Asymptotic properties for an M-estimator of the regression function with truncation and dependent data

Abstract

Asymptotic normality Consistency a b s t r a c tIn this paper, we construct a nonparametric M-estimator of a regression function for a left truncated model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. Under appropriate assumptions, we establish weak and strong consistency of the estimator as well as its asymptotic normality. Finite sample behavior of the estimators shows that the M-estimator is more robust than the Nadaraya-Watson type estimator.

Strong uniform consistency rate of an M-estimator of regression function for incomplete data under α-mixing condition

“…The aim of this paper is to establish weak consistency and asymptotic normality for the Mestimator from truncated and censored data, under α-mixing assumption, by considering weaker conditions that those given to get strong consistency. So we complete our first work on strong consistency, and we extend the result of Wang and Liang (2012) from truncated to censored and truncated data, by lightening the conditions on the mixing coefficient, and the result of Lemdani and Ould Said (2017) from the censored independent data to the censored dependent data (as particular case).…”

“…), which are used together with M to get the expression of the Variance of Ψ x (x, θ) . Note that condition R1 is similar to condition M9 given in Lemdani and Ould Said (2017), and conditions R2 and R3 are used in Wang and Liang (2012).…”

“…Assumption B1 is needed to make the bias term negligible. B2 is needed to use a Doob's technique to get asymptotic normality and is used in Masry (2005), Wang and Liang (2012) and others. Note that condition B2 can be satisfied if we choose…”

“…However, all this researches consider the case when a complete data is available, while in most works where the survival time Y is the variable of interest, referred here as the lifetime, two different problems appear : The first one, if the time origin of the lifetime precedes the start of the study, only subjects that fail after the beginning of the study are being followed, otherwise they are left truncated. Wang and Liang (2012) defined a new robust estimator adapted to this type of data based on the estimator of classical regression function proposed by Ould Said and Lemdani (2006) and established its weak and strong consistency (without rate), as well as its asymptotic normality for α-mixing processes. The second problem which can appear in survival data is right censoring, which arises when a subