1. Introduction
Let us consider a data model which lives time where the event of interest is a failure (or death) due to the 
event, 
and the non-zero integer m, the number of possible causes. By convention, 
corresponds to the state of functioning (or of life) of the observed individual. It is assumed that the observation is stopped when a failure (or death) occurs, but this observation may be right-censored in a non-informative way. Some examples of this situation corresponds to the case where the event of interest is due to another cause, or withdrawal of the individual from the study or at the end of the study. In the case of right censoring time, the time of failure of year for individuals and their causes are not known to the experimenter. A data model as described above is commonly called “competing risks model” (or competitors) and is studied in fields such as medical control, demography, actuarial science, economics or industrial reliability. In Andersen et al. [4] , an illustration and details of mathematics techniques on competing risks in biomedical applications are developed. For example in the study of AIDS, the different competitive risks can be 1) death due to AIDS, 2) death due to tuberculosis or 3) death due to other causes and in this case 
(see Figure 1).
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 [5] , Heckman and Honoré [6] ). Issues to be resolved include virtually the underlying functions for different causes and effects of covariates on the rate of occurrence of competing risks. One of the problems we may face is that the information on the cause of failure of the individual observation can only be known after the autopsy, while we don’t know anything about individuals censored in monitoring. In addition, the incident distributions (due to specific causes) do not allow to describe satisfactorily the probabilities of the various marginal (failures 
case
) in competing risks models. Assumptions of independence of competing risks can help ensure observability in some cases, but they are not reasonable only in such models.
1.1. Related Works
The estimators of Nelson-Aalen and Kaplan-Meier [3] are generally studied in the literature following two approaches: firstly, the method of martingale (Aalen [1] [2] ; Andersen et al. [4] ; Fleming and Harrington [7] , Prentice et al. [8] ) and secondly the law of the iterated logarithm (Breslow and Crowley [9] , Földes and Rejtö [10] or Major and Rejtö [11] , Földes and Rejtö [12] , Gill [13] , Csörgö and Horváth [14] , Ying [15] and Chen and Lo [16] ). Recently, applications have been made in the context of competing risks (Latouche [17] ; Belot [18] ). Latouche [17] states that during the planification of clinical trials, the evaluation of the number of patients to be included is a critical issue because such a formulation does not exist in the Fine and Gray’s [19] model. For this purpose, he therefore computes the number of patients within the context of competition for an inference based function on cumulative incidence and then, he studies the properties of the model of Fine and Gray when it is wrongly specified. Belot [18] presents the data got from randomized clinical tests on prostate cancer patients who died for several reasons.
1.2. Contributions
In this paper, the stochastic processes developed by Aalen [1] [2] are adapted to Nelson-Aalen and KaplanMeier estimators [3] in a context of competing risks (e.g. Aalen and Johansen [20] , Andersen et al. [4] ). We focus only on the complete probability distributions of downtime individuals whose causes are known and which bring us to consider a partition of individuals sub-groups for each cause. We provide a new proof of the consistency of the Nelson-Aalen estimator in the context of competing risks by using the method of martingale. Under the regularity assumptions for the sequence 
(
is a sequence of integers such that 
and 
is the number of observable samples) we obtain an almost-safe speed estimator of Kaplan-Meier [3] which is the same as that obtained by Giné and Guillou [21] which is 
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.
2. Preliminary Results
Lifetime analysis (also referred to as survival analysis) is the area of statistics that focuses on analyzing the time
Figure 1. Example of 3 risks competing model.
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.) 
that describes the length of time from a time origin until an event of interest occurs. We will suppose throughout that 
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 
also called lifetime distribution or the failure distribution, is the probability that the failure time of an individual is less or equal than the value 
It is given for 
by: 
The function 
is right-continuous, nondecreasing and satisfies 
and 
We denote by 
the left-continuous function obtained from 
in the following way:
The distribution of 
may equivalently be dealt with in terms of the survival function which is given, for 
by:
The cumulative hazard function is defined for 
by:
When 
is continuous, the relation 
is valid for all 
We can then call 
the log-survival function.
If 
admits a derivative with respect to Lebesgue measure on 
the probability density function exists and is defined for 
by:
Heuristically, the function 
may be seen as the instantaneous probability of experiencing the event.
With the same hypothesis of differentiability, the hazard function exists and is defined for 
by:
The quantity 
can be interpreted as the instantaneous probability that an individual dies at time 
conditionally on he or she having survived until that time.
For an extensive introduction to lifetime analysis, the reader is referred e.g. to the books of Cox and Oakes [22] and Kalbfleisch and Prentice [23] .
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 
be the nonnegative r.v. with distribution function 
that stands for the censoring time of the individual. As before, the nonnegative r.v. 
with distribution function 
denotes the failure time of the individual. If 
is censored, instead of 
we observe 
which gives the information that 
is greater than 
In any case, the observable r.v. consists of 
, where 
denotes the indicator function. The nonnegative r.v. 
stands for the observed duration of time which may correspond either to the event of interest 
or to a censoring time 
As a sequel to above, it is assumed that 
and 
are independent. Consequently, the random variable 
has the distribution function 
given by
The following subdistribution functions of 
will be needed:
and
The relation
is valid for any 
The relations that connect the subdistribution functions 
and to the distribution functions 
and 
are given by:
and
The cumulative hazard function of 
can be expressed as:
Kaplan and Meier [3] introduced the product-limit estimator for the survival distribution function. The estimator of the cumulative hazard function is the Nelson-Aalen estimator introduced by Nelson [24] [25] and generalized by Aalen [1] [2] .
Let 
for 
be 
independent copies of the random vector 
Let 
be the order statistics associated to the sample 
If there are ties between a failure time (or several failure times) and a censoring time, then the failure time(s) is (are) ranked ahead of the censoring time(s).
We define the empirical counterparts of 
and 
by:
The Kaplan-Meier product-limit estimator is defined for 
by:
The Nelson-Aalen estimator for 
is then defined for 
by:
The following relations are valid for 
where 
the Kaplan-Meier estimator of
, is defined for 
by:
Let 
be a sequence of integers between 
and 
In order to always have asymptotical results, we suppose that the sequence 
satisfies the following hypothesis:
for 
large enough, the sequence 
is non-increasing and 
for 
large enough, the sequence 
is non-increasing and there exists a constant 
such that 
with 
is a non-increasing sequence such that:
Condition 
is required when applying the results of Gin? and Guillou [21] while Condition 
is required when applying the results of Cs
rgö [26] .
The following result formulates the laws of the iterated logarithm-type (LIL-type) result on the mentioned increasing intervals.
Theorem 1 (Csörgö [26] ; Giné and Guillou [21] ) Let 
be a sequence of integers such that 
and, for the almost sure results, satisfying 
We have1:
If, in addition, 
is assumed continuous, then we also have:
Proof. See Csörgö [26] ; Giné and Guillou [21] . 
The continuity of 
is required to linearize the Kaplan-Meier process. Indeed, if 
is continuous, then 
can be approximated by 
on the random interval 
Precisely, we have the following result.
Proposition 1 (Giné and Guillou [21] ) Let 
be a sequence of integers satisfying 
and Hypothesis
. If 
is continuous, then
Proof. See Giné and Guillou [21] . 
3. Evaluation of the Conditional Functions of Distribution to the Specific Causes
Let 
be a continuous random variables representing respectively the lifetimes in each of the 
risks competing, 
be the set of index cause, where 0 corresponds to the condition of the individual observed, 
the random variable of the event of interest and 
the random variable case, where 
if 
for all 
is the distribution function of 
the survival function such that 
the random variable C of the event censoring right, 
and for technical reasons, 
such that 
if (
and
) and 
if
.
We notice that 
and 
are observable and 
is so only for 
uncensored.
We assume that censorship is not informative. The joint law 
is completely specified by the specific incident distributions cause 
defined by
(1)
which are none other than the sub-distributions of the specific cause of failure 
The cumulative hazard rate of specific-cause 
corresponding to 
is given by
(2)
Let 
be n-sample of observable triplet 
where 
and 
, with 
and where 
represent the time that an individual 
is subject to the cause 
If 
and 
are independent, the random variable 
admits distribution function 
defined by 
Then the Nelson-Aalen estimator of 
is given for 
by (see e.g. in Andersen et al. [4] )
(3)
with
and where
(4)
is the counting of the number of failures observed in case of 
the time interval 
and
(5)
is the number of individuals in the sample observation that survive beyond time 
Thus, for any 
(6)
represents the number of individuals who may fall down specific cause 
or be censored.
