^{1}

^{2}

Yu <i>et al</i>. (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.

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

be i.i.d. copies of_{R}, F_{V} and _{T} (f_{R} or

(see [_{2} is also called a likelihood. We shall call Λ the full likelihood and Λ_{2} a simplified one. Since our sample

where

and the integrals are Lebesgure integrals. We say that a function

Example 1.1. Consider 3 cases of right censoring:

Case (1).

Case (2).

Case (3). _{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 _{R} is non-informative (not informative) about F_{T} in case (1), as F_{T} and F_{R} are both independent parameters.

If

Proposition 1.1. The full likelihood

Example 1.1 (continued). In case (1), _{T} and condition Equation (4) fails.

_{T} based on _{j} that is associated with_{T} is continuous), while the GMLE based on Λ is _{T} is the PLE

Remark 1.1. Example 1.1 indicates that if Equation (4) is not valid then the MLE based on so-called “likelihood” _{R} is non-informative about F_{T}.

Williams and Lagakos (W&L) [

where

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., [

W&L Theorem (Theorem 3.1 in [

By the W&L Theorem, one can easily make use of the existing results about the PLE under the assumption

Since

Equation (7) holds, so

On the other hand, case (3) in Example 1.1 shows that the PLE can be inconsistent for

A1

Notice that

A2

Definition 1. If _{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

Example 1.2.

Yu et al. [

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

We shall first show that A1 and A2 are a sufficient condition of Equation (4), extending a result in [

Theorem 2.1. Equation (4) holds if A1 and A2 hold.

Proof. Since_{T} by A2. Moreover, by A1

_{T} by A1 and A2, as

The next example and lemma help us to understand the constant-sum model (5).

Example 2.1. Suppose

Lemma 2.1.

Theorem 2.2. If

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

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)

In our theorem, there are two modifications to the W&L Theorem.

1) Equation (5) with continuous

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”

Example 3.1. Suppose that

Thus

random variable such that

Example 3.2. A random sample of complete data

Proof of Theorem 3.1. It suffice to show (Þ) part. Since

Remark 3.1. In the previous proof, let

Corollary 3.1. If A1 holds then

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

tablished in the literature as follows.

Theorem 3.2 (Yu et al. [

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 [

by

since

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

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 [

Without continuity assumptions, Gu and Zhang [

case of the DC model, their results imply that (8) and (9) also hold if

either

Theorem 3.3. Equations (8) and (9) are valid if A1 holds and if either T is continuous or (1)

The answer to the question is “No” in general. We shall explain through several examples.

Example 4.1. Suppose that

where _{T}. One can verify that possible observations I_{i}’s are

Thus the parametric model satisfies the N&S condition Equation (4). But in view of

rify that the PLE of

but the MLE which maximizes

Then the MLE is the one that

Since

Example 4.2. Suppose that

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

Example 4.3. Suppose that T is continuous,

This defines a parametric family of a continuous random variable with parameter p. The possible observations I_{i}’s are

Thus both Q and G in Equation (4) are not functions of p or F_{T} and Equation (4) holds. Hence in this example, A1 is not a necessary condition of Equation (4).

Example 4.4. Suppose that

ple 4.3, then A1 fails and Equation (4) holds for the random vector

It shows that if

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 [

We thank the Editor and the referee for their valuable comments.

Qiqing Yu,Kai Yu, (2016) Equivalence between the Dependent Right Censorship Model and the Independent Right Censorship Model. Open Journal of Statistics,06,209-219. doi: 10.4236/ojs.2016.62018

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 2.1WLOG, one can assume that u satisfies

If T is a continuous random variable, then the previous equation and Equation (6) yield

Assume that

Since

where

iff

iff

iff

iff for almost all r (w.r.t.

iff

If

1)

2)

3)

Thus