Yu et al. (2012) considered a certain dependent right censorship model. We show that this model is equivalent to the independent right censorship model, extending a result with continuity restriction in Williams and Lagakos (1977). Then the asymptotic normality of the product limit estimator under the dependent right censorship model follows from the existing results in the literature under the independent right censorship model, and thus partially solves an open problem in the literature.
Received 8 February 2016; accepted 9 April 2016; published 12 April 2016
In this paper we study various dependent right censorship (RC) models and their relation to the independent RC model in the literature. The definitions of these RC models are given in Definition 1.
Right censored data occur quite often in industrial experiments and medical research. A typical example in medical research is a follow-up study; a patient is enrolled and has a certain treatment within the study period. If the patient dies within the study period, we observe the exact survival time T; otherwise, we only know that the patient survives beyond the censoring time R. Thus the observable random vector is, where
() and, the indicator function of the event. Let
be i.i.d. copies of. Let be the cumulative distribution function (cdf) of T and. Denote FR, FV and the cdf’s of R, V and, respectively, and the conditional cdf of R given T and. Let fT (fR or) be the density function of T (R or) (with respect to (w.r.t.) some measure). The common right censorship model assumes T and R are independent (). Then the likelihood (function) for RC data is often defined as
(see  ), where is a collection of all cdf’s if under the non-parametric set-up, or a parametric cdf family with a parameter, say and is the parameter space, and f is the density of F. Recall that the formal definition of the likelihood (function) for a sample is,. More- over, if , where does not depend on θ, and Λ2 is also called a likelihood. We shall call Λ the full likelihood and Λ2 a simplified one. Since our sample,
and the integrals are Lebesgure integrals. We say that a function is non-informative about the function with if it is not assumed that H is a function of or θ. We shall further clarify what “non- informative” means in the next example.
Example 1.1. Consider 3 cases of right censoring:
Case (1). with the parameter space (see Equation (1)).
Case (2). with.
Case (3). with parameters,. FR is informative about
FT in cases (2) and (3), as it is a function of FT in case (2) and a function of in case (3). However, FR is non-informative (not informative) about FT in case (1), as FT and FR are both independent parameters.
If, Λ in Equation (2) may be simplified as as in Equation (1) due to the non-informative property by the well-known result as follows.
Proposition 1.1. The full likelihood can be simplified as (see Equation (1)) iff
Example 1.1 (continued). In case (1), is a likelihood function, as Equation (4) holds. In case (2), is informative about FT and condition Equation (4) fails.
is not a likelihood, but can be viewed as a partial likelihood. The generalized maximum likelihood (GMLE)
of ST based on is still the PLE, i.e., , where is the i-th order statistic of’s and is the δj that is associated with. The variance satisfies (if ST is continuous), while the GMLE based on Λ is (as and) and by the delta method. Thus the PLE is not efficient. In case (3), is not a likelihood function. The full likelihood is (as ). If one treats as a (partial) likelihood, then its GMLE of ST is the PLE. Let,. Then, i.e., the PLE is not consistent at 2. The GMLE based on Λ is.
Remark 1.1. Example 1.1 indicates that if Equation (4) is not valid then the MLE based on so-called “likelihood” as in Equation (1) can be inconsistent, or can be less efficient than the MLE based on Λ due to loss of information on. However, it is difficult to verify Equation (4) in practical applications, thus people propose some sufficient conditions. A typical sufficient condition of Equation (4) is that and FR is non-informative about FT.
Williams and Lagakos (W&L)  point out that is often un-realistic. They further propose a constant-sum model (which allows) as follows.
In the literature, there are many studies on the asymptotic properties of the PLE by weakening the assumptions in the independent RC model over the years (see, e.g.,  -  ). It is conceivable that the asymptotic properties of the PLE is difficult under the continuous constant-sum model in Equation (5). However, the next theorem makes it trivial.
W&L Theorem (Theorem 3.1 in  ). W&L (1977)). Suppose that is a continuous random vector. Then Equation (5) holds iff a random vector such that (1), (see Equation (6)) and, where (2) if, and (3) if.
By the W&L Theorem, one can easily make use of the existing results about the PLE under the assumption to establish asymptotic properties of the PLE under the continuous constant-sum model. Indeed, by (2) and (3) of the W&L Theorem,
Since, the PLE based on from (see Equation (7)) satisfies
a.s., where (see  ). By the W&L Theorem, and
Equation (7) holds, so under the continuous RC model given in Equation (5), even if.
On the other hand, case (3) in Example 1.1 shows that the PLE can be inconsistent for under a dependent RC model. Hence the W&L Theorem is quite significant. Yu et al.  show that the PLE is consistent under the dependent RC model considered in  -  , etc., which assumes A1 and A2 as follows.
A1 for all r, or equivalently, a.e. in t (w.r.t.) on the set.
Notice that is well defined if and undefined if. We define if. Notice that A1 says that is constant in, thus is well defined if.
A2 is non-informative about, with.
Definition 1. If and FR is non-informative about FT, then we call the RC model the independent RC model. The dependent RC model considered in this paper assumes that A1 and A2 hold.
Next example and Example 3.1 in Section 3 are examples that satisfies A1 but.
Example 1.2., and T has a binomial distribution () with parameter.
Yu et al.  show that A1 and A2 are the necessary and sufficient (N&S) condition of Equation (4) under the non-parametric set-up. Then we may ask the following questions:
1) Are A1 and A2 the N&S condition of Equation (4) under the parametric set-up?
2) What is the relation between the constant-sum model (5) and A1?
3) Can the W&L Theorem be extended by eliminating the continuity restriction?
We give answers to the 3 questions. In Section 2, we show that A1 and A2 are a sufficient condition for Equation (4) under both non-parametric set-up and non-parametric set-up (see Theorem 2.1). Our study suggests that the constant sum model (5) is a special case of A1. In Section 3, we extend the W&L Theorem to the case that A1 holds (rather than the case that Equation (5) holds), which allows being discontinuous. As a consequence, we establish the asymptotic normality of the PLE under the dependent RC model and under certain regularity conditions, making use of the existing results in the literature about the PLE under the independent RC model. In Section 4, we show that under the parametric set-up, A1 and A2 are not a necessary condition of Equation (4). Section 5 is a concluding remark. Some detailed proofs are relegated to Appendix.
2. The Relation between Equation (4), Equation (5) and A1
We shall first show that A1 and A2 are a sufficient condition of Equation (4), extending a result in  under the non-parametric set-up. Then we shall show that if is continuous, the constant sum model is the same as A1; otherwise, these two models are different.
Theorem 2.1. Equation (4) holds if A1 and A2 hold.
Proof. Since, it is non-informative about FT by A2. Moreover, by A1. Thus
is non-informative about FT by A1 and A2, as and are equivalent. Then Equation (4) holds. ,
The next example and lemma help us to understand the constant-sum model (5).
Example 2.1. Suppose, and. Then A1 holds, but not the constant-sum model assumption (5), as Equation (6) yields, and . Thus, violating Equation (5).
Lemma 2.1. and, if is continuous, where
Theorem 2.2. If is continuous, then A1 and Equation (5) are equivalent.
The proofs of Lemma 2.1 and Theorem 2.2 are very technical but not difficult. For a better presentation, we relegate them to Appendix (see Section A.1 and Section A.2).
Remark 2.1. Example 2.1 shows that A1 is not a special case of Equation (5) (or the constant-sum model). However, if is continuous, A1 and the constant-sum model are equivalent. Thus the continuous constant-sum model is a special case of A1. Since Yu et al.  show that under the non-parametric set-up, A1 and A2 are the N&S condition that Equation (4) holds, it is desirable to extend W&L Theorem to the model that assumes A1 rather than the constant-sum model by eliminating the continuity assumption.
3. Extension of the W&L Theorem
In the next theorem, we extend the W&L Theorem from the continuous constant-sum model to A1.
Theorem 3.1. A1 holds iff there exist extended random variables Z and Y such that 1) and
, where, 2) if, and 3) if.
In our theorem, there are two modifications to the W&L Theorem.
1) Equation (5) with continuous is replaced by A1 without continuity assumptions.
2) The random vector is replaced by the extended random vector.
In fact, W&L Theorem is not accurate as stated, unless a random variable is allowed to take “values” (see Examples 3.1 and 3.2 below). However, by the common definition of a random variable, it does not take values. Thus the random variables in their theorem should be referred to the extended random variables.
Example 3.1. Suppose that and T has a uniform distribution
, then A1 holds and is a continuous random vector, but. By Theorem 2.2, it satisfies the constant-sum model. Consequently, the assumptions in the W&L Theorem are satisfied. In particular, R does not take the value. If the W&L Theorem were correct, according to their definition, there would be a random variable Y with a cdf defined in Equation (6). However, for (the proof is given in Appendix (see Section A.3)).
Thus is not a proper cdf as claimed in the W&L Theorem. Y should be modified to be an extended
random variable such that
Example 3.2. A random sample of complete data from T which has the exponential distribution can be viewed as a special case of the RC data. But is not even defined for a random variable R. However, if we consider extended random variables in A1, that is, R may take values, then we can define. Since,. Thus Theorem 3.1 is trivially true in such case.
Proof of Theorem 3.1. It suffice to show (Þ) part. Since is a conditional distribution, defines a “cdf” on. Denote and let be the Borel s-field on Ω. Without loss of generality (WLOG), one can assume that is the probability space such that . Let W be the joint cdf defined by . By the Kolmogorov consistency theorem, induces a random vector on Ω by . Note that and. Verify that if; if. Verify that as. Thus con- ditions (1), (2) and (3) hold. ,
Remark 3.1. In the previous proof, let be the support of and. Then may not be defined on A, but may be defined on A. Thus it is necessary to create a new random variable Z.
Corollary 3.1. If A1 holds then and.
The asymptotic properties of the PLE under the continuous constant-sum model are obtained by making use of the W&L Theorem and the existing results in the literature on the PLE under the continuous independent RC
model. Denote The consistency of the PLE under assumption A1 is es-
tablished in the literature as follows.
Theorem 3.2 (Yu et al.  ). Under A1,.
Now by Theorem 3.1 and Corollary 3.1, we can construct another proof of the consistency of the PLE as follows.
Corollary 3.2. Under A1, where
Proof. Yu and Li  show that if, then, where
Under A1, may not be true, but by Theorem 3.1,
, ,. Thus the observation’s are i.i.d. from, which can be viewed as being generated from, as well as. Thus replacing and
by and, respectively, in the previous equation yields. Now
since, the proof is done. ,
Remark 3.2. Notice that the statements in Theorem 3.2 is slightly different from the statements in Corollary 3.2. One is based on, and the other is based on.
The asymptotic normality of the PLE under A1 without continuity assumption has not been established in the literature. It can be done now by making use of Theorem 3.1 and the existing results in the literature on the PLE under the independent RC model. In particular, assuming T is continuous, Breslow and Crowley  and Gill  show that
Without continuity assumptions, Gu and Zhang  and Yu and Li  among others established asymptotic normality of the GMLE under the double censorship (DC) model. Since the independent RC model is a special
case of the DC model, their results imply that (8) and (9) also hold if , and if
either or. The next result follows from Theorem 3.1 and Corollary 3.1, which partially solves the open problem in  about the asymptotic normality of the PLE under the dependent RC model.
Theorem 3.3. Equations (8) and (9) are valid if A1 holds and if either T is continuous or (1)
and (2) either or, where the random variable Y is defined in Theorem 3.1.
4. Are A1 and A2 the N&S Condition of Equation (4) under the Parametric Set-Up?
The answer to the question is “No” in general. We shall explain through several examples.
Example 4.1. Suppose that, , , and
where and. This defines a parametric family of dis- crete distribution functions FT. One can verify that possible observations Ii’s are, , , ,. Write, then is either Q or G in Equation (3). In particular, , , which lead to
Thus the parametric model satisfies the N&S condition Equation (4). But in view of, A1 fails. Note that the PLE maximizes over. It is important to notice that with is not a likelihood. However, with is a likelihood by Proposition 1.1. Ve-
rify that the PLE of, thus the PLE is not consistent,
but the MLE which maximizes over is consistent, as expected. In fact,
which is of the form.
Then the MLE is the one that. Verify that
Since, the MLE a.s. as expected. That is, the MLE of p based on is consistent.
Example 4.2. Suppose that and. This specifies a parametric family of discrete distributions with parameter subject to the constraint and. Then A1 and A2 are the N&S condition of Equation (4) (see Section A.4 in Appendix).
Remark 4.1. In Example 4.1, since A1 fails, the W&L Theorem does not hold.
Both Examples 4.1 and 4.2 are parametric cases, but A1 and A2 are the N&S condition of Equation (4) only in one case. In both cases the MLE’s based on the simplified likelihood as in Equation (1) are consistent. They indicate that in general under the parametric set-up, A1 is not the necessary condition of Equation (4). Since the two examples are discrete case, we also discuss two continuous examples.
Example 4.3. Suppose that T is continuous,
This defines a parametric family of a continuous random variable with parameter p. The possible observations Ii’s are and. A1 is violated due to the table for. The and in Equation (3). satisfy
Thus both Q and G in Equation (4) are not functions of p or FT and Equation (4) holds. Hence in this example, A1 is not a necessary condition of Equation (4).
Example 4.4. Suppose that and. Define as in Exam-
ple 4.3, then A1 fails and Equation (4) holds for the random vector. Now define
,. Then does not satisfy A1 but Equation (4) holds.
It shows that if, A1 is not a necessary condition of Equation (4) though Equation (4) can hold under proper assumptions on. The idea can be extended to the other continuous parametric families e.g., , Weibull, Gamma etc.
5. Concluding Remark
We have established the equivalence between the standard RC model and the dependent RC model. The result simplifies the study on the properties of the estimators under the dependent RC model. The results in this paper may have applications in linear regression with right-censored data. For instance, the model assumption considered in  can be weekend. It is also of interest to study whether the result can be extended to the double censorship model  and the mixed interval censorship model  .
We thank the Editor and the referee for their valuable comments.
We shall give the proofs of Lemma 2.1 and Theorem 2.2 and the proofs in some examples of the paper here.
A1. Proof of Lemma 2.1
WLOG, one can assume that u satisfies. By Equation (6),
If T is a continuous random variable, then the previous equation and Equation (6) yield
A2. Proof of Theorem 2.2
Assume that is continuous. Then by Lemma 2.1, Equation (5) holds iff a.e. in u w.r.t., iff a.e. in u w.r.t..
Since is continuous,. By Lemma 2.1,
where. Thus, Equation (5) holds iff a.e. in u (w.r.t.);
iff a.e. in u (w.r.t.);
iff a.e. in u;
iff a.e. in u (w.r.t.);
iff for almost all r (w.r.t.), is constant in t a.e. w.r.t. on;
iff, is constant in t a.e. w.r.t. on (which is A1). ,
A3. Proof of the Equation for in Example 3.1
A4. Proof of Example 4.2
If is not constant a.e. (w.r.t.) in, such that
3) is fixed.
Thus, contradicting by assumption (2). ,