In this paper, we study the simple convergence and the uniform convergence of the Nelson-Aalen nonparametric estimator studied in Njamen $\&$ Ngatchou (Njamen $\&$ Ngatchou, 2014) in a contest of competiting risks.

This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.

Introduction:In this article, we only focus on the probability distributions of the breakdown time whose causes are known, and we consider a partition of the observations into subgroups according to each of the causes as defined in Njamen and Ngatchou [1]. By adapting the stochastic processes developed by Aalen [2, 3], we derive a Kaplan-Meier [4] nonparametric estimator for the survival function in competiting risks.Result & Discussion:In a region where there is at least one observation, we prove on one hand that this new nonparametric estimator is unbiased in competiting risk and on the other hand, using the Lenglart inequality, we establish its uniform consistency in competiting risks.

This work is concerned with multivariate conditional heteroscedastic autoregressive nonlinear (CHARN) models with an unknown conditional mean function, conditional variance matrix function and density function of the distribution of noise. We study the kernel estimator of the latter function when the former are either parametric or nonparametric. The consistency, bias and asymptotic normality of the estimator are investigated. Confidence bound curves are given. A simulation experiment is performed to evaluate the performance of the results.

In this paper, inspired by the estimator of the cumulated specific incidence proposed by Marubini and Vasecchi [23], we obtain the Kaplan-Meier estimator of the survival function of all causes of death combined by summing the estimator of fj(t) (j ∈ {1, . . . , m}) obtained by plug-in. We establish that the Kaplan-Meier estimator of the survival function overestimates the cumulative incidence in the presence of competitive events. By making the product of all the contributions of the studied system, we establish the likelihood function of the specific risk function for the competitive risk model. Finally, under the assumptions of Dinse and Larson [13], and using the delta-method, we establish the variance of the cumulative incidence function in competiting risks.