\in K,\ y\in T\\}$ and
$K$ will be called an origin-symmetric (or centrally symmetric) convex set if
$K=-K$. The convex hull of a set $S$ is the smallest convex set $K$ such that
$S\subseteq K$, and is denoted by ${\rm conv}\\{S\\}$. The support function of
a convex set $K$ is $h_{K}(x)=\sup_{y\in K}\langle x,y\rangle$. We denote
$[n]=\\{1,...,n\\}$ for $n\in\mathbb{N}$.

The study of Gaussian correlation inequalities started with the works of Šidák
[16, 17] and Khatri [10], who independently proved the case of correlation
between one random Gaussian and $n-1$ others (or, in the geometric
formulation, between a convex body and a slab). It was conjectured by Das
Gupta, Eaton, Olkin, Perlman, Savage, and Sobel in [7], and later proved by
Royen in [14], that for $K,T\subseteq\mathbb{R}^{d}$ centrally symmetric,

$\gamma_{d}(K\cap T)\geq\gamma_{d}(K)\gamma_{d}(T).$ (7)

This inequality is known as the Gaussian Correlation Inequality.

The history of this problem is as follows. Pitt [12] published a proof of
inequality (7) in two dimensions, and introduced the decoupling approach to
the problem. However, at the time Pitt’s results could not be translated to
higher dimension, and so proofs of special cases were given: in [15]
Schechtman, Schlumprecht, and Zinn proved the inequality for two ellipsoids,
for a body and a ball, and in some other cases. In [8], Hargé proved the
inequality for a body and an ellipsoid, and in [3], Cordero-Erasquin used the
Cafarelli contraction theorem to give another proof of this case.

Finally, Royen proved the Gaussian correlation conjecture in [14], using
Pitt’s decoupling approach together with some fine analysis. He also proved
that the only equality cases in (7) are for ‘orthogonal’ sets, i.e. for
$K=K_{1}\times\mathbb{R}^{d_{2}},T=\mathbb{R}^{d_{1}}\times T_{2}$ in
$\mathbb{R}^{d}=\mathbb{R}^{d_{1}+d_{2}}$. Royen’s proof is actually
applicable to to a large family of distributions, for a simplified reading of
his proof see [11].

Since Royen’s proof, there has been some work on expanding and refining the
conjecture. Eskenazis, Nayar and Tkocz [5] have extended the Gaussian
correlation inequality to Gaussian mixtures, thus also proving a spherical
correlation inequality (with respect to convex sets on the sphere).

The version of Conjecture 4 presented in this note was proposed by Tehranchi
in [19], who also strengthened a result of Schechtman, Schlumprecht, and Zinn
[15], and showed that

$\gamma_{d}(K)\gamma_{d}(T)\leq(1-s)^{-d/2}\gamma_{d}\left(\sqrt{\frac{2(1-s)}{1+t}}(K\cap
T)\right)\gamma_{d}\left(\sqrt{\frac{1-s}{2(1-t)}}(K+T)\right)$

for all $\sqrt{s}\leq t<1$. Conjecture 4 is also motivated by the fact the a
similar inequality holds for the Lebesgue measure, namely for $K,T$ origin-
symmetric,

${\rm{Vol}}(K+T){\rm{Vol}}(K\cap T)\geq{\rm{Vol}}(K){\rm{Vol}}(T),$

as shown by Rogers and Shephard in [13] (for more on this see [2, Section
1.5]).

A final motivation for this work is the quantitative version of the Gaussian
Correlation Inequality. In [4, Theorem 31], De, Nadimpalli and Servedio
defined a function $c(K,T)\geq 0$ and determined that

$\gamma_{d}(K\cap T)-\gamma_{d}(K)\gamma_{d}(T)\geq c(K,T).$

Conjecture 4 is another example of a quantitative bound, as it can be
rewritten as

$\gamma_{d}(K\cap
T)-\gamma_{d}(K)\gamma_{d}(T)\geq\left(\frac{1}{\gamma_{d}(K+T)}-1\right)\gamma_{d}(K)\gamma_{d}(T).$

The probabilistic version of the strong Gaussian Correlation conjecture,
Conjecture 2, is new, as far as we know. However, it is equivalent to 4:

######  Proposition 7.

Conjecture 4 holds if and only if Conjecture 2 holds.

###### Proof.

Assume that Conjecture 4 holds, and denote by $X=(X_{1},\dots,X_{n})$ the
centered Gaussian vector whose indices are the Gaussian variables of
Conjecture 2. The Gaussian vector $X$ can be expressed as a linear image of a
standard Gaussian $Y\in\mathbb{R}^{d}$ for some $d\in[n]$, i.e. there exist
$u_{1},\dots,u_{n}\in\mathbb{R}^{d}$ such that $X_{i}=\langle Y,u_{i}\rangle$.
Construct the following two convex bodies in $\mathbb{R}^{d}$,

$K=\\{y:|\langle y,u_{i}\rangle|\leq s_{i}\ \forall i\in[n]\\},\ \
T=\\{y:|\langle y,u_{i}\rangle|\leq t_{i}\ \forall i\in[n]\\}.$ (8)

Note that $K+T\subseteq\\{y:|\langle y,u_{i}\rangle|\leq s_{i}+t_{i}\ \forall
i\in[n]\\}$, so applying (6), we get that

$\displaystyle\Pr\left(|X_{i}|\leq s_{i}+t_{i}\ \forall
i\in[n]\right)\cdot\Pr\left(|X_{i}|\leq\min\\{s_{i},t_{i}\\}\ \forall
i\in[n]\right)$ $\displaystyle\quad\geq\gamma_{d}(K+T)\gamma_{d}(K\cap
T)\geq\gamma_{d}(K)\gamma_{d}(T)$ $\displaystyle\quad=\Pr\left(|X_{i}|\leq
s_{i}\ \forall i\in[n]\right)\cdot\Pr\left(|X_{i}|\leq t_{i}\ \forall
i\in[n]\right).$

In the other direction, assume Conjecture 2 holds. Let $K$ and $T$ be origin-
symmetric convex bodies in $\mathbb{R}^{d}$ for some $d\in[n]$. Assume that
$K,T$ are polytopes; the inequality for general origin-symmetric convex bodies
follows by a standard approximation argument.

Let $\\{u_{i}\\}_{i=1}^{n}$ denote the set of outer unit normals of facets of
$K+T$. Note that any outer unit normal of a facet of $K$, $T$, or $K\cap T$ is
already in this set. Taking $Y$ a $d$-dimensional standard Gaussian, we
construct $X_{1},\dots,X_{n}$ jointly Gaussian random variables by taking
$X_{i}=\langle Y,u_{i}\rangle$. Then applying (4) with
$\\{X_{i}\\}_{i=1}^{n}$, $s_{i}=h_{K}(u_{i})$, and $t_{i}=h_{T}(u_{i})$, we
recall that $h_{K+T}=h_{K}+h_{T}$ and $h_{K\cap T}=\min\\{h_{K},h_{T}\\}$ to
get the following:

$\displaystyle\gamma_{d}(K+T)\cdot\gamma_{d}(K\cap T)$
$\displaystyle\quad=\Pr\left(|X_{i}|\leq s_{i}+t_{i}\ \forall
i\in[n]\right)\cdot\Pr\left(|X_{i}|\leq\min\\{s_{i},t_{i}\\}\ \forall
i\in[n]\right)$ $\displaystyle\quad\geq\Pr\left(|X_{i}|\leq s_{i}\ \forall
i\in[n]\right)\cdot\Pr\left(|X_{i}|\leq t_{i}\ \forall
i\in[n]\right)=\gamma_{d}(K)\gamma_{d}(T).$

∎

######  Remark 8.

This proof is quite standard, and a similar variant of it has been used to
show the equivalence of Equations (3) and (7).

##  3 Proofs

###  3.1 The refined Šidák-Khatri inquality

To prove Theorem 1 we will use a lemma similar to Giannopoulos’s proof [6] of
the Šidák-Khatri inquality:

######  Lemma 9.

Let $d_{1},d_{2}\in\mathbb{N}$ and $d=d_{1}+d_{2}$. Let $K$ be an origin-
symmetric convex body in
$\mathbb{R}^{d}=\mathbb{R}^{d_{1}}\times\mathbb{R}^{d_{2}}$ and let
$T=T_{1}\times\mathbb{R}^{d_{2}}$ be a product body, where $T_{1}$ is a convex
body in $\mathbb{R}^{d_{1}}$. Denote by
$\pi_{i}:\mathbb{R}^{d}\to\mathbb{R}^{d_{i}}$, $i=1,2$, the orthogonal
projections.

For any $s\in\mathbb{R}^{d_{1}}$ define the function
$f(s)=\gamma_{d_{2}}(K\cap\pi_{1}^{-1}(s))$, and assume that for any $z>0$,
$\left\\{s\in\pi_{1}(K)\ \middle|\ f(s)\geq z\right\\}$ and $T_{1}$ satisfy
(5) in $\mathbb{R}^{d_{1}}$.

Then $K$ and $T$ also satisfy $\eqref{eq:geometric-sidak}$ in
$\mathbb{R}^{d}$.

We leave the proof of this lemma for later, and show that it immediately gives
a proof of Theorem 3:

###### Proof of Theorem 3.

Assume without loss of generality that $u\in{\rm
sp}\\{e_{1}\\}\subset\mathbb{R}^{d}$, where $e_{1}$ is the first standard
basis vector. In the notation of Lemma 9, take $d_{1}=1,d_{2}=d-1$, then
$T_{1}$ is a symmetric interval $[-a,a]$ (with $a=1/\|u\|$) and for any $z>0$,
$\left\\{s\in\pi_{1}(K)\ \middle|\ f(s)\geq z\right\\}$ is also some symmetric
interval, $[-b(z),b(z)]$, by log-concavity of $f$. These intervals clearly
satisfy (5), since for $d=1$, Theorem 3 holds trivially. ∎

Using Theorem 3 we prove the refined Šidák-Khatri inequality.

###### Proof of Theorem 1.

Denote $X=(X_{1},\dots,X_{n})$. The Gaussian vector $X$ can be expressed as a
linear image of a standard Gaussian $Y\in\mathbb{R}^{d}$ for some $d\in[n]$,
i.e. there exist $u_{1},\dots,u_{n}\in\mathbb{R}^{d}$ such that $X_{i}=\langle
Y,u_{i}\rangle$. Taking $s_{1}=1+a_{1},s_{2}=1,\dots,s_{n}=1$ and
$t_{1}=1,t_{2}=\infty,\dots,t_{n}=\infty$, we construct the two bodies

$K=\\{y\in\mathbb{R}^{d}:|\langle y,u_{i}\rangle|\leq s_{i}\ \forall
i\in[n]\\},\ \ T=\\{y\in\mathbb{R}^{d}:|\langle y,u_{i}\rangle|\leq t_{i}\
\forall i\in[n]\\},$

and note that $T$ is a symmetric slab in direction $u_{1}$. Applying Theorem
3, and using the fact that ${\rm conv}(K\cup T)\subseteq\\{y:|\langle
y,u_{1}\rangle|\leq 1+a_{1}\\}$, we find that

$\displaystyle\Pr(|X_{1}|\leq 1+a_{1})\cdot\Pr(|X_{i}|\leq 1\ \forall
i\in[n])$ $\displaystyle\quad=\gamma_{d}({\rm conv}(K\cup
T))\cdot\gamma_{d}(K\cap T)\geq\gamma_{d}(K)\cdot\gamma_{d}(T)$
$\displaystyle\quad=\Pr(|X_{1}|\leq 1)\cdot\Pr(|X_{1}|\leq
1+a_{1},\dots,|X_{n}|\leq 1).$

This proves (1), and since the choice of the first coordinate $X_{1}$ was
arbitrary, we may apply the inequality inductively on other coordinates and
find that for all $a_{1},\dots,a_{n}>0$,

$\dfrac{\Pr(|X_{i}|\leq 1\ \forall i\in[n])}{\prod_{i=1}^{n}\Pr(|X_{i}|\leq
1)}\geq\dfrac{\Pr(|X_{i}|\leq 1+a_{i}\ \forall
i\in[n])}{\prod_{i=1}^{n}\Pr(|X_{i}|\leq 1+a_{i})}.$

Since the inequality holds for all $\\{a_{i}\\}$, we prove (2). ∎

######  Remark 10.

Our refined Šidák inequality may be reinterpreted for $Y_{1},\dots,Y_{n}\sim
N(0,1)$ standard Gaussians (Taking $Y_{i}=\frac{X_{i}}{\sigma_{i}}$) as

$\dfrac{\Pr(|Y_{1}|\leq c_{1},\dots,|Y_{n}|\leq
c_{n})}{\prod_{i=1}^{n}\Pr(|Y_{i}|\leq
c_{i})}\geq\sup_{a_{1},\dots,a_{n}\in(0,\infty)}\dfrac{\Pr(|Y_{1}|\leq
c_{1}+a_{1},\dots,|Y_{n}|\leq c_{n}+a_{n})}{\prod_{i=1}^{n}\Pr(|Y_{i}|\leq
c_{i}+a_{i})},$

for any choice of $c_{i}>0$ for all $i$. In Šidák’s original paper, he uses
the same inequality with 1 on the left hand side to obtain a confidence
rectangle for $Y_{1},\dots,Y_{n}$. In particular, for confidence level of
$(1-\alpha)$ one takes $c_{i}=c$ such that
$\Phi(c)=\frac{1}{2}(1+(1-\alpha)^{1/k})$.

In the refined inequality, if one may find any $a>0$ such that

$A=\frac{\Pr(|Y_{i}|\leq c+a\ \forall i\in[n])}{\prod_{i=1}^{n}\Pr(|Y_{i}|\leq
c+a)}>1$

then the same choice of $c_{i}=c$ would improve the confidence level to
$A(1-\alpha)$.

Finally, let us return to the proof of the lemma from the beginning of this
subsection.

###### Proof of Lemma 9.

Note that ${\rm conv}\\{K\cup T\\}=\pi_{1}({\rm conv}\\{K\cup
T\\})\times\mathbb{R}^{d_{2}}$. Then by Fubini’s theorem,

$\displaystyle\gamma_{d}({\rm conv}\\{K\cup T\\})\gamma_{d}(K\cap T)$
$\displaystyle=\gamma_{d_{1}}(\pi_{1}({\rm conv}\\{K\cup
T\\}))\int_{\mathbb{R}^{d_{1}}}f(s)\mathbbm{1}_{T_{1}}(s)\mathop{}\\!d\gamma_{d_{1}}(s)$
$\displaystyle=\gamma_{d_{1}}({\rm conv}\\{\pi_{1}(K)\cup
T_{1}\\})\int_{0}^{\infty}\gamma_{d_{1}}(\left\\{s\in T_{1}\ \middle|\
f(s)\geq z\right\\})\mathop{}\\!dz$
$\displaystyle=\int_{0}^{\infty}\gamma_{d_{1}}(\\{f\geq z\\}\cap
T_{1})\cdot\gamma_{d_{1}}({\rm conv}\\{\pi_{1}(K)\cup T_{1}\\})\mathop{}\\!dz$
$\displaystyle\geq\int_{0}^{\infty}\gamma_{d_{1}}(\\{f\geq z\\}\cap
T_{1})\cdot\gamma_{d_{1}}({\rm conv}\\{\\{f\geq z\\}\cup
T_{1}\\})\mathop{}\\!dz,$

where the last inequality holds since for any $z>0$, $\\{f\geq
z\\}\subseteq\pi_{1}(K)$ by definition. By the assumption, for any $z>0$,

$\displaystyle\gamma_{d_{1}}(\\{f\geq z\\}\cap T_{1})\cdot\gamma_{d_{1}}({\rm
conv}\\{\\{f\geq z\\}\cup T_{1}\\})\geq\gamma_{d_{1}}(\\{f\geq
z\\})\cdot\gamma_{d_{1}}(T_{1}).$

Therefore,

$\displaystyle\gamma_{d}({\rm conv}\\{K\cup T\\})\gamma_{d}(K\cap
T)\geq\int_{0}^{\infty}\gamma_{d_{1}}(\\{f\geq
z\\})\mathop{}\\!dz\cdot\gamma_{d_{1}}(T_{1})=\gamma_{d}(K)\cdot\gamma_{d_{1}}(T_{1}).$

Noting that $\gamma_{d}(T)=\gamma_{d_{1}}(T_{1})$, the proof is completed. ∎

######  Remarks 11.

  * •

The only property of $\gamma_{d}$ used in the proof is that it is a product
probability measure with respect to the splitting
$\mathbb{R}^{d}=\mathbb{R}^{d_{1}}\times\mathbb{R}^{d_{2}}$.

  * •

If the assumption that $\left\\{s\in\pi_{1}(K)\ \middle|\ f(s)\geq z\right\\}$
and $T_{1}$ satisfy (5) in $\mathbb{R}^{d_{1}}$ is replaced with the weaker
assumption that the sets satisfy (6), then we can conclude, with the same
proof, that $K$ and $T$ satisfy (6) in $\mathbb{R}^{d}$.

###  3.2 Stronger Gaussian correlation inequality

We begin with a simple proof of Theorem 5, which shows that (6) holds for
unconditional convex sets. Recall that a convex set is called unconditional if
it is symmetric with respect to all coordinate hyperplanes. Following [15], we
use the Karlin-Rinott Theorem. Denote by $\preccurlyeq$ the product partial
order on $\mathbb{R}^{n}$, i.e. $x\preccurlyeq y$ if and only if $\forall
i:x_{i}\leq y_{i}$. This is a lattice; denote by $\lor,\land$ join and meet
operations (i.e. coordinate-wise maximum and minimum).

######  Theorem 12 ([9, Theorem 2.1]).

Let $\mu$ be a product measure on $\mathbb{R}^{d}$ and let $f_{i}$, $1\leq
i\leq 4$, be nonnegative functions on $\mathbb{R}^{d}$ satisfying

$f_{1}(x)f_{2}(y)\leq f_{3}(x\lor y)f_{4}(x\land y).$

Then,

$\int f_{1}d\mu\int f_{2}d\mu\leq\int f_{3}d\mu\int f_{4}d\mu.$

We mention that this theorem is a continuous version of the celebrated four-
function theorem [1].

###### Proof of Theorem 5.

Let $Q=\\{x\in\mathbb{R}^{d}\ |\ 0\preccurlyeq x\\}$ be the positive orthant.
Note that by uncoditionality of $K$, we have that if $0\preccurlyeq
x\preccurlyeq y\in K$, then also $x\in K$, and similarly for $T$.

By unconditionality of $K$ and $T$, if $x\in K\cap Q$ and $y\in T\cap Q$, then
since $x\land y\preccurlyeq x,y$ we have $x\land y\in K\cap T\cap Q$. By the
positivity of elements in $Q$ we have $x\lor y\preccurlyeq x+y$ which implies
that $x\lor y\in(K+T)\cap Q$. It follows that $\gamma_{d}$ and the functions

$f_{1}=1_{K\cap Q},\ f_{2}=1_{T\cap Q},\ f_{3}=1_{(K+T)\cap Q},\
f_{4}=1_{K\cap T\cap Q}$

satisfy the assumptions of Theorem 12, and we find that

$\gamma_{d}(K\cap Q)\gamma_{d}(T\cap Q)\leq\gamma_{d}(K\cap T\cap
Q)\gamma_{d}((K+T)\cap Q).$

Since $\gamma_{d}(K)=2^{n}\gamma_{d}(K\cap Q)$, and similarly for $T,K\cap T$
and $K+T$, the theorem follows. ∎

######  Remark 13.

Theorem 5 holds for any symmetric product measure, and the proof is verbatim.

