Asymptotic Normality of the Nelson-Aalen and the Kaplan-Meier Estimators in Competing Risks

This paper studies the asymptotic normality of the Nelson-Aalen and the Kaplan-Meier estimators in a competing risks context in presence of independent right-censorship. To prove our results, we use Robelledo’s theorem which makes it possible to apply the central limit theorem to certain types of particular martingales. From the results obtained, confidence bounds for the hazard and the survival functions are provided.


{ }
1, , j m ∈  , the failure time from the j th cause is a non-negative random variable (r.v.) j τ . The competing risks model postulates that only the smallest failure time is observable, it is given by the r.v.
( ) 1 min , , m T τ τ =  with distribution function (d.f.) denoted by F. The cause of failure associated to T is then indicated by a r.v. η which takes value j if the failure is due to the j th cause for a The following modeling technique is extracted in Njamen and Ngatchou [10]: we assume that T is, in its turn, at risk of being independently right-censored by a non-negative r.v. C with d.f. G. Consequently, the observable r.v. are Johansen [14] were the first to extend the Kaplan The asymptotic properties of the Kaplan-Meier estimator on the distribution function have been studied by several authors (see Peterson [15], Andersen and al. [16], Shorack and Wellner [17], Breslow and Crowley [18]).
In this paper, in a region where there is at least one observation, we are interested in providing asymptotic properties of the Nelson-Aalen and Kaplan-Meier nonparametric estimators of the functions in the presence of independent right-wing censorship in the context of competitive risks set out in Njamen and Ngatchou ([10], [11]).
The rest of the paper is organized as follows: Section 2 describes preliminary results and rappels used in the paper. In Section 3, we obtain two laws: In Section 3.1, we give limit law of Nelson-Aalen's nonparametric estimator for competing risks as defined in Njamen and Ngatchou [10] and Njamen [12]. In Sect. 3.2, we give limit law of Kaplan-Meier's nonparametric estimator in competing risks as defined in Njamen and Ngatchou [10] and Njamen [13]. In Section 4, we give the trust Bands, including the Hall-Wellner trust Bands and the Nair precision equal bands.

Preliminary and Rappels
For 0 t ≥ , we introduce the following subdistribution functions since the different risks are mutually exclusive. The relation The cumulative hazard function of T and the partial cumulative hazard function of T related to cause j for are given for 0 t ≥ respectively by the following expressions: Nelson-Aalen estimators of Λ and of ( ) The Aalen-Johansen estimator for ( ) For all 0 t ≥ , the following equalities hold:

Results
In this section, we continue the works of Njamen and Ngatchou [10], Njamen [12] and Njamen and Ngatchou [11]. In fact, Njamen and Ngatchou ([10], p. 9), studies the consistency of Nelson-Aalen's non-parametric estimator in competing risks, while Njamen ( [12], pp. 11-12) studies respectively the simple convergence and the uniform convergence in probability of Nelson-Aalen's nonparametric estimator in competing risks; and Njamen and Ngatchou ( [11], p. 13) study the bias and the uniform convergence of the non-parametric estimator survival function in a context of competing risks. It is also shown there that this estimator is asymptotically unbiased. For this purpose, we use the martingale approach as the authors mentioned above.

Limit Law of Nelson-Aalen's Nonparametric Estimator for Competing Risks
In what follows, we study the asymptotic normality of Nelson-Aalen's non-parametric estimator in competitive risks. For that, considering, for all and 0 t ≥ , one has the Nelson-Aalen type cumulative hazard function estimator (Nelson,[19]; Aalen, [20], Njamen and Ngatchou, [10]) defined by . The cumulative risk in a region where there is at least one observation is given for all , by (see Njamen, [12]. p. 9) which indicates whether the individual i is still at risk just before time t (the individual has not yet undergone the event). Its estimator was defined in Njamen and Ngatchou ( [10], p. 7).
The following theorem gives the limit law of the Neslson-Aalen estimator in competing risks of Njamen (2017, p. 9). This is the first fundamental result of this article. Theorem 1.
In a region where there is at least one observation, it is assumed that is a centered Gaussian martingale of variance such that: where for all 0 s ≥ , To prove this theorem, we need the Robelledo theorem. In fact, the Rebolledo theorem below makes it possible to apply the central limit theorem for certain types of particular martingales. .
Suppose that n f and f are predictable and locally bounded Suppose also that the processes , , which in turn can be written in terms of ( ) which finally, can be rewritten as can be seen as a random noise process.
where there is at least one observation, the survival function of ( ) min , is defined for all 0 t ≥ by: Otherwise, from which one obtains (see Theorem 3, p. 11 of Njamen, [12]), Consequently, the increasing process of   Also, for all 0 t ≥ and for all  is a martingale. We apply the central limit theorem for the martingales (Rebolledo's Theorem). In this purpose, we show that the condition of this theorem is satisfied by One has, for all and also by the proof of the Theorem 3 of Njamen ( [12], p. 11), we have: We have to show that as n → ∞ , , n t Z  converges to 0 in probability.
One has, for all 0 t ≥ ,

Limit Law of Kaplan-Meier's Nonparametric Estimator in Competing Risks
The Kaplan-Meier estimator of the survival function (Kaplan and Meier, [23]) is defined by The variance of ( ) ( ) ( ) ( ) j j n S t S t approximated by that of ( ) ( ) ( ) ( ) j j S t S t * is: The estimator of the corresponding variance of ( ) ( ) j n S t is given by The following result concerning the asymptotic law of nonparametric Kaplan-Meier estimator and constituted the second fundamental result of this paper: Theorem 3.
In an area where there is at least one observation, if we assume that for all , the non-parametric estimator and condition C implies: when n → ∞ , we deduce that:

Confidence Intervals
For ( ) 0,1 α ∈ , we wish to find two random functions L b and U b such that at Nair, to retrieve the suitable critical value.
In particular, because of the joined character, for a given t their extent is wider than that of the corresponding point IC. In what follows we give the expressions obtained in the absence of transformation.

The Hall-Wellner Confidence Bands
Under the assumption of continuity of survival functions where L x and U x are given by

The Nair Precision Equal Bands
Using a weighted Brownian bridge will notably modify the bounds to IC. For  (12) and (14), we see that the bounds relating to Nair ([25]) bands are proportional to the bounds IC and simply correspond to a risk adjustment threshold used in the past.

Conclusions and Perspectives
In this paper we have studied the asymptotic normality of Nelson-Aalen and Kaplan-Meier type estimators in the presence of independent right-censorship as defined in Njamen and Ngatchou ([10], [11]) and Njamen [12] using Robelledo's theorem that allows applying the central limit theorem to certain types of particular martingales. From the results obtained, confidence bounds for the hazard and the survival functions are provided.
As a perspective, obtaining actual data would allow us to perform numerical simulations to gauge the robustness of our obtained estimators.