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In this paper, stochastic processes developed by Aalen [1] [2] are adapted to the Nelson-Aalen and Kaplan-Meier [3] estimators in a context of competing risks. We focus only on the probability distributions of complete downtime individuals whose causes are known and which bring us to consider a partition of individuals into sub-groups for each cause. We then study the asymptotic properties of nonparametric estimators obtained.

Let us consider a data model which lives time where the event of interest is a failure (or death) due to the

It is important to note that in most data models in competing risks, the functions that characterize the probability distribution of the variable of interest and the marginal are not always observable (see Tsiatis [

The estimators of Nelson-Aalen and Kaplan-Meier [

In this paper, the stochastic processes developed by Aalen [

The rest of the paper is organized as follows: Section 2 describes preliminary results and notations used in the paper and Section 3 evaluates the conditional functions of distribution to the specific causes. Section 4 contains the main results of the paper as well as some properties of our estimators obtained. The last section concludes the paper.

Lifetime analysis (also referred to as survival analysis) is the area of statistics that focuses on analyzing the time

duration between a given starting point and a specific event. This endpoint is often called failure and the corresponding length of time is called the failure time or survival time or lifetime.

Formally, a failure time is a nonnegative random variable (r.v.)

The most basic quantities used to summarize and describe the time elapsed from a starting point until the occurrence of an event of interest are the distribution function and the hazard function. The cumulative distribution function at time

The function

The distribution of

The cumulative hazard function is defined for

When

If

Heuristically, the function

With the same hypothesis of differentiability, the hazard function exists and is defined for

The quantity

For an extensive introduction to lifetime analysis, the reader is referred e.g. to the books of Cox and Oakes [

The main difficulty in the analysis of lifetime data lies in the fact that the actual failure times of some individuals may not be observed. An observation is right-censored if it is known to be greater than a certain value, provided the exact time is unknown. Let

As a sequel to above, it is assumed that

The following subdistribution functions of

and

The relation

is valid for any

The relations that connect the subdistribution functions

and

The cumulative hazard function of

Kaplan and Meier [

Let

We define the empirical counterparts of

The Kaplan-Meier product-limit estimator is defined for

The Nelson-Aalen estimator for

The following relations are valid for

where

Let

Condition

The following result formulates the laws of the iterated logarithm-type (LIL-type) result on the mentioned increasing intervals.

Theorem 1 (Csörgö [^{1}:

If, in addition,

Proof. See Csörgö [

The continuity of

Proposition 1 (Giné and Guillou [

Proof. See Giné and Guillou [

Let

We notice that

We assume that censorship is not informative. The joint law

which are none other than the sub-distributions of the specific cause of failure

The cumulative hazard rate of specific-cause

Let

with

and where

is the counting of the number of failures observed in case of

is the number of individuals in the sample observation that survive beyond time

represents the number of individuals who may fall down specific cause

Estimator similar

and with

The relation between the cumulative hazard rate _{j} is given by^{2}

A nonparametric estimator of the distribution function

is given by

The size

and for

Let

For a given

Therefore, if an individual

Naturally, it appears that we considered the information provided over time as a filter, which is used to describe the fact that past information is contained in the current information, hence we have the natural filtration

For

If

since, the quantity

For a given

which indicates whether the individual

• if

where

writing made possible because

Thus, we have

The stochastic process defined for

is the martingale associated with the subject at risk

Theorem 2 Let

For

is a

for

Proof. See Breuils ([

For a given

Proposition 2 For a given

is the martingale associated with the subject specific cause

Proof.

The martingale

The first result of this paper concerns the consistency of the Nelson-Aalen estimator for the competing risks based on martingale approach.

Theorem 3 For

Proof.

where the expectation of the martingale

Hence, we arrive at result.

Using the fact that

we have:

It follows that

Our second LIL-type result provides almost sure and in probability rates of convergence of

Following Giné and Guillou [

Theorem 4 Let

where

where

Both results of Theorem above always provides a rate in probability of uniform convergence of

To prove Theorem 4, we have drawn from results based on the inference of empirical processes, given that in order to linearize the Kaplan-Meier process, it is necessary to impose continuity condition on

Lemma 1 Let

Proof. The proof of this result follows straightforwardly from the proof of the first part of Theorem 1 concerning the supremum of

Proof of Theorem 4. The following decomposition is obtained for

Equality (14) entails that:

Notice that the assumption of continuity of

In this paper, we have adapted the stochastic processes of Aalen [

I would like to thank Prof. Nicolas Gabriel ANDJIGA, Prof. Celestin NEMBUA CHAMENI, Prof. Eugene Kouassi for their support and their advices. I would also like to thank specially Prof. Kossi Essona GNEYOU for his collaboration and his cooperation during the preparation of this paper.