Next, we see that unlike in Theorem 3, in the unconditional case the Minkowski
sum in Conjecture 4 cannot be replaced with a smaller set such as the convex
hull of $K,T$.

######  Example 14.

Let $K,T\subseteq\mathbb{R}^{2}$ and $N\in\mathbb{R}_{+}$, with
$K=[-1/N,1/N]\times[-N,N]$, and $T=[-N,N]\times[-1/N,1/N]$. Denoting $L={\rm
conv}\\{\pm e_{1},\pm e_{2}\\}$ where $\\{e_{1},e_{2}\\}$ is the standard
orthonormal basis we have

$\displaystyle\gamma_{2}(K)\gamma_{2}(T)$
$\displaystyle=\gamma_{1}([-1/N,1/N])^{2}\gamma_{1}([-N,N])^{2},$

while

$\displaystyle\gamma_{2}(K\cap T)\gamma_{2}({\rm conv}(K\cup T))$
$\displaystyle=\gamma_{1}([-1/N,1/N])^{2}\gamma_{2}({\rm conv}(K\cup T))$
$\displaystyle\leq\gamma_{1}([-1/N,1/N])^{2}\gamma_{2}((N+1/N)L)$
$\displaystyle=\gamma_{1}([-1/N,1/N])^{2}\gamma_{1}\left(\left[-\frac{(N+1/N)}{\sqrt{2}},\frac{(N+1/N)}{\sqrt{2}}\right]\right)^{2}$

where the inequality is due to the inclusion $K,T\subseteq(N+1/N)L$, and the
volume of the square $(N+1/N)L$ can be computed by the rotation invariance of
$\gamma_{2}$. The inequality

$\gamma_{2}(K)\gamma_{2}(T)\leq\gamma_{2}(K\cup T)\gamma_{2}({\rm conv}(K\cup
T))$

would hence imply that

$\gamma_{1}([-N,N])\leq\gamma_{1}\left(\left[-\frac{(N+1/N)}{\sqrt{2}},\frac{(N+1/N)}{\sqrt{2}}\right]\right)$

which is false for $N\geq 3$.

We end this subsection with the proof of Theorem 6, which is a standard
tensorization argument.

###### Proof of Theorem 6.

Let $K,T\subseteq\mathbb{R}^{d}$ be origin-symmetric sets. Denote by
$K^{N}=\prod_{i=1}^{N}K\subseteq(\mathbb{R}^{d})^{N}=\mathbb{R}^{Nd}$, and
similarly
$T^{N}=\prod_{i=1}^{N}T\subseteq(\mathbb{R}^{d})^{N}=\mathbb{R}^{Nd}$. Then,
by the assumption, for all $N\in\mathbb{N}$,

$\gamma_{Nd}(K^{N}+T^{N})\gamma_{Nd}(K^{N}\cap T^{N})\geq
c_{Nd}\cdot\gamma_{Nd}(K^{N})\gamma_{Nd}(T^{N}).$

Note that $K^{N}+T^{N}=(K+T)^{N}$, and $K^{N}\cap T^{N}=(K\cap T)^{N}$. Since
$\gamma_{Nd}$ is a product measure, $\gamma_{Nd}(K^{N})=(\gamma_{d}(K))^{N}$,
$\gamma_{Nd}(T^{N})=(\gamma_{d}(T))^{N}$,
$\gamma_{Nd}(K^{N}+T^{N})=(\gamma_{d}(K+T))^{N}$ and $\gamma_{Nd}(K^{N}\cap
T^{N})=(\gamma_{d}(K\cap T))^{N}$ . Plugging these into the inequality above
and taking $N$-th root gives

$\gamma_{d}(K+T)\gamma_{d}(K\cap T)\geq
c_{Nd}^{1/N}\gamma_{d}(K)\gamma_{d}(T).$

As $d$ is constant we may apply our assumption on $c_{N}$ and take
$N\to\infty$ in the inequality above to conclude the proof. ∎

## References

  * [1] Rudolf Ahlswede and David E. Daykin, _An inequality for the weights of two families of sets, their unions and intersections_ , Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 43 (1978), 183–185.
  * [2] Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman, _Asymptotic geometric analysis, Part i_ , vol. 261, American Mathematical Society, 2021.
  * [3] Dario Cordero-Erausquin, _Some applications of mass transport to gaussian-type inequalities_ , Archive for rational mechanics and analysis 161 (2002), 257–269.
  * [4] Anindya De, Shivam Nadimpalli, and Rocco A. Servedio, _Quantitative correlation inequalities via semigroup interpolation_ , 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference (James R. Lee, ed.), LIPIcs, vol. 185, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, pp. 69:1–69:20.
  * [5] Alexandros Eskenazis, Piotr Nayar, and Tomasz Tkocz, _Gaussian mixtures: entropy and geometric inequalities_ , Annals of Probability 46 (2018), no. 5, 2908–2945.
  * [6] Apostolos A. Giannopoulos, _On some vector balancing problems_ , Studia Mathematica 122 (1997), 225–234.
  * [7] Somesh Das Gupta, Morris L. Eaton, Ingram Olkin, Michael Perlman, Leonard J. Savage, and Milton Sobel, _Inequalities on the probability content of convex regions for elliptically contoured distributions_ , Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), vol. 2, 1972, pp. 241–265.
  * [8] Gilles Hargé, _A particular case of correlation inequality for the gaussian measure_ , Annals of probability (1999), 1939–1951.
  * [9] Samuel Karlin and Yosef Rinott, _Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions_ , Journal of Multivariate Analysis 10 (1980), no. 4, 467–498.
  * [10] Chinubhai G. Khatri, _On certain inequalities for normal distributions and their applications to simultaneous confidence bounds_ , The Annals of Mathematical Statistics (1967), 1853–1867.
  * [11] Rafał Latała and Dariusz Matlak, _Royen’s proof of the gaussian correlation inequality_ , Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2014–2016, Springer, 2017, pp. 265–275.
  * [12] Loren D. Pitt, _A gaussian correlation inequality for symmetric convex sets_ , The Annals of Probability 5 (1977), no. 3, 470–474.
  * [13] Claude A. Rogers and Geoffrey C. Shephard, _The difference body of a convex body_ , Archiv der Mathematik 8 (1957), no. 3, 220–233.
  * [14] Thomas Royen, _A simple proof of the gaussian correlation conjecture extended to multivariate gamma distributions_ , arXiv preprint arXiv:1408.1028 (2014).
  * [15] Gideon Schechtman, Thomas Schlumprecht, and Joel Zinn, _On the Gaussian measure of the intersection_ , Annals of probability (1998), 346–357.
  * [16] Zbyněk Šidák, _Rectangular confidence regions for the means of multivariate normal distributions_ , Journal of the American Statistical Association 62 (1967), no. 318, 626–633.
  * [17] Zbynek Šidák, _On multivariate normal probabilities of rectangles: their dependence on correlations_ , The Annals of Mathematical Statistics 39 (1968), no. 5, 1425–1434.
  * [18] Stanislaw J. Szarek and Elisabeth Werner, _A nonsymmetric correlation inequality for gaussian measure_ , Journal of multivariate analysis 68 (1999), no. 2, 193–211.
  * [19] Michael R. Tehranchi, _Inequalities for the Gaussian measure of convex sets_ , Electronic Communications in Probability 22 (2017), 1 – 7.