Equivalence between the Dependent Right Censorship Model and the Independent Right Censorship Model

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.


Introduction
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 ( ) , the indicator function of the event { } , , , , f ) be the density function of T (R or R T ) (with respect to (w.r.t.) some measure).The common right censorship model assumes T and R are independent ( T R ⊥ ).Then the likelihood (function) for RC data is often defined as (see [1]), where  is a collection of all cdf's if under the non-parametric set-up, or a parametric cdf family with a parameter, say θ ∈ Θ and Θ is the parameter space, d , , , 0, , and the integrals are Lebesgure integrals.We say that a function ( ) . F R is informative about F T in cases (2) and (3), as it is a function of F T in case (2) and a function of ( ) , , , θ θ θ θ in case (3).However, F R is non-informative (not informative) about F T in case (1), as F T and F R are both independent parameters.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 Λ can be simplified as ( ) Example 1.1 (continued).In case (1), ( ) is informative about F T and condition Equation (4) fails. ( . is not a likelihood, but can be viewed as a partial likelihood.The generalized maximum likelihood (GMLE) Ŝ of S T based on ( ) by the delta method.Thus the PLE is not efficient.In case (3), ( ) ).If one treats ( ) as a (partial) likelihood, then its GMLE of S T is the PLE 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 θ ∈ Θ .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 T R ⊥ and F R is non-informative about F T .
Williams and Lagakos (W&L) [2] point out that T R ⊥ is often un-realistic.They further propose a constant-sum model (which allows T R ⊥ / ) as follows.
( ) ( ) ( ) 1 a.e.w.r.t. the measure induced by , 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., [3]- [10]).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.
Then Equation (5) holds iff ∃ a random vector ( ) By the W&L Theorem, one can easily make use of the existing results about the PLE under the assumption

T R ⊥
to establish asymptotic properties of the PLE under the continuous constant-sum model.Indeed, by ( 2) and (3) of the W&L Theorem, (see [10]).By the W&L Theorem, T Z F F = and Equation ( 7) holds, so On the other hand, case (3) in Example 1.1 shows that the PLE can be inconsistent for t τ < under a depen- dent RC model.Hence the W&L Theorem is quite significant.Yu et al. [11] show that the PLE is consistent under the dependent RC model considered in [12]- [15], etc., which assumes A1 and A2 as follows.A1 Notice that and F R is non-informative about F T , 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 T R ⊥ / .
Example 1.2.( ) and T has a binomial distribution ( ( ) [11] 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 ( ) , T R 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.

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 [11] under the non-parametric set-up.Then we shall show that if ( ) , T R is continuous, the constant sum model is the same as A1; otherwise, these two models are different.
Proof.Since  ( ) and T R ⊥ .Then A1 holds, but not the constant-sum model assumption (5), as Equation (6) yields ( ) ( ) However, if ( ) , T R 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. [11] 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.

Extension of the W&L Theorem
In the next theorem, we extend the W&L Theorem from the continuous constant-sum model to A1.
In our theorem, there are two modifications to the W&L Theorem.
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 ( ) , then A1 holds and ( ) , T R is a continuous random vector, but T R ⊥ / .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 ( ) B ⋅ defined in Equation (6).However, ( ) Thus ( ) B ⋅ is not a proper cdf as claimed in the W&L Theorem.Y should be modified to be an extended random variable such that ( ) ( ) ( ) from T which has the exponential distribution ( ) T Exp µ can be viewed as a special case of the RC data.But | R T f 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 R = ∞ .Since ( ) , Ω = −∞ ∞ and let  be the Borel σ-field on Ω.Without loss of generality (WLOG), one can assume that ( ) otherwise.
The consistency of the PLE under assumption A1 is established in the literature as follows.
Theorem 3.2 (Yu et al. [11]).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, ( ) ( ) Proof.Yu and Li [10] show that if T R ⊥ , then . Thus the observation ( ) which can be viewed as being generated from ( ) , T R , as well as ( ) 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 [3] and Gill [6] show that ( ) , for , Without continuity assumptions, Gu and Zhang [16] and Yu and Li [17] 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 that (9) also hold if , where the random variable Y is defined in Theorem 3.1.

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.: . This defines a parametric family of discrete distribution functions F T .One can verify that possible observations I i 's are ( )

.
= ≤ be the cumulative distribution function (cdf) of T and 1 T T S F = − .Denote F R , F V and , T R F the cdf's of R, V and ( ) , T R , respectively, and | Let f T (f R or | R T

GF
⋅ is non-informative about F T by A1 and A2, as | are equivalent.Then Equation (4) holds. The next example and lemma help us to understand the constant-sum model (5).Example 2.1.Suppose ( ) ~, T bin n p ,

Theorem 3 . 1 .⋅
A1 holds iff there exist extended random variables Z and Y such that 1) Z Y ⊥ and , where ( )

Remark 3 . 1 .A.
Thus conditions (1), (2) and (3) hold. In the previous proof, let  be the support of , = Ω  .Then ( ) , T R may not be defined on A, but ( ) , Z Y may be defined on A. Thus it is necessary to create a new random variable Z. Corollary 3.1.If A1 holds then V T R Y Z = 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

3 . 2 . 2 .
proof is done. Remark Notice that the statements in Theorem 3.2 is slightly different from the statements in Corollary 3.One is based on ( ) The asymptotic normality of the PLE under A1 without continuity assumption has not been established in the literature.
a c b c + < − by assumption(2). 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).
The next result follows from Theorem 3.1 and Corollary 3.1, which partially solves the open problem in [11] 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 ( We shall explain through several examples. 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.This defines a parametric family of a continuous random variable with parameter p.The possible observationsI i 's are { } t and ( )1, ∞ .A1 is violated due to the table for Thus both Q and G in Equation (4) are not functions of p or F T and Equation (4) holds.Hence in this example, Q. Q. Yu, K. Yu