# On the Role of Volterra Integral Equations in Self-Consistent, Product-
Limit, Inverse Probability of Censoring Weighted, and Redistribution-to-the-
Right Estimators for the Survival Function

Robert L. Strawderman111Corresponding author:
<EMAIL_ADDRESS>Department of Biostatistics and
Computational Biology, University of Rochester, Rochester, NY, USA, 14642
Benjamin R. Baer  Department of Biostatistics and Computational Biology,
University of Rochester, Rochester, NY, USA, 14642  School of Mathematics and
Statistics, University of St Andrews, St Andrews, Fife, Scotland, KY16 9SS

###### Abstract

This paper reconsiders several results of historical and current importance to
nonparametric estimation of the survival distribution for failure in the
presence of right-censored observation times, demonstrating in particular how
Volterra integral equations of the first kind help inter-connect the resulting
estimators. The paper begins by considering Efron’s self-consistency equation,
introduced in a seminal 1967 Berkeley symposium paper. Novel insights provided
in the current work include the observations that (i) the self-consistency
equation leads directly to an anticipating Volterra integral equation of the
first kind whose solution is given by a product-limit estimator for the
censoring survival function; (ii) a definition used in this argument
immediately establishes the familiar product-limit estimator for the failure
survival function; (iii) the usual Volterra integral equation for the product-
limit estimator of the failure survival function leads to an immediate and
simple proof that it can be represented as an inverse probability of censoring
weighted estimator (i.e., under appropriate conditions). Finally, we show that
the resulting inverse probability of censoring weighted estimators, attributed
to a highly influential 1992 paper of Robins and Rotnitzky, were implicitly
introduced in Efron’s 1967 paper in its development of the redistribution-to-
the-right algorithm. All results developed herein allow for ties between
failure and/or censored observations.

Key words: Kaplan-Meier estimator; Product integration; Right-censoring

##  1 Introduction

Efron, (1967) introduces the concept of self-consistency to motivate an
iteratively specified estimator for the left-continuous version of the
survival function under random right censoring. Assuming that a solution to
this iterative scheme exists, it is shown to be unique and satisfy a recursive
equation (Efron,, 1967, Thm. 7.1). In a key corollary to this result, Efron,,
Cor. 7.1 shows that the left-continuous version of the Kaplan-Meier estimator
(Kaplan and Meier,, 1958) coincides with the self-consistent estimator over a
range defined by the observed data. Proof by induction is used to establish
both results. Efron, then introduces the redistribution-to-the-right (RTTR)
algorithm as an alternative way to understand the self-consistent (hence
Kaplan-Meier) estimator, demonstrating how the result produced by running this
algorithm equals the self-consistent estimator.

Satten and Datta, (2001), motivated by developments in Robins and Rotnitzky,
(1992), later prove that the Kaplan-Meier estimator for the failure
distribution function can be represented as an inverse-probability of
censoring weighted (IPCW) estimator in which the denominator uses a Kaplan-
Meier estimator for the censoring survival function. They remark that this
result is “somewhat unsatisfactory” due to the use of the latter in deriving
this equivalence. A constructive proof using “coupled” inverse-probability
weighted estimators is then given that avoids using the Kaplan-Meier estimator
of the censoring survival function.

The overarching intent of this work is to show how these results are easily
established, and directly interconnected, through the Volterra theory of
product integration as formulated in Gill and Johansen, (1990) and summarized
in Andersen et al., (1993, Sec. II.6). In particular, after reviewing the
self-consistency equation for the survival function for failure in Section
2.1, the following results are established in Section 2.2: (i) it is shown
that Efron’s self-consistency equation directly generates a Volterra integral
equation for the survival function of the censoring distribution, the solution
being given by its corresponding product-limit estimator; then, using this
result, (ii) a transparent non-inductive proof is given that shows the
product-limit estimator of the survival function for failure, hence Kaplan-
Meier estimator, solves Efron’s self-consistency equation. Of particular
interest in (i) is our use of the anticipating form of Volterra integral
equation (e.g., Gill and Johansen,, 1990, p. 1526), as it allows us to
establish the indicated result for general failure and censoring
distributions. In Section 2.3, we continue studying the product-limit
estimator, where (iii) it is shown that the IPCW estimator for the cumulative
distribution function for failure (e.g., Satten and Datta,, 2001) follows
easily from the corresponding Volterra integral equation that yields the
product-limit estimator for the survival function for failure; and, (iv) a
necessary and sufficient condition is given for the latter to also be
represented as an IPCW estimator. Finally, Section 2.4 reconsiders Efron’s
RTTR algorithm; there, it is shown that the resulting estimator at convergence
leads directly to an IPCW estimator for the failure survival function,
arguably being the earliest such example of IPCW estimation.

This paper makes several pedagogically novel contributions in the areas of
self-consistency, product-limit estimation, and IPCW estimation. Indeed,
although the results that the product-limit estimator solves Efron’s self-
consistency equation and that the product-limit estimator can be represented
as an IPCW estimator are both well-known, the approach taken to establish
these results is new, and elegantly ties all known non-parametric estimators
for the failure survivor function together in a transparent and interesting
way.

##  2 Main Results

###  2.1 A brief review of Efron’s self-consistency equation

Suppose $T>0$ and $U>0$ are independent random variables having respective
survival functions $S(t)=P\\{T>t\\}$ and $K(t)=P\\{U>t\\}$ for $t\geq 0$.
Define $X=\min\\{T,U\\}$ and $D=I\\{T\leq U\\},$ and assume that the available
data takes the form of an independent and identically distributed sample
$(X_{i},D_{i}),i=1\ldots n.$ Define

$\phi_{S(\cdot)}(t;X,D):=I\\{X>t\\}+(1-D)I\\{X\leq t\\}\frac{S(t)}{S(X)},$

as a random function of $t$; under the usual convention that 0/0 = 0, to be
assumed throughout, this function is well-defined for $t\geq 0.$ Assuming that
$P\\{S(U)>0\\}>0$, an easy calculation also shows that

$\phi_{S(\cdot)}(t;X,D)=P[T>t\mid X,D]$ (1)

and hence that $E[\phi_{S(\cdot)}(t;X,D)]=S(t)$. This last identity motivates
the “self-consistent” estimator for $S(t)$ as originally proposed in Efron,
(1967).

Specifically, define for $t\geq 0$ the naive estimator
$\widehat{S}_{0}(t):=n^{-1}\sum_{i=1}^{n}I\\{X_{i}>t\\}.$ Let ${\cal
D}_{0}=\\{t\geq 0:\widehat{S}_{0}(t)>0\\}$ and assume that $t\in{\cal D}_{0}.$
Then, for $k\geq 1$, one can recursively define the sequence of estimators

$\widehat{S}_{k}(t):=\frac{1}{n}\sum_{i=1}^{n}\phi_{\widehat{S}_{k-1}(\cdot)}(t;X_{i},D_{i}).$
(2)

A limit of this sequence exists as $k\rightarrow\infty$ (Rogers,, 1980); call
this limit $\widehat{S}(t)$. Since $\widehat{S}_{0}(t)>0,$ $\widehat{S}(t)$ is
positive and satisfies the fixed-point identity, or self-consistency equation,

$\widehat{S}(t)=\widehat{S}_{0}(t)+\frac{1}{n}\sum_{i=1}^{n}(1-D_{i})I\\{X_{i}\leq
t\\}\frac{\widehat{S}(t)}{\widehat{S}(X_{i})}.$ (3)

The estimator $\widehat{S}(t)$ solving (3) above has been defined for
$t\in{\cal D}_{0};$ this set includes $t=0,$ where (3) implies the trivial
solution $\widehat{S}(0)=1.$ Technically, equation (3) also remains well
defined for $t>0$ such that $\widehat{S}_{0}(t)=0$ under the convention that
$0/0=0$. This is consistent with the iterative scheme in (2), which trivially
stays at zero when it is started at zero. It may be noted, however, that the
derivation of (1) that motivates (2), hence (3), itself requires that
$P(S(U)>0)>0.$ Consequently, the fact that (3) also characterizes the solution
when $\widehat{S}(t)=0$ stems primarily from (2) and the indicated convention.
For later reference, we define $\widehat{S}_{\mathit{SC}}(\cdot)$ as the self-
consistent estimator for $t\geq 0$; in particular,
$\widehat{S}_{\mathit{SC}}(t)=\widehat{S}(t),$ the non-trivial solution to (3)
for $t\in{\cal D}_{0},$ and $\widehat{S}_{\mathit{SC}}(t)=0$ for $t\in{\cal
D}_{0}^{C}=\\{t\geq 0:\widehat{S}_{0}(t)=0\\}.$

######  Remark.

Although the development of self-consistent estimation above is for the right-
continuous function $S(t)$, Efron, (1967) actually introduces the concept of
self-consistency in the context of estimating its left-continuous version
$S(t-)=P(T\geq t)$. In particular, in the absence of ties between failures
and/or censored observations, Efron, defines the self-consistent estimator as
the solution to

$\widehat{S}(t-)=\widehat{S}_{0}(t-)+\frac{1}{n}\sum_{i=1}^{n}(1-D_{i})I\\{X_{i}<t\\}\frac{\widehat{S}(t-)}{\widehat{S}(X_{i}-)}.$

Efron, argues that $\widehat{S}(t-)=0$ for
$t>X_{(n)}=\max_{i=1,\ldots,n}X_{i}$, and proves that the solution coincides
with the left-continuous version of the Kaplan-Meier estimator for $S(t-)$
when $t\leq X_{(n)}$ (Efron,, 1967, Cor. 7.1). These results imply that the
solution to (3) agrees with the Kaplan-Meier estimator for $t<X_{(n)},$ a fact
proved using different tools, and in greater generality, in the next section.

###  2.2 Efron’s self-consistency equation generates a Volterra integral
equation having a product-limit estimator as its solution

In this section, we characterize the solution to (3) using results from the
theory of Volterra integral equations (Gill and Johansen,, 1990; Andersen et
al.,, 1993). In the process, we also establish a transparent connection
between the self-consistent and product-limit estimators.

To allow for the possibility of ties, we assume that there exists $m$ unique
observed survival times $X_{i}$ such that $0<X_{(1)}<X_{(2)}<\cdots<X_{(m)}$,
where $m\leq n$. It will also be convenient in the coming developments to
employ counting process notation. For $i=1\ldots n$ and $t\geq 0$, let
$N_{i}(t)=I\\{X_{i}\leq t,D_{i}=1\\}$, $C_{i}(t)=I\\{X_{i}\leq t,D_{i}=0\\}$,
and $Y_{i}(t)=I\\{X_{i}\geq t\\}$. Let $N(t)=\sum_{i=1}^{n}N_{i}(t)$,
$C(t)=\sum_{i=1}^{n}C_{i}(t)$, and $Y(t)=\sum_{i=1}^{n}Y_{i}(t);$ in addition,
define $\Delta N(t)=N(t)-N(t-)$ and $\Delta C(t)=C(t)-C(t-)$ for every $t>0$.
Finally, let $Y^{\dagger}_{i}(t)=I\\{X_{i}\geq t,D_{i}=0\text{ or
}X_{i}>t,D_{i}=1\\}$, and define
$Y^{\dagger}(t)=\sum_{i=1}^{n}Y^{\dagger}_{i}(t)$.

The modified at-risk process $Y^{\dagger}_{i}(t)$ was recently studied by Baer
et al., (2023). It is easily seen that
$Y^{\dagger}_{i}(t)=Y_{i}(t)-I\\{X_{i}=t,\Delta_{i}=1\\};$ hence,
$Y^{\dagger}(t)=Y(t)-\Delta N(t).$ When $\Delta N(t)=0$,
$Y^{\dagger}(t)=Y(t),$ but not otherwise. This highlights the fact $Y(t)$ and
$Y^{\dagger}(t)$ are respectively the proper choice of at-risk processes for
failure and censoring when ties between failure and censoring times can exist.
The reason these processes are not necessarily equal stems from the asymmetric
nature of $D=I\\{T\leq U\\},$ which classifies any instance of $T=U$ as a
failure. As will become clear below, the process $Y^{\dagger}(t)$ specifically
reflects the fact that one is no longer considered to be at risk for censoring
when a failure occurs at the same time.

Equation (3) trivially implies $\widehat{S}(0)=1$ and that $\widehat{S}(t)=0$
on ${\cal D}_{0}^{c}=\\{t:\widehat{S}_{0}(t)=0\\}.$ Since
$\widehat{S}_{0}(t)=n^{-1}Y(t+)$ and $Y(t+)=0$ for $t\geq X_{(m)},$ it
therefore suffices to characterize the solution to the self-consistency
equation (3) for $t\in(0,X_{(m)}).$ To do so, and using the aforedefined
notation, we first rewrite (3) for such $t$ as

$\widehat{S}(t)=\frac{Y(t+)}{n}+\frac{\widehat{S}(t)}{n}\int_{(0,t]}\frac{\mathrm{d}C(u)}{\widehat{S}(u)},$
(4)

where the integral on the right-hand side is a Lebesgue-Stieltjes integral
(e.g., Carter and van Brunt,, 2012). Any solution $\widehat{S}(t)$ to (4) is
necessarily non-increasing; moreover, since $t<X_{(m)}$ implies that
$Y(t+)>0$, it must also satisfy $\widehat{S}(t)>0$ there. It follows that the
integral on the right-hand side is well-defined since $C(u)$ is a right-
continuous step-function and $\widehat{S}(u)>0$ for $u\leq t$.

Because $\widehat{S}(t)>0$ for $t<X_{(m)},$ we can divide (4) through by
$\widehat{S}(t)$ to obtain

$1=\frac{Y(t+)}{n\widehat{S}(t)}+\frac{1}{n}\int_{(0,t]}\frac{\mathrm{d}C(u)}{\widehat{S}(u)}.$
(5)

Next, define for $t<X_{(m)}$ the function

$\widehat{K}(t):=\frac{Y(t+)}{n\widehat{S}(t)},$ (6)

where $\widehat{S}(\cdot)$ is defined through (4). We stress here that (6) is
merely a definition; that is, the notation on the left-hand side is being
defined through the expression on the right-hand side, the latter depending
directly on the (unknown) solution $\widehat{S}(t)$ that solves (4). Using
this definition and equation (5), equation (4) can now be rewritten

$\widehat{K}(t)=1-\int_{(0,t]}\widehat{K}(u)\frac{\mathrm{d}C(u)}{Y(u+)}.$ (7)

The integral on the right-hand side is again a Lebesgue-Stieltjes integral,
and is well-defined since $Y(u+)$ and $C(u)$ are right-continuous monotone
functions with left-hand limits and $1/Y(u+)$ is positive for $u\leq
t<X_{(m)}.$ Importantly, the resulting expression (7) is an example of an
_anticipating_ Volterra integral equation (Helton,, 1977, 1978); see also Gill
and Johansen, (1990, p. 526) for a brief review. Using this fact, the
proposition below characterizes the solution to (7) as a product integral
(Gill and Johansen,, 1990).

######  Proposition 1.

Under (6), $\widehat{K}(0)=1;$ moreover, $\widehat{K}(t)$ satisfies the
anticipating Volterra integral equation (7) for $0<t<X_{(m)}.$ For such $t,$
equation (7) has the unique right-continuous solution

$\widehat{K}(t)=\Prodi_{(0,t]}\left\\{1-\frac{\mathrm{d}C(u)}{Y^{\dagger}(u)}\right\\},$
(8)

where $Y^{\dagger}(u)=\sum_{i=1}^{n}I\\{X_{i}\geq u,D_{i}=0\text{ or
}X_{i}>u,D_{i}=1\\}$ and $\prodi$ denotes the product integral. Moreover, if
$\Delta N(u)\Delta C(u)=0$ for $u<X_{(m)}$, then (8) reduces to

$\widehat{K}(t)=\Prodi_{(0,t]}\left\\{1-\frac{\mathrm{d}C(u)}{Y(u)}\right\\}$
(9)

and $\widehat{K}(t)$ necessarily also satisfies the non-anticipating Volterra
integral equation

$\widehat{K}(t)=1-\int_{(0,t]}\widehat{K}(u-)\frac{\mathrm{d}C(u)}{Y(u)}.$

The estimators (8) and (9) are product-limit estimators for the censoring
survival function. The latter arises as a special case of (8) when $\Delta
N(u)\Delta C(u)=0$ for $u<X_{(m)};$ that is, under a “no common
discontinuities” condition (e.g., such as when $m=n$, where ties of any kind
are absent). However, when this condition fails, only (8), which employs the
censoring at-risk process $Y^{\dagger}(\cdot)$ is a suitable estimator for the
censoring survival function $K(t)=P(U>t)$ (cf. Gill,, 1980, p. 36); see also
Baer et al., (2023, Lemma 2).

######  Remark.

The assumption that $\Delta N(u)\Delta C(u)=0$ for $u<X_{(m)}$ holds almost
surely when at least one of $T$ and $U$ are absolutely continuous random
variables. More generally, while
$\\{N_{i}(u)-N_{i}(u-)\\}\\{C_{i}(u)-C_{i}(u-)\\}=0$ for $u\geq 0$ and every
$i$ by construction, this does not imply $\Delta N(u)\Delta C(u)=0;$
counterexamples are easily constructed when $T$ and $U$ are each discrete with
common support.

######  Remark.

A commonly recommended procedure for estimating $K(\cdot)$ is to (i) swap
$D_{i}$ with $1-D_{i}$ for all observations in a data set (i.e., so that
observed censored observations become “failures” and vice versa); then, (ii)
calculate the product-limit estimator for “failure” as defined in this
modified dataset. Doing so returns $\widehat{K}(t)$ as defined in (9), which
only consistently estimates $K(t)$ under independent censoring provided
$\Delta N(u)\Delta C(u)=0$ for $u\in[0,t]$.

The results above show that the self-consistency equation for $\widehat{S}(t)$
directly generates a Volterra integral equation for $\widehat{K}(t)$ that has
the product-limit estimator (8) as its solution (i.e., an estimator of the
censoring survival function). This observation is interesting for at least two
reasons: (i) it highlights a fundamentally important connection between
estimating the failure and censoring survival functions; and, (ii) equation
(4) itself does not generate an obvious Volterra integral equation for
$\widehat{S}(t).$ Combining Proposition 1 with the definition of
$\widehat{K}(t)$ in (6) yields the unique estimator for $S(t)$ that solves the
self-consistency equation.

######  Proposition 2.

The product-limit estimator

$\widehat{S}_{\mathit{PL}}(t):=\Prodi_{(0,t]}\left\\{1-\frac{\mathrm{d}N(u)}{Y(u)}\right\\}$
(10)

is the unique solution to (4) for $0<t<X_{(m)}.$ For $t\geq 0,$ it follows
that the self-consistent estimator $\widehat{S}_{\mathit{SC}}(t)$ may be
written

$\widehat{S}_{\mathit{SC}}(t)=\widehat{S}_{\mathit{PL}}(t)I\\{t<X_{(m)}\\}.$
(11)

A version of Proposition 2 was originally proved in Efron, (1967) using
inductive arguments assuming all observed survival times are unique (i.e., no
ties). The current result is more general and is obtained as a consequence of
existing theory for Volterra integral equations. This novel derivation
highlights the important connections between the product-limit estimators for
the survival and censoring distributions (cf. Gill,, 1980).

######  Remark.

Consider $t<X_{(m)}.$ Substitution of (10) into the right-hand side of (6)
necessarily results in the general form of $\widehat{K}(\cdot)$ given in (8).
In practice, this means that $K(\cdot)$ is easily estimated using (6) once
$\widehat{S}_{\mathit{PL}}(t)$ in (10) is computed, avoiding the need for data
manipulation like that described in the previous remark.

As pointed out in Andersen et al., (1993, p. 260), the behavior of commonly
used nonparametric estimators for $S(t)$ typically differ only in the extreme
right tail. The self-consistent estimator $\widehat{S}_{\mathit{SC}}(t)$ in
(11) equals the product-limit estimator $\widehat{S}_{\mathit{PL}}(t)$ in (10)
for $t<X_{(m)}.$ However, the estimator $\widehat{S}_{\mathit{PL}}(t)$ is
itself well-defined for all $t>0$ under the convention that 0/0 = 0 (Gill,,
1980; Andersen et al.,, 1993). Using this fact, it can be easily shown that
$\widehat{S}_{\mathit{PL}}(t)=\widehat{S}_{\mathit{SC}}(t)=0$ for $t\geq
X_{(m)}$ when $\Delta N(X_{(m)})=Y(X_{(m)})$ (i.e., all observations with
$X_{i}=X_{(m)}$ have $D_{i}=1$, hence are failures). However, when $\Delta
N(X_{(m)})<Y(X_{(m)}),$ the estimator $\widehat{S}_{\mathit{PL}}(t)>0$ remains
constant for $t\geq X_{(m)},$ exceeding $\widehat{S}_{\mathit{SC}}(t)=0$
there. Finally, as originally defined, the Kaplan-Meier estimator
$\widehat{S}_{KM}(t)$ agrees with both $\widehat{S}_{\mathit{PL}}(t)$ and
$\widehat{S}_{\mathit{SC}}(t)$ for $t\geq 0$ when $\Delta
N(X_{(m)})=Y(X_{(m)}).$ However, when $\Delta N(X_{(m)})<Y(X_{(m)}),$ we have
$\widehat{S}_{KM}(t)=\widehat{S}_{\mathit{PL}}(t)=\widehat{S}_{\mathit{SC}}(t)$
only for $t<X_{(m)};$ Kaplan and Meier leave $\widehat{S}_{KM}(t)$ undefined
for $t\geq X_{(m)}$ beyond assuming that it respects the natural range
restriction $\widehat{S}_{KM}(t)\in[0,\widehat{S}_{\mathit{PL}}(X_{(m)})]$
(cf. Kaplan and Meier,, 1958, p. 463). Succinctly, it follows that
$\widehat{S}_{\mathit{SC}}(t)\leq\widehat{S}_{KM}(t)\leq\widehat{S}_{\mathit{PL}}(t)$
for $t\geq 0,$ with equality holding for $t<X_{(m)},$ and also for $t\geq
X_{(m)}$ when $\Delta N(X_{(m)})=Y(X_{(m)}).$

###  2.3 The product-limit estimator for $S(t)$ solves a Volterra integral
equation that generates an IPCW estimator

Theorems II.6.1 and II.6.6 in Andersen et al., (1993, Sec. II.6) imply that
$\widehat{S}_{\mathit{PL}}(t)$ in (10) uniquely solves the (non-anticipating)
Volterra integral equation

$\widehat{S}_{\mathit{PL}}(t)=1-\int_{(0,t]}\widehat{S}_{\mathit{PL}}(u-)\frac{\mathrm{d}N(u)}{Y(u)}$
(12)

for all $t>0$. Below, we establish how and under which conditions (12) leads
to natural IPCW estimators of the cumulative distribution function and the
survival function of $T$.

Define the IPCW estimator (cf. Robins and Rotnitzky,, 1992)

$\widehat{F}_{\mathit{IPCW}}(t):=\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}I\\{X_{i}\leq
t\\}}{\widehat{K}(X_{i}-)}$ (13)

with domain $t>0$, where $\widehat{K}(\cdot)$ is defined in (8). This
estimator is is well-defined since $\widehat{K}(\cdot)$ is non-increasing and
$\widehat{K}(X_{(m)}-)>0.$ A version of this IPCW estimator of $F(t)=P\\{T\leq
t\\}$ using $\widehat{K}(\cdot)$ as defined in (9) was studied in Satten and
Datta, (2001), who also note the need for its replacement with
$\widehat{K}(\cdot)$ in (8) when ties can exist. Below, we show that (12)
leads directly to $\widehat{F}_{\mathit{IPCW}}(t);$ we then consider the
problem of directly estimating $S(t)$ using IPCW and its connections to both
$\widehat{F}_{\mathit{IPCW}}(t)$ and $\widehat{S}_{\mathit{PL}}(t)$.

Proceeding informally for the moment, the key relation (6) that defined
$\widehat{K}(\cdot)$ together with results summarized in previous propositions
implies that $\widehat{K}(u-)\widehat{S}_{\mathit{PL}}(u-)=Y(u)/n$ and hence
that $\widehat{S}_{\mathit{PL}}(u-)=n^{-1}Y(u)/\widehat{K}(u-)$ for $u\leq
X_{(m)}$. Similarly to $\widehat{S}_{\mathit{PL}}(\cdot),$ it is easy to see
that the product-limit estimator $\widehat{K}(t)$ in (8) is also well-defined
for $t\geq X_{(m)};$ under the convention that 0/0 = 0, inspection of the tail
behavior of both $\widehat{K}(\cdot)$ and $\widehat{S}_{\mathit{PL}}(\cdot)$
then shows that $\widehat{S}_{\mathit{PL}}(u-)=n^{-1}Y(u)/\widehat{K}(u-)$ for
$u>0$. Substitution of this expression for $\widehat{S}_{\mathit{PL}}(u-)$
into the right-hand side (12) now gives the relationship

$1-\widehat{S}_{\mathit{PL}}(t)=\frac{1}{n}\int_{(0,t]}\frac{\mathrm{d}N(u)}{\widehat{K}(u-)}.$

Simplifying the resulting integral leads to the desired result, formalized
below.

######  Proposition 3.

Let $t\geq 0$ and define
$\widehat{F}_{\mathit{PL}}(t):=1-\widehat{S}_{\mathit{PL}}(t)$ as an estimator
for the distribution function of $T$, where $\widehat{S}_{\mathit{PL}}(t)$
satisfies (12) for $t>0$. Then,
$\widehat{F}_{\mathit{PL}}(t)=\widehat{F}_{\mathit{IPCW}}(t).$

That is, the product-limit estimator and IPCW estimators for $F(t)=1-S(t)$
agree for $t\geq 0.$ Since
$\widehat{S}_{\mathit{PL}}(t)=1-\widehat{F}_{\mathit{PL}}(t)$ by definition,
the result of this proposition further implies that
$\widehat{S}_{\mathit{PL}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t);$ that is,

$\widehat{S}_{\mathit{PL}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t)=1-\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}+\widetilde{S}_{\mathit{IPCW}}(t),$
(14)

where

$\widetilde{S}_{\mathit{IPCW}}(t):=\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}I\\{X_{i}>t\\}}{\widehat{K}(X_{i}-)}.$

The IPCW estimator $\widetilde{S}_{\mathit{IPCW}}(t)$ is an obvious analog to
$\widehat{F}_{\mathit{IPCW}}(t)$ for estimating $S(t),$ and it is also
frequently equated to $\widehat{S}_{\mathit{PL}}(t)$ in the literature.
However, the identity in (14) clearly shows that
$\widetilde{S}_{\mathit{IPCW}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t),$ hence
$\widetilde{S}_{\mathit{IPCW}}(t)=\widehat{S}_{\mathit{PL}}(t),$ if and only
if

$\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}=1.$ (15)

The following result gives a useful identity involving the summation appearing
on the left-hand side of (15), and leads to a a simple necessary and
sufficient condition for (15), hence for
$\widetilde{S}_{\mathit{IPCW}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t),$ to hold.

######  Proposition 4.

The following identity holds:

$\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}+\frac{\Delta
C(X_{(m)})}{n\widehat{K}(X_{(m)}-)}=1.$ (16)

Therefore, (15) holds if and only if $\Delta N(X_{(m)})=Y(X_{(m)})$ (i.e.,
$\Delta C(X_{(m)})=0$). Consequently,
$\widetilde{S}_{\mathit{IPCW}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t)$ for $t\geq
0$ if and only if $\Delta N(X_{(m)})=Y(X_{(m)})$.

The condition that $\Delta N(X_{(m)})=Y(X_{(m)})$ is equivalent to requiring
that all observations with $X_{i}=X_{(m)}$ have $D_{i}=1$; this means that all
observations occurring at the last observation time are failure times and
results in
$\widetilde{S}_{\mathit{IPCW}}(t)=\widehat{S}_{\mathit{PL}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t)=0$
for $t\geq X_{(m)}$. Under this same condition, it is not difficult to show
that $\widehat{K}(t)>0$ for $t\geq X_{(m)}$. Finally, when the conditions
specified in Proposition 4 do hold, the IPCW estimator
$\widetilde{S}_{\mathit{IPCW}}(t)$ agrees with $\widehat{S}_{\mathit{SC}}(t)$
for $t\geq 0.$ However, when $\Delta N(X_{(m)})<Y(X_{(m)})$, the IPCW
estimator $\widetilde{S}_{\mathit{IPCW}}(t)$ does not agree with any of the
other estimators, at least in finite samples.

###  2.4 The RTTR Estimator is an IPCW Estimator

In addition to $\widehat{S}_{\mathit{SC}}(t-),$ Efron, (1967, p. 842-3)
introduced the redistribute-to-the-right (RTTR) estimator, defined as the
limit of the RTTR algorithm. Under the setting where $m=n$ (i.e., no ties),
Efron, (1967) argues that the RTTR estimator for $S(t-)=P\\{T\geq t\\},$ say
$\widehat{S}_{\mathit{RTTR}}(t-),$ equals $\widehat{S}_{\mathit{SC}}(t-)$ for
$t>0.$ The key to this result lies in equation (7.19) of Efron, (1967, p.
843), where it is further argued that
$\widehat{S}_{\mathit{RTTR}}(t-)-\widehat{S}_{\mathit{RTTR}}(t)=(7.19)$ when
(i) $t$ is a failure time or (ii) $t=X_{(n)}$ regardless of failure or
censoring status; otherwise,
$\widehat{S}_{\mathit{RTTR}}(t-)-\widehat{S}_{\mathit{RTTR}}(t)=0.$ Written in
terms of our notation, and for the case where $m=n$ so that the ordered
observation times are given by $X_{(1)}<\cdots<X_{(n)},$ one can write

$\widehat{S}_{\mathit{RTTR}}(X_{(k)}-)-\widehat{S}_{\mathit{RTTR}}(X_{(k)})=J_{(k)}D_{(k)}I\\{k<n\\}+J_{(n)}I\\{k=n\\}$
(17)

where $D_{(k)}$ is the failure indicator associated with $X_{(k)}$ and

$J_{(k)}=\frac{1}{n}\prod_{j=1}^{k-1}\left\\{1+\frac{1}{n-j}\right\\}^{1-D_{(j)}}=\frac{1}{n}\prod_{j:X_{j}<X_{(k)}}\left\\{1+\frac{\Delta
C(X_{j})}{Y(X_{j}+)}\right\\}.$ (18)

Notice the special status given to the largest observation in (17): this
observation contributes to a jump in $\widehat{S}_{\mathit{RTTR}}(\cdot)$
regardless of failure status. Considering the last expression on the right-
hand side of (18), it can be seen that

$J_{(k)}=\frac{1}{n}\Prodi_{(0,X_{(k)})}\left\\{1+\frac{\mathrm{d}C(u)}{Y(u+)}\right\\}=\frac{1}{n\widehat{K}(X_{(k)}-)},$
(19)

where the last equality follows immediately from earlier calculations; see, in
particular, the proof of Proposition 1. To the authors’ knowledge, the
earliest paper to recognize the connection between the increments (17) and
IPCW weighting appears to be Malani, (1995);

When ties can exist between failure and/or censoring times, the increments
(17) require modification. In particular, if $m\leq n$ and
$X_{(1)}<\cdots<X_{(m)}$ are the unique observation times, a natural extension
of Efron’s arguments leading to his equation (7.19) gives

$\widehat{S}_{\mathit{RTTR}}(X_{(k)}-)-\widehat{S}_{\mathit{RTTR}}(X_{(k)})=J_{(k)}\Delta
N(X_{(k)})I\\{k<m\\}+J_{(m)}Y(X_{(m)})I\\{k=m\\}$ (20)

where $J_{(k)}$ is given by (19). It is easily seen that (20) reduces to (17)
when $m=n,$ and again assigns special status to the largest observation time
$X_{(m)}$. To the author’s knowledge, the extension (20) in the case of ties
appears to be new.

Now, let $\widehat{H}(t)$ be any right-continuous function such that
$\widehat{H}(t)=1$ for $t\in[0,X_{(1)}),$ and for $r=1,\ldots,m,$ that
$\widehat{H}(X_{(r)}-)-\widehat{H}(X_{(r)})\geq 0$ and
$\widehat{H}(t-)-\widehat{H}(t)=0$ for $t\neq X_{(r)}.$ Then, $\widehat{H}(t)$
is a non-increasing right-continuous step function where jumps can only (but
do not always) occur at $X_{(r)},r=1,\ldots,m.$ Letting
$k_{t}=\max\\{j:X_{(j)}\leq t\\}$ for $t\geq 0$ (i.e., with $X_{(0)}=0$), it
follows that

$\widehat{H}(t)=1-\sum_{r=1}^{k_{t}}\\{\widehat{H}(X_{(r)}-)-\widehat{H}(X_{(r)})\\}.$

Because $\widehat{S}_{\mathit{RTTR}}(t)$ is an example of $\widehat{H}(t),$ we
can therefore represent

$\displaystyle\widehat{S}_{\mathit{RTTR}}(t)$
$\displaystyle=1-\sum_{r=1}^{k_{t}}\\{\widehat{S}_{\mathit{RTTR}}(X_{(r)}-)-\widehat{S}_{\mathit{RTTR}}(X_{(r)})\\}$
$\displaystyle=1-\sum_{r=1}^{k_{t}}\left\\{J_{(r)}\Delta
N(X_{(k)})I\\{r<m\\}+J_{(m)}I\\{r=m\\}\right\\}.$ (21)

The following proposition extends the known result that
$\widehat{S}_{\mathit{RTTR}}(t)=\widehat{S}_{\mathit{SC}}(t)$ for $t\geq 0$ to
the case of tied data; the results in Efron, (1967) for the case of no ties
(i.e., $m=n$) are easily obtained as a special case. In addition, this result
clarifies the respective connections between $\widehat{S}_{\mathit{RTTR}}(t)$
and the estimators $\widehat{F}_{\mathit{IPCW}}(t)$ and
$\widetilde{S}_{\mathit{IPCW}}(t)$ defined in the previous section.

######  Proposition 5.

Suppose $m\leq n$ (i.e., ties may exist), and let
$\widehat{S}_{\mathit{RTTR}}(t)$ be the estimator of $S(t)$ obtained using the
RTTR algorithm, the jump sizes being given by (20). Then,
$\widehat{S}_{\mathit{RTTR}}(t)=\widehat{S}_{\mathit{SC}}(t)$ for $t\geq 0.$
Moreover, defining
$\widehat{F}_{\mathit{RTTR}}(t)=1-\widehat{S}_{\mathit{RTTR}}(t)$, we have
$\widehat{F}_{\mathit{RTTR}}(t)=\widehat{F}_{\mathit{IPCW}}(t)$ for
$t<X_{(m)}.$ This relationship holds for $t\geq 0$ if $\Delta
N(X_{(m)})=Y(X_{(m)}),$ in which case
$\widehat{S}_{\mathit{RTTR}}(t)=\widetilde{S}_{\mathit{IPCW}}(t)$ for $t\geq
0.$

Proposition 5 shows that the relationship between the IPCW and RTTR estimators
for the cumulative distribution function $F(t)=P\\{T\leq t\\}$ holds in
general provided we consider values of $t$ bounded away from the tail;
equivalence for all $t\geq 0$ only follows under the additional condition that
$\Delta N(X_{(m)})=Y(X_{(m)}).$ Interestingly, the result further shows that
$\widehat{S}_{\mathit{RTTR}}(t)$ does not have a simple IPCW representation
unless this same tail condition holds; rather, when $t<X_{(m)},$ the identity
(14) implies

$\widehat{S}_{\mathit{RTTR}}(t)=1-\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}+\widetilde{S}_{\mathit{IPCW}}(t).$

The result that
$\widehat{S}_{\mathit{RTTR}}(X_{(m)})=\widehat{S}_{\mathit{SC}}(X_{(m)})=0$
together with (21) provides a useful interpretation of the identity (16) in
Proposition 4; in particular, the first summand on the left-hand side of (16)
represents the total mass placed on observations where failures occur, and the
second summand represents the total mass placed on the last observation, or
equivalently, the total mass placed on censored observations. This
interpretation reflects the fact that the RTTR algorithm assigns observations
$X_{(j)}<X_{(m)}$ positive mass only when there is at least one failure
occurring at $X_{(j)};$ hence, the only possible place where any censored
observation(s) can receive mass is at $X_{(m)}.$

Although not explicitly recognized in Efron, (1967), Proposition 5
demonstrates that the RTTR estimator for $F(t)$ is perhaps the first
construction of an IPCW estimator, predating the seminal work of Robins and
Rotnitzky, (1992) by 25 years. Interestingly, in later work, Satten and Datta,
(2001) characterize the RTTR estimator as an imputation rather than an IPCW
estimator, despite the latter arguably providing a more natural
characterization.

##  3 Discussion

The results of this paper reconsider several important estimators for the
survival function and establish direct connections between them in a novel
way. The results presented in Propositions 1-4 highlight the central
importance of the Volterra integral equation of the first kind, as well as the
strong interdependence between the problems of nonparametric estimation for
$S(t)$ and $K(t)$. The novel and transparent proofs make use of known theory
for Volterra equations, and avoid inductive arguments; a supplementary
appendix gives distinct, alternative proofs of Propositions 3 and 4 for
interested readers. We also shed new insight on the form of the estimator
obtained from Efron’s redistribution-to-the-right algorithm, respectively
establishing a closed form representation in the presence of ties, conditions
under which it can be represented as an IPCW estimator, and a new and simple
induction-free proof that this estimator reduces to that generated by the
self-consistency equation (3).

Self-consistency has been considered as a way to motivate estimators in
numerous problems, including problems involving more complex forms of
censoring; see, for example, Turnbull, (1974, 1976), Mykland and Ren, (1996),
Huang and Wellner, (1996), and Mangalam et al., (2008). It would be
interesting to investigate the extent to which analogous arguments may help in
characterizing the solutions to other problems that satisfy a version of the
self-consistency property, and whether there are analogous connections to
appropriate Volterra integral equations and/or IPCW-like estimators. Tsai and
Crowley, (1985) show in general censoring problems that self-consistent
estimators are nonparametric maximum likelihood estimators and hence satisfy
certain asymptotic optimality criteria (Groeneboom and Wellner,, 1992).
Although self-consistency is neither the most convenient nor contemporary
approach to estimator computation, it remains an interesting and useful
approach to characterizing estimators that may have good efficiency
properties.

## Acknowledgement

This revision of an earlier work due to Strawderman, (2023) was initiated
during BRB’s postdoctoral study at the University of Rochester. RLS and BRB
contributed equally to its final content. RLS and BRB thank David Oakes for a
helpful conversation that led to the supplemental proof of Proposition 3; BRB
also thanks David Oakes for an introduction to the redistribute-to-the-right
algorithm.

## References

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## Appendix

### Proof of Proposition 1

###### Proof.

To prove (8), we begin by noting that $t<X_{(m)}$ guarantees that $Y(u+)>0$
for $u\leq t$. Hence, using results briefly summarized in Gill and Johansen,
(1990, p. 1526) for general forms of anticipating Volterra integral equations,
the unique solution to (7) is given by

$\widehat{K}(t)=\left[\Prodi_{(0,t]}\left\\{1+\frac{\mathrm{d}C(u)}{Y(u+)}\right\\}\right]^{-1}=\left[\prod_{(0,t]}\left\\{1+\frac{\Delta
C(u)}{Y(u+)}\right\\}\right]^{-1}.$ (22)

Here, the finite product form given in the last equality on the right-hand
side arises because $C(\cdot)$ can only jump at a finite set of times. To see
that (22) reduces to (8), observe that straightforward algebra yields

$1/\left\\{1+\frac{\Delta C(u)}{Y(u+)}\right\\}=\frac{Y(u+)}{Y(u+)+\Delta
C(u)}=1-\frac{\Delta C(u)}{Y(u+)+\Delta C(u)}.$

Now, we can write $Y(u)-Y(u+)=\Delta N(u)+\Delta C(u);$ hence, $Y(u+)+\Delta
C(u)=Y(u)-\Delta N(u)=Y^{\dagger}(u)$ and it follows immediately that

$\left[\prod_{(0,t]}\left\\{1+\frac{\Delta
C(u)}{Y(u+)}\right\\}\right]^{-1}=\prod_{(0,t]}\left\\{1-\frac{\Delta
C(u)}{Y^{\dagger}(u)}\right\\}=\Prodi_{(0,t]}\left\\{1-\frac{\mathrm{d}C(u)}{Y^{\dagger}(u)}\right\\},$
(23)

establishing (8).

To see that result (9) follows under the assumption that $\Delta N(u)\Delta
C(u)=0$ for $u<X_{(m)}$, we simply note

$\widehat{K}(t)=\prod_{(0,t]}\left\\{1-\frac{\Delta
C(u)}{Y^{\dagger}(u)}\right\\}=\prod_{(0,t]}\left\\{1-\frac{\Delta
C(u)}{Y(u)-\Delta N(u)}\right\\}=\prod_{(0,t]}\left\\{1-\frac{\Delta
C(u)}{Y(u)}\right\\},$

which is equivalent to (9). This latter form is also the known unique solution
to the integral equation for $\widehat{K}(t)$ stated in the theorem for the
case where $\Delta N(u)\Delta C(u)=0$ for $u<X_{(m)};$ see, for example,
Theorem II.6.1 in Andersen et al., (1993, Sec. II.6). ∎

### Proof of Proposition 2

###### Proof.

The result (11) follows immediately for $t=0$ and $t\geq X_{(m)}$ by previous
arguments; hence, we must only show that $\widehat{S}_{\mathit{SC}}(t)$ equals
$\widehat{S}_{\mathit{PL}}(t)$ given in (10) for $t\in(0,X_{(m)}).$

Suppose $\widehat{S}(t)$ solves (4), equivalently (3), for $t\in(0,X_{(m)}).$
Proposition 1 and the definition of $\widehat{K}(t)$ in (6) then imply that
the solution $\widehat{S}(t)$ to (4) necessarily must also satisfy

$\widehat{S}(t)\times\prod_{(0,t]}\left\\{1-\frac{\Delta
C(u)}{Y^{\dagger}(u)}\right\\}=\frac{Y(t+)}{n}.$

The crux of the proof thus involves determining a suitable relationship
between $n^{-1}Y(t+)$ and the product-limit form of $\widehat{K}(t)$ in
Proposition 1.

We begin by finding a comparable finite-product representation for
$n^{-1}Y(t+)$. Using a telescoping product and the fact that $Y(0+)=n$, we can
write

$\frac{Y(t+)}{n}=\frac{Y(t+)}{Y(0+)}=\prod_{0<u\leq
t}\frac{Y(u+)}{Y(u)}=\prod_{0<u\leq t}\left\\{1-\frac{\Delta
N(u)}{Y(u)}-\frac{\Delta C(u)}{Y(u)}\right\\},$

where the last equality follows from $Y(u+)=Y(u)-\Delta N(u)-\Delta C(u)$.
Now, since $Y^{\dagger}(u)=Y(u)-\Delta N(u),$ observe that

$\frac{\Delta C(u)}{Y(u)}=\frac{Y^{\dagger}(u)}{Y(u)}\frac{\Delta
C(u)}{Y^{\dagger}(u)}=\left\\{1-\frac{\Delta N(u)}{Y(u)}\right\\}\frac{\Delta
C(u)}{Y^{\dagger}(u)};$

hence,

$1-\frac{\Delta N(u)}{Y(u)}-\frac{\Delta C(u)}{Y(u)}=\left\\{1-\frac{\Delta
N(u)}{Y(u)}\right\\}\left\\{1-\frac{\Delta C(u)}{Y^{\dagger}(u)}\right\\}.$

Taking the product of both sides over all jumps, we see that

$\displaystyle\frac{Y(t+)}{n}$
$\displaystyle=\prod_{(0,t]}\left\\{1-\frac{\Delta
N(u)}{Y(u)}\right\\}\left\\{1-\frac{\Delta C(u)}{Y^{\dagger}(u)}\right\\}$
$\displaystyle=\prod_{(0,t]}\left\\{1-\frac{\Delta
N(u)}{Y(u)}\right\\}\times\prod_{(0,t]}\left\\{1-\frac{\Delta
C(u)}{Y^{\dagger}(u)}\right\\}$
$\displaystyle=\Prodi_{(0,t]}\left\\{1-\frac{\mathrm{d}N(u)}{Y(u)}\right\\}\times\widehat{K}(t).$
(24)

Therefore, since $\widehat{K}(t)\widehat{S}(t)=n^{-1}Y(t+),$ we have

$\widehat{S}(t)=\frac{n^{-1}Y(t+)}{\widehat{K}(t)}=\Prodi_{(0,t]}\left\\{1-\frac{\mathrm{d}N(u)}{Y(u)}\right\\}=\widehat{S}_{\mathit{PL}}(t),$

the latter being given in (10) and proving the desired result. We note here
that the relationship derived in (24) is that originally given in Gill, (1980,
p. 36); to see this, note that (i) $Y^{\dagger}(t)=Y(t)-\Delta N(t);$ and,
(ii) $Y(t+)=Y(t)-\Delta N(t)-\Delta C(t)$ gives $\Delta C(t)=Y(t)-Y(t+)-\Delta
N(t).$ ∎

### Proof of Proposition 3

###### Proof.

The result is trivial for $t=0$. For $t>0,$ previous results imply that the
relation (6) can be rewritten as
$\widehat{K}(u)\widehat{S}_{\mathit{PL}}(u)=Y(u+)/n;$ as seen in (24), this
relation is in fact valid for $u>0.$ Taking the limit of this equation from
the left, it follows that
$\widehat{K}(u-)\widehat{S}_{\mathit{PL}}(u-)=Y(u)/n$ for $u>0$. Since
$Y(u)>0$ for $u\leq X_{(m)}$, we have $\widehat{K}(u-)>0$ and
$\widehat{S}_{\mathit{PL}}(u-)>0$ for $u\leq X_{(m)}$. Substitution of
$\widehat{S}_{\mathit{PL}}(u-)=n^{-1}Y(u)/\widehat{K}(u-)$ into the right-hand
side of the integral equation (12) then gives

$\widehat{F}_{\mathit{PL}}(t)=1-\widehat{S}_{\mathit{PL}}(t)=\frac{1}{n}\int_{(0,t]}\frac{\mathrm{d}N(u)}{\widehat{K}(u-)}.$

This last expression remains well-defined for $t>X_{(m)}$ since
$\widehat{K}(X_{(m)}-)>0$ and $\Delta N(u)=0$ for $u>X_{(m)}$. The desired
result now follows immediately upon evaluating the integral. ∎

### Proof of Proposition 4

###### Proof.

The proof of Proposition 3 and (14) imply

$\widehat{S}_{\mathit{PL}}(X_{(m)})=1-\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}-\widetilde{S}_{\mathit{IPCW}}(X_{(m)});$

since $X_{(m)}$ is the largest observation time and $\widehat{K}(X_{i}-)>0$
for each $i,$ it is trivially true that
$\widetilde{S}_{\mathit{IPCW}}(X_{(m)})=0.$ Consequently,

$\widehat{S}_{\mathit{PL}}(X_{(m)})=1-\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)},$

and (16) therefore holds if and only if

$\widehat{S}_{\mathit{PL}}(X_{(m)})=\frac{\Delta
C(X_{(m)})}{n\widehat{K}(X_{(m)}-)}.$

Towards this end, note that

$\widehat{S}_{\mathit{PL}}(X_{(m)})=\widehat{S}_{\mathit{PL}}(X_{(m)}-)\left(1-\frac{\Delta
N(X_{(m)})}{Y(X_{(m)})}\right)=\widehat{S}_{\mathit{PL}}(X_{(m)}-)\frac{\Delta
C(X_{(m)})}{Y(X_{(m)})},$

the last equality following from the fact that $Y(X_{(m)})=\Delta
N(X_{(m)})+\Delta C(X_{(m)})$. Since
$\widehat{K}(u-)\widehat{S}_{\mathit{PL}}(u-)=Y(u)/n$ for $u>0,$ we have for
$u=X_{(m)}$ that

$\frac{\widehat{S}_{\mathit{PL}}(X_{(m)}-)}{Y(X_{(m)})}=\frac{1}{n\widehat{K}(X_{(m)}-)},$

proving (16). In view of the fact that $Y(X_{(m)})=\Delta N(X_{(m)})+\Delta
C(X_{(m)}),$ we also see that (15) holds if and only if $\Delta C(X_{(m)})=0$
$\Leftrightarrow$ $Y(X_{(m)})=\Delta N(X_{(m)}).$ Finally, it is immediate
from (14) that
$\widetilde{S}_{\mathit{IPCW}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t)$ for $t\geq
0$ if and only if (15) holds. ∎

### Proof of Proposition 5

###### Proof.

Because the RTTR algorithm starts with a total mass of one, the preservation
of mass through redistribution implies that

$\sum_{k=1}^{m-1}J_{(k)}\Delta N(X_{(k)})+Y(X_{(m)})J_{(m)}=1.$ (25)

Hence, using (21), we find that

$\displaystyle\widehat{S}_{\mathit{RTTR}}(t)$ $\displaystyle=$ $\displaystyle
1-\sum_{r=1}^{k_{t}}\left(J_{(r)}\Delta
N(X_{(r)})I\\{r<m\\}+J_{(m)}Y(X_{(m)})I\\{r=m\\}\right)$ $\displaystyle=$
$\displaystyle\begin{cases}1-\sum_{r=1}^{m-1}J_{(r)}\Delta
N(X_{(r)})-J_{(m)}Y(X_{(m)})&t\geq X_{(m)}\\\
1-\sum_{r=1}^{k_{t}}J_{(r)}\Delta N(X_{(r)})&t<X_{(m)}\end{cases}$
$\displaystyle=$ $\displaystyle\begin{cases}0&t\geq X_{(m)}\\\
1-\sum_{r=1}^{k_{t}}\frac{\Delta
N(X_{(r)})}{n\widehat{K}(X_{(r)}-)}&t<X_{(m)}\end{cases},$

the last representation respectively following from (25) and (20). Focusing on
the case where $t<X_{(m)},$ it is easily seen that

$\sum_{r=1}^{k_{t}}J_{(r)}\Delta N(X_{(r)})=\sum_{i=1}^{n}I\\{X_{i}\leq
t\\}\frac{D_{i}}{n\widehat{K}(X_{i}-)}=\widehat{F}_{\mathit{IPCW}}(t),$

where $\widehat{F}_{\mathit{IPCW}}(t)$ is defined in (13). Therefore,

$\widehat{S}_{\mathit{RTTR}}(t)=\begin{cases}0&t\geq X_{(m)}\\\
1-\widehat{F}_{\mathit{IPCW}}(t)&t<X_{(m)}\end{cases}\,=\,\begin{cases}0&t\geq
X_{(m)}\\\ \widehat{S}_{\mathit{PL}}(t)&t<X_{(m)}\end{cases},$

the last equality following by Proposition 3 and (14). The result (11) in
Proposition 2 now establishes the desired equivalence, that is,
$\widehat{S}_{\mathit{RTTR}}(t)=\widehat{S}_{\mathit{SC}}(t)$ for $t\geq 0.$

Moreover, these same calculations prove that
$\widehat{F}_{\mathit{RTTR}}(t)=1-\widehat{S}_{\mathit{RTTR}}(t)=\widehat{F}_{\mathit{IPCW}}(t)$
for $t<X_{(m)};$ by Proposition 4, this relationship between the IPCW and RTTR
estimator holds for $t\geq 0$ under the additional condition that $\Delta
N(X_{(m)})=Y(X_{(m)})$ (i.e., all observations at the last observation time
are failures), in which case
$\widehat{S}_{\mathit{RTTR}}(t)=\widetilde{S}_{\mathit{IPCW}}(t)$ for $t\geq
0.$

The arguments above are predicated on the validity of (25); we now provide
independent verification that (25) holds. Recalling the fact that
$Y(X_{(m)})=\Delta N(X_{(m)})+\Delta C(X_{(m)}),$ we can rewrite the left-hand
side of (25) as

$\sum_{k=1}^{m}J_{(k)}\Delta N(X_{(k)})+\Delta C(X_{(m)})J_{(m)}.$

Using the definition of $J_{(k)},$ it is easy to show that

$\sum_{k=1}^{m}J_{(k)}\Delta
N(X_{(k)})=\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}$

and

$\Delta C(X_{(m)})J_{(m)}=\frac{\Delta C(X_{(m)})}{n\widehat{K}(X_{(m)}-)};$

hence, the left-hand side of (25) reduces to

$\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}+\frac{\Delta
C(X_{(m)})}{n\widehat{K}(X_{(m)}-)},$

which is indeed equal to one by (16). ∎

## Supplementary Appendix

In this supplementary appendix, we respectively provide alternative proofs of
Propositions 3 and 4 that may be of independent interest to readers.

### Alternative proof of Proposition 3

The following proof uses a different approach than that given in the main
Appendix, and represents a minor generalization of a proof communicated by
David Oakes in the case that $\Delta N(\cdot)\Delta C(\cdot)=0$. Under this
condition, the proof below can be considerably simplified and made more
transparent by only considering $t>0$ such that
$\Delta\widehat{S}_{\mathit{PL}}(t)=\widehat{S}_{\mathit{PL}}(t-)-\widehat{S}_{\mathit{PL}}(t)>0$.

###### Proof.

Consider $t<X_{(m)}$. We may see that

$\Delta\left\\{\frac{Y(t+)}{n}\right\\}=\frac{\Delta N(t)+\Delta C(t)}{n}$
(26)

and

$\displaystyle\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\widehat{K}(t)\right\\}$
$\displaystyle=\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\right\\}\widehat{K}(t)+\widehat{S}_{\mathit{PL}}(t-)\Delta\left\\{\widehat{K}(t)\right\\}$
(27)
$\displaystyle=\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\right\\}\widehat{K}(t-)\left\\{1-\frac{\Delta
C(t)}{Y^{\dagger}(t)}\right\\}+\widehat{S}_{\mathit{PL}}(t-)\widehat{K}(t-)\frac{\Delta
C(t)}{Y^{\dagger}(t)}$
$\displaystyle=\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\right\\}\widehat{K}(t-)\frac{Y(t+)}{Y^{\dagger}(t)}+\frac{Y(t)}{n}\frac{\Delta
C(t)}{Y^{\dagger}(t)}.$

Since $Y(u+)/n=\widehat{S}_{\mathit{PL}}(u)\widehat{K}(u)$ for all $u>0$, the
increments $n^{-1}\Delta Y(t+)$ and
$\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\widehat{K}(t)\right\\}$ must be
equal. Equating these and rearranging terms, we have derived that

$\displaystyle
n\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\right\\}\widehat{K}(t-)$
$\displaystyle=\frac{Y^{\dagger}(t)}{Y(t+)}\left\\{\Delta N(t)+\Delta
C(t)-\frac{Y(t)}{Y^{\dagger}(t)}\Delta C(t)\right\\}$
$\displaystyle=\frac{Y^{\dagger}(t)}{Y(t+)}\left\\{\Delta N(t)-\frac{\Delta
C(t)\Delta N(t)}{Y^{\dagger}(t)}\right\\}$ $\displaystyle=\Delta
N(t)\frac{Y^{\dagger}(t)}{Y(t+)}\left\\{1-\frac{\Delta
C(t)}{Y^{\dagger}(t)}\right\\}$ $\displaystyle=\Delta N(t).$

Therefore

$\displaystyle\widehat{S}_{\mathit{PL}}(t)$ $\displaystyle=1-\sum_{u\leq
t\,:\,\Delta
N(u)>0}\Delta\left\\{\widehat{S}_{\mathit{PL}}(u)\right\\}=1-\sum_{u\leq
t\,:\,\Delta N(u)>0}\frac{\Delta
N(u)}{n\widehat{K}(u-)}=1-\frac{1}{n}\int_{(0,t]}\frac{\mathrm{d}N(u)}{\widehat{K}(u-)}.$

Evaluating the integral as in the main proof, we obtain
$\widehat{S}_{\mathit{PL}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t)$, as desired.
The behavior for $t\geq X_{(m)}$ follows as in the previous proof. ∎

We now show how the proof can be simplified in the special case that $\Delta
N(\cdot)\Delta C(\cdot)=0$. Consider $t\in(0,X_{(m)})$ such that
$\Delta\widehat{S}_{\mathit{PL}}(t)=\widehat{S}_{\mathit{PL}}(t-)-\widehat{S}_{\mathit{PL}}(t)>0$.
Then it must be that $\Delta N(t)>0$ and hence that $\Delta C(t)=0$. The first
equation, (26), simplifies to

$\Delta\left\\{\frac{Y(t+)}{n}\right\\}=\frac{\Delta N(t)}{n}$

and the second equation, (27), simplifies to

$\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\widehat{K}(t)\right\\}=\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\right\\}\widehat{K}(t),$

since $\Delta\left\\{\widehat{K}(t)\right\\}=\widehat{K}(t-)\Delta
C(t)/Y^{\dagger}(t)=0$. Notice that no calculation was required for either of
these equalities. The conclusion that
$n\Delta\left\\{\widehat{S}_{\mathit{PL}}(t)\right\\}\widehat{K}(t-)=\Delta
N(t)$ follows immediately.

### Alternative proof of Proposition 4

Below, we present an alternative proof of Proposition 4, specifically the
derivation of conditions under which (15) holds, as well as the identity (16).

###### Proof.

Let $X_{(f)}\leq X_{(m)}$ be the largest observation time such that there is
at least one observation $X_{i}$ with $D_{i}=1;$ substitution of $t=X_{(f)}$
into (14) gives

$\widehat{S}_{\mathit{PL}}(X_{(f)})=1-\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}-\widetilde{S}_{\mathit{IPCW}}(X_{(f)}).$

Observe that the definition of $X_{(f)}$ implies

$\widetilde{S}_{\mathit{IPCW}}(X_{(f)})=\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}I\\{X_{i}>X_{(f)}\\}}{\widehat{K}(X_{i}-)}=0$

since $\widehat{K}(X_{i}-)>0$ and $D_{i}I\\{X_{i}>X_{(f)}\\}=0$ for
$i=1,\ldots,n.$ Consequently,

$\widehat{S}_{\mathit{PL}}(X_{(f)})=1-\frac{1}{n}\sum_{i=1}^{n}\frac{D_{i}}{\widehat{K}(X_{i}-)}.$

The last equality above clearly implies (15) holds if and only if
$\widehat{S}_{\mathit{PL}}(X_{(f)})=0.$ Since

$\widehat{S}_{\mathit{PL}}(X_{(f)})=\left[\Prodi_{(0,X_{(f)})}\left\\{1-\frac{\mathrm{d}N(u)}{Y(u)}\right\\}\right]\left\\{1-\frac{\Delta
N(X_{(f)})}{Y(X_{(f)})}\right\\}$

and the first term in the product on the right-hand side never vanishes, it
follows that $\widehat{S}_{\mathit{PL}}(X_{(f)})=0$ if and only if $\Delta
N(X_{(f)})=Y(X_{(f)}).$ However, such an equality can only occur when (i)
$X_{(f)}=X_{(m)}$ and (ii) that all observations with $X_{i}=X_{(m)}$ have
$D_{i}=1.$ In general, $X_{(f)}=X_{(m)}$ is necessary, but not sufficient, for
$\Delta N(X_{(m})=Y(X_{(m)})$ since both failure and censoring can occur at
$X_{(m)}.$

To establish (16), we first note that
$\widehat{S}_{\mathit{PL}}(X_{(f)})=\widehat{S}_{\mathit{PL}}(X_{(m)}).$ This
result is immediate when $f=m,$ so consider $f<m.$ In this case, by definition
of $X_{(f)},$ we must have $\Delta N(X_{(j)})=0$ for $j=f+j,$
$j=1,\ldots,m-f.$ Consequently, by considering the behavior of the product
integral representation for $\widehat{S}_{\mathit{PL}}(\cdot),$ it is easily
shown that
$\widehat{S}_{\mathit{PL}}(X_{(f)})=\widehat{S}_{\mathit{PL}}(X_{(m)}).$
Establishing (16) now follows as in the main proof; in particular, since

$\widehat{S}_{\mathit{PL}}(X_{(m)})=\frac{\Delta
C(X_{(m)})}{n\widehat{K}(X_{(m)}-)}$

and $\widehat{S}_{\mathit{PL}}(X_{(f)})=\widehat{S}_{\mathit{PL}}(X_{(m)}),$
the identity (16) must hold. That
$\widetilde{S}_{\mathit{IPCW}}(t)=1-\widehat{F}_{\mathit{IPCW}}(t)$ for $t\geq
0$ if and only if (15) holds also follows as in the main proof. ∎