Estimator similar 
analogue to (2) and on the sub-group 
individuals crashing case 
is given by
(7)
and with 
and
The relation between the cumulative hazard rate 
and survival 
in the subgroup Aj is given by2
(8)
A nonparametric estimator of the distribution function 
of time life in subgroups 
is defined by
(9)
is given by
(10)
The size 
of the subgroup 
individuals is not observable due to the inaccessibility of all subgroups of specific causes 
Nevertheless, we can assign a probability 
to each of the individuals belonging to one of the 
subgroups. Thus, one can estimate the size 
by 
given by ( see e.g. in Satten and Datta [27] or Datta and Satten [28] ) 
where 
is the estimator of the probability that the individual n˚
in the sample subgroup
, subset of risk of specific-cause
. Thus, the final estimators for the cumulative hazard rate 
due to the specific cause 
and the corresponding distribution function 
have the respective expressions
(11)
and for 
(12)
4. Main Results
Let 
be a positive random variable and 
be a censoring variable such that 
and 
In this model of random censorship, for a sample 
subject to a specific causes 
we can observe the couple 
where 
and 
with 
and where 
is the time that an individual 
is subject to the cause 
For a given 
and an individual 
with 
the counting process is defined by:
Therefore, if an individual 
undergoes event before time 
then 
otherwise 
We can also define the counting process
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 
where
For 
and for 
we have
If 
denotes the left boundary at 
of 
we have
since, the quantity 
takes only the values 0 and 1.
For a given 
we define the function
which indicates whether the individual 
is still at risk just before time 
(the individual has not yet undergone the event). Therefore• if 
then, 
and
• if 
then,
where 
is the natural filtration (all information available at time
), where the notation 
refers to formal writing of the stochastic integral
writing made possible because 
is a growing process. The expression of 
in function of the counting process 
is given by
Thus, we have 
The stochastic process defined for 
and 
by
is the martingale associated with the subject at risk 
Thereafter 
is the compensating process 
because it is the integral of the product of two predictable processes.
Theorem 2 Let 
be an absolutely continuous lifetime and 
be a censoring variable for any arbitrary distribution 
Let 
be the risk function associated with 
Let’s put 
and 
.
For 
the process defined by
is a 
martingale if and only if
for 
such that 
Proof. See Breuils ([30] , p. 25) and Fleming and Harrington ([7] , p. 26). 
For a given 
and a given
, the expressions of
, 
and 
are those of formulas (4), (5) and (2) respectively. Using these notations, we can directly obtain the following preliminary result:
Proposition 2 For a given 
and a given
, the stochastic processes defined by
(13)
is the martingale associated with the subject specific cause 
Proof.
The martingale 
represents the difference between the number of failures due to a specific cause 
observed in the time interval
, i.e.
, and the number of failures predicted by the model for the 
cause. This definition fulfills the Doob-Meyer decomposition.
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 
such that 
we have
Proof.
where the expectation of the martingale 
(specific for 
cause) is equal to zero and where 
Indeed,
Hence, we arrive at result.
Using the fact that
we have:
It follows that 
is an asymptotically unbiased estimator of 
Hence, we arrived at result. 
Our second LIL-type result provides almost sure and in probability rates of convergence of 
to 
for 
uniformly over the random increasing intervals
. (See is Deheuvels and Einmahl [31] [32] for very fine results of the model law iterated logarithm functional and available in a point or on a compact strictly included in the support of H). This result is consistent with that of Stute [33] which constitutes a compromise between the results of Breslow and Crowley [9] , Földes and Rejtö [10] or Major and Rejtö [11] , and those of Földes and Rejtö [12] , Gill [13] , Csörgö and Horváth [14] , Ying [15] and Chen and Lo [16] .
Following Giné and Guillou [34] , we say that a non-increasing sequence 
of numbers is regular if there exists a constant 
such that for all 
We denote by 
the following hypothesis:
for 
large enough, the sequence 
is regular non-increasing and there exists a constant 
such that 
with 
is a non-increasing sequence such that
Theorem 4 Let 
be a sequence of integers such that 
for all 
and which satisfies hypothesis 
for the almost-sure part. For all 
we assume that 
is alway continuous. Therefore,
where 
is the Landau in almost sure sense, and
where 
is the Landau in probability.
Both results of Theorem above always provides a rate in probability of uniform convergence of 
to 
for all 
through a random growing intervals 
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 
Firstly, under the Hypothesis 
we have the following result:
Lemma 1 Let 
be a sequence of integers such that 
and, for the almost-sure results, such that 
is satisfied. The rate of convergence of 
to 
is given by
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 
by means of integration by parts:
(14)
Equality (14) entails that:
Notice that the assumption of continuity of 
for 
ensures that 
is continuous according to proposition 1. We then conclude with Theorem 1 and Lemma 1. 
5. Conclusion
In this paper, we have adapted the stochastic processes of Aalen [1] [2] to the Nelson-Aalen and Kaplan-Meier [3] estimators in a context of competing risks. We have focused particularly 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 have also provided some asymptotic properties of nonparametric estimators obtained.
Acknowledgements
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.
NOTES
1
is the Landau in almost sure sense and 
is the Landau in probability.
2
denote the product integral (see Gill & Johansen ).