The model of competiting risks have been widely studied in the literature, see e.g. Tsiatis [17A. Tsiatis, "A nonidentifiability aspect of the problem of competing risks", Proc. Natl. Acad. Sci. USA, vol. 72, no. 1, pp. 20-22.
[http://dx.doi.org/10.1073/pnas.72.1.20] [PMID: 1054494] ], Elandt-Johnson and Johnson [18R.J. Elandt, and N.L. Johnson, "Survival models and data analysis", John Wiley and Sons, New York, 1980.], Andersen and al [19P.K. Andersen, Ø. Borgan, R.D. Gill, and N. Keiding, "Statistical Models based on Counting Processes", Springer Series in Statistics, Spring-Verlag, New York, 1993.
[http://dx.doi.org/10.1007/978-1-4612-4348-9] ], Crowder [20M.J. Crowder, "Classical competing risks", CRC Press, 2001.
[http://dx.doi.org/10.1201/9781420035902] ]. Competiting risks problems are often formulated in terms of potential or latent failure times corresponding to the different failure types. Let m be the number of risks acting on the population. For j = 1, let T_j be the r.v. representing the lifetime of an individual exposed to the risk of death from cause j only. We assume that the T_j are proper, so that even in the absence of any other risk, an individual will experience an event of cause j eventually. The random variables (T₁,...,T_m) are called the net lifetimes corresponding to the m postulated possible causes of failure. We assume that each failure is due to a single cause and that the occurrence of an event of a given type precludes the other events from occurring. Consequently, for each individual, we cannot observe (T₁) jointly but only the smallest T_j. For each individual, the observable r.v. consists of the overall failure time T and the cause of failure C.

The independent risks model postulates that the T_j are stochastically independent of each other. Several authors have shown that based on data from competiting risks, the assumption of independence of the different risks is not testable because there is no way to distinguish between independent or dependent latent lifetimes. Tsiatis [17A. Tsiatis, "A nonidentifiability aspect of the problem of competing risks", Proc. Natl. Acad. Sci. USA, vol. 72, no. 1, pp. 20-22.
[http://dx.doi.org/10.1073/pnas.72.1.20] [PMID: 1054494] ], showed that without the hypothesis of independence of the different risks, the model of latent lifetimes is unidentifiable. Indeed, the set of crude distribution functions is consistent with an infinity of joint distributions of latent lifetimes. To each dependent-risks model, there corresponds a unique independent-risks model with the same subdistribution functions. But each independent-risks model has a whole class of satellite dependent-risks models. When the risks are dependent, Klein and Moeschberger [21M.L. Moeschberger, and J.P. Klein, "Consequences of departures from independence in exponential series systems", Technometrics, vol. 26, no. 3, pp. 277-284.
[http://dx.doi.org/10.1080/00401706.1984.10487965] ] showed that the product-limit estimator for the marginal distribution function pertaining to a given risk converges with probability one to a function which may not be the same as the marginal function of interest.

Historically, the main aim of competing risks was seen as the estimation of the marginal distributions. This consists of taking data in which the risks act together and trying to infer how some of them would act in isolation i.e. to make inference free of the “nuisance” aspects. This is an attempt to make inference about the net risks from observations on the crude risks. It amounts to deriving the F_j from the observations of (T, C) This approach is justified if the independence of the risks can be assumed. Otherwise, the counter-argument is that the F_j do not describe events that physically occur, they only describe failures from some isolated cause in situations in which all the other risks have been removed somehow. If the risks are dependent, it is the cause-specific distribution function F_j, not the net distribution function T_j, that is relevant to the real situation describing failures from cause j that can occur in the presence of all the other risks. Prentice and al [22R.L. Prentice, J.D. Kalbfleisch, A.V. Peterson Jr, N. Flournoy, V.T. Farewell, and N.E. Breslow, "The analysis of failure times in the presence of competing risks", Biometrics, vol. 34, no. 4, pp. 541-554.
[http://dx.doi.org/10.2307/2530374] [PMID: 373811] ]. emphasized these points in their criticism of the traditional approach of competiting risks. Cause-specific subdistribution functions are the basic estimable quantities in the dependent competiting risks framework. But this does provide no information about the joint distribution of the lifetimes.

Consider a population in which each subject is exposed to m mutually exclusive competing risks which may be dependent. For j{1,...,m}, the failure time from the j-th cause is a non-negative random variable (r.v.) T_j. The competiting risks model postulates that only the smallest failure time is observable, it is given by the r.v. T = min(T₁,...,T_m) with distribution function (d.f.) denoted by F. The cause of failure associated to T is then indicated by a r.v. C which takes value j if the failure is due to the j-th cause for a j {1,...,m} i.e. if T = T_j. 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. is

where δ(.) denotes the indicator function. As T and C are independent, the r.v. Z has d.f. H given by 1 - H = (1 - F)(1 - G). Let τ_H = sup{t:H(t) < 1} denotes the rightendpoint of H beyond which no observation is possible. The subdistribution functions F^(j) pertaining to the different risks or causes of failure are defined for j {1,..,m} and t ≥ 0 by

For t ≥ 0, the quantity F^(j)(t) represents the probability that an event of type j will occur before time t and that the other types of event have not yet occurred at this time. The functions F^(j) are improper distribution functions since they are not worth 1 at infinity, i.e. lim_t → ∞ F^(j) (t) < 1.

When the independence of the different competiting risks may not be assumed, the functions F^(j) for j {1,..,m} are the basic estimable quantities.

In this paper, we are interested supplying the consistency of the nonparametric Kaplan-Meier estimator of the survival function S^j = 1 - F^(j) for j = {1,...,m}in the presence of independent right-censoring in a context of competiting risks.

The Kaplan-Meier estimator was developed for situations in which only one cause of failure and the independent right-censoring are considered. Aalen and Johansen [23O.O. Aalen, and S. Johansen, "An empirical transition matrix for non-homogeneous Markov chains based on censored observations", Scand. J. Stat., vol. 5, no. 3, pp. 141-150.] were the firsts to extend the Kaplan-Meier estimator to several causes of failure in the presence of independent censoring. In the present situation, the d.f. F may be consistently estimated by the Kaplan-Meier estimator denoted by . For j {1,..,m}, the subdistribution functions F^(j) may be consistently estimated by means of the Aalen-Johansen estimators denoted respectively by , hence the nonparametric estimator of the survival function = 1 - for j {1,..., m}. Indeed, when the process of the states occupied by an individual in time is a time-inhomogeneous Markov process, Aalen and Johansen [23O.O. Aalen, and S. Johansen, "An empirical transition matrix for non-homogeneous Markov chains based on censored observations", Scand. J. Stat., vol. 5, no. 3, pp. 141-150.] introduced an estimator of the transition probabilities between states in presence of independent random right-censoring. The competiting risks set-up corresponds to the case of a time-inhomogeneous Markov process with only one transcient state and several absorbing states (that can be labeled 1, ..., m).

If the data is not censored, S_n(t) can be estimated by the c.d.f.

(2)

where F_n(t) is the empirical c.d.f. of T_i. If observations (T_i) with censures are available, estimate S only by the data (T_i), i = 1,...,n uncensored (δ_i = 1) provides a biased estimate. Indeed if we define an estimator ,

(3)

then by the law of large numbers we have:

(4)

because

Thus, in the case of censored data, the estimation of the survival function S requires specific tools. There are several non-parametric methods among which the well known is the one of Kaplan-Meier. First, we define the non-parametric estimator of the survival function, which can be deduced immediately from the estimation of cumulative risk function, also called the cumulative hazard rate Λ, defined by Λ(t) = .

By the relationship = exp(-Λ) and by estimating Λ by the Nelson-Aalen estimator defined in Njamen and Ngacthou [1D.A. Njamen, and J. N. Wandji, "Nelson-Aalen and Kaplan-Meier estimators in competing risks", Appl. Math. (Irvine), vol. 5, no. 4, pp. 765-776.
[http://dx.doi.org/10.4236/am.2014.54073] ], We obtain the non-parametric estimator of the survival function of S given by:

(5)

It can also be written as the following:

where m(t_i) denotes the number of “events” observed up to the moment t_i and r(t_i) is the number of “individuals” at risk at time t_i

In 1958, Kaplan and Meier proposed an estimator of the survival function S, also called a product-limit estimator. It is based on the following idea: an “individual” is alive after the instant t means to be alive just before the instant t and not to “die” at t.

Definition 3 At any point, the estimator of KM of the survival S of the lifetime variable T_i is, with the above definitions:

(6)

with .

If there is no ex-aequo (i.e. identical death times for several subjects), we can simply write (t) in the following form:

(7)

The Kaplan-Meier non-parametric estimator G_n of G is obtained as the following:

Let T₍₁₎ ≤...≤T_(n) be the associated statistics order, then the Kaplan-Meier estimator defined in (7) is rewritten for all t by

(8)

where R_i = n-i + 1.

The identity (8) comes from the following property:

The Kaplan-Meier non-parametric estimator is widely used in several fields in the society (demography, actuarial science, psychology, etc.) and in the basic sciences, particularly in epidemiology, to show the effectiveness of a treatment compared to another one.

Example 2 (Friereich and et al.,[27J. Freireich, "Emil, Gehan, and al. “The effect of 6-mercaptopurine on the duration of steroid-induced remissions in acute leukemia: A model for evaluation of other potentially useful therapy", Blood, vol. 21, no. 6, pp. 699-716.]) In 1963, Freireich et al. conducted a therapeutic trial to compare the remission durations in weeks of leukemia according to whether they received 6-mercaptopyne or placebo (The control group received placebo and the trial was double-blind).

Remission duration, in the week, according to treatment:

The numbers followed by the sign + correspond to patients who were lost of sight or “censored” on the date considered. They are therefore excluded “alive” from the study and we know only from them that their survival duration is greater than the one indicated. For example, the fourth patient treated with 6M-P had a remission duration of more than 6 weeks.

The calculations give us:

The Kaplan-Meier method consists of estimating the survival probability over a period of time, in other words, the probability of being alive at the end of the interval if we were at the beginning of the interval. Therefore it is necessary to calculate the number of patients presenting the event in this interval divided by the number of patients exposed to the risk of the event during this interval.

Tables 1 and 2 provide general information about the two treatments.

Table 1
Summary in R of all the information about the 6MP.

Table 2
Summary in R of all the information about the 6MP.

The two tables above, obtained from the R code > summary(s), allow us to have not only the survival of all the events at risk, but also Greenwood’s approximations of the variance of survival, as well as the confidence interval in the survival curve.

The following graphs, obtained by the R software, allow us to compute the curve of treatment with 6-mercaptopuria and the curve of treatment with Placebo.

Figs. (1 and 2) represent respectively the survival curve with the evolution of the number of patients at risk during the follow-up, whose the survival curve is the graphical representation of the survival function, i.e. the probability of survival as a function of time. At the beginning of each curve, 100% of patients are alive (probability 1), it’s a stepped curve with a step corresponding to each event. The height of the step is proportional to the number of events on the interval. The lost-of-view is represented by vertical bars throughout the step of the staircase. If the censors are too many and taint the graph, they are not respected, which is not the case for the two graphs above. The accuracy of the estimation of those survival curves Figs. (1 and 2) is represented by a confidence interval of 95%. This interval takes the form of two curves corresponding to the upper limit and the lower limit (see in the dashed lines in the graphs above). This interval has a probability of 95% to include the true survival curve. The two survival curves above are under the form of steps, continuous by lump and jumps at each point of discontinuity.

Fig. (1) Estimation of K-M for 6-mercaptopurnie.

Fig. (2) Estimation of K-M for Placebo.

These two curves can also be obtained in the same graph, which will make it possible to obtain a more precise and concise analysis:

Note that Fig. (3) graph above contains two curves representing the follow-up of treatment with by 6-mercaptopurine and placebo treatment. This graph is obtained by software R. These are staircase curves with the corresponding margin for each event. The height of the margin is proportional to the number of events on the interval. The lost-of-view is represented by vertical bars throughout the staircase. These two survival curves are in the form of a staircase, continuous by piece and have jumped at each point of discontinuity. Finally, we note that the 6-MP treatment curve is higher than that of the Placebo effect. Hence the 6-MP treatment effective.

Fig. (3) Curve of K-M of the survival function for Freireich data.

The asymptotic properties of the KM estimator have been studied by several authors (see e.g Peterson [28J.L. Peterson, "Petri nets", ACM Comput. Surv., vol. 9, no. 3, pp. 223-252. CSUR.
[http://dx.doi.org/10.1145/356698.356702] ], Andersen et al. [19P.K. Andersen, Ø. Borgan, R.D. Gill, and N. Keiding, "Statistical Models based on Counting Processes", Springer Series in Statistics, Spring-Verlag, New York, 1993.
[http://dx.doi.org/10.1007/978-1-4612-4348-9] ], Shorack and Wellner [16G.R. Shorack, and J.A. Wellner, Empirical processes with applications to statistics, Wiley: New-York, .]). A consistency property of the KM estimator is given by:

Theorem 1 If the survival T of the distribution function F and the censure C of the distribution function G are independent, then

Proof See Shorack and Wellner ([16G.R. Shorack, and J.A. Wellner, Empirical processes with applications to statistics, Wiley: New-York, .], p. 304).

The first strong representation of F_n-F by an average of v.a. i.i.d. was obtained by Burke and et al [29M.D. Burke, S. Csörgõ, and L. Horvarth, "Strong approximation of some biometric estimates under random censorship", Z. Wahrsh. View. Gebiete, vol. 56, pp. 87-112.
[http://dx.doi.org/10.1007/BF00531976] , 30M.D. Burke, S. Csörgõ, and L. Horvarth, "A correction to improvmement of strong approximation of some biometric estimates under random censorship", Probab. Theory Relat. Fields, vol. 79, pp. 51-57.
[http://dx.doi.org/10.1007/BF00319103] ], with a remainder of order O(n^-1/2logn²n). This result is based on the work of Komlos, et al. [31J. Kolmos, P. Major, and G. Tusnady, "Weak convergence and embedding limit theorems of probability theory", Colloq. Math. Soc. Janos. Bolyai, North Holland, vol. 11, pp. 149-165.], on the approximation of empirical processes. A second type of approximation was established by Lo and Singh [32H. S. Lo, "The product-limit estimator and bootstrap: some asymptotic representation", Probab. Theory Relat. Fields, vol. 71, pp. 455-465.
[http://dx.doi.org/10.1007/BF01000216] ], with a negligible term of order O((n^-1logn)^3/4). This rate was then improved to O(n^-1logn) by Lo et al [33H.S. Lo, Y.P. Mack, and J.L. Wang, "Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator", Probab. Theory Relat. Fields, vol. 80, pp. 461-473.
[http://dx.doi.org/10.1007/BF01794434] ].

The modeling done in this work is that obtained from Njamen and Ngatchou [1D.A. Njamen, and J. N. Wandji, "Nelson-Aalen and Kaplan-Meier estimators in competing risks", Appl. Math. (Irvine), vol. 5, no. 4, pp. 765-776.
[http://dx.doi.org/10.4236/am.2014.54073] ]. We recall the main lines of this modeling: let τ₁, τ₂,...,τ_m be a continuons random variables representing respectively the lifetimes in each of the m risks competing, J = {1,2,...,m} {0} be the set of index cause, where 0 corresponds to the condition of the individual observed, T = min(τ₁, τ₂,...,τ_m) the random variable case, where η = j if T = τ_j , for all j = 1,2,...,m, F is the distribution function of T, S = 1 - F the survival function such that S(t) = [T > t], the random variable C of the event censoring right, δ = 11_{T ≤ C} and for technical reasons, ξ = η δ such that ξ = j if (T ≤ C and η = j) and ξ = 0 if T > C.

We notice that δ and ξ are observable and η is so only for T uncensored.

We assume that censorship is not informative. The joint law (T,η) is completely specified by the specific incident distributions cause j ,F_(j) (t) defined by

(9)

which are none other than the sub-distribution of the specific cause of failure j = 1,...,m. The cumulative hazard rate of specific-cause j (j = 1,...,m) corresponding to (1) is given by

(10)

Let (Z₁, δ₁, ξ₁)...(Z_n, δ_n, ξ_n) be n- sample of observable triplet (Z₁, δ₁, ξ₁) where Z_i = min(T_i, C_i) and δ_i = 11_{Ti≤Ci}, which T_i = min(τⁱ₁,...,τⁱ_m) and where τⁱ_j represents the time that an individual i is subject to the cause j. If T_i and C_i are independent, the random variable Z_i admits distribution function H_i defined by 1 - H_i = (1 - F_i) (1 - G_i). Nelson-Aalen’s estimator of Λ_j is given for j = 1,...,m by (see e.g in Andersen et al [19P.K. Andersen, Ø. Borgan, R.D. Gill, and N. Keiding, "Statistical Models based on Counting Processes", Springer Series in Statistics, Spring-Verlag, New York, 1993.
[http://dx.doi.org/10.1007/978-1-4612-4348-9] ].)

(11)

which

(12)

and where

(13)

is the counting of the number of failures observed in case of j the time interval [0, t] and

(14)

is the number of individuals in sample observation that survive beyond time t. Thus, for any j {1,..,m}

(15)

The relation between the cumulative hazard rate Λ^*(j) and survival S^*(j) = 1 - F^*(j) in the subgroup A_j is given by

(16)

is given by

(17)

Since (Z_i) is a Markov process comprising a state transcendant and m absorbent states, the F_(j) functions are estimable using the corresponding Aalen-Johansen [23O.O. Aalen, and S. Johansen, "An empirical transition matrix for non-homogeneous Markov chains based on censored observations", Scand. J. Stat., vol. 5, no. 3, pp. 141-150.], estimate defined for t ≥ 0 by:

(18)

where H^-_n is the continuous left-hand modeling of the empirical distribution function H_n defined for t ≥ 0 by

The expression of the Aalen-Johansen the estimator implies equally the modification continue to the left of Kaplan-Meier the estimator of F while is defined for all t ≥ 0 by:

(19)

The final estimator obtained for the cumulative hazard rate Λ^*(j)(t) due to the specific cause j and the corresponding distribution function F^*(j) (t) given by (13) and (16) are written by

(20)

and

(21)

respectively.

The relation between the Kaplan-Meier estimator and Λ^*(j)_n in the context of competiting risks is given for all t ≥ 0 by:

(22)

where

is a sub-distribution of H_n(t), and where for all t ≥ 0,

is the Nelson-Aalen estimator for the specific cause of the cumulative hazards Λ^(j).

Theorem 2 (Aalen and Johansen [23O.O. Aalen, and S. Johansen, "An empirical transition matrix for non-homogeneous Markov chains based on censored observations", Scand. J. Stat., vol. 5, no. 3, pp. 141-150.]; Andersen et al [19P.K. Andersen, Ø. Borgan, R.D. Gill, and N. Keiding, "Statistical Models based on Counting Processes", Springer Series in Statistics, Spring-Verlag, New York, 1993.
[http://dx.doi.org/10.1007/978-1-4612-4348-9] ].) Letσ < τ_H. Assuming that the subdistribution functions F ^*(j)are continuous for j = 1,..., . Asn→ ∞ we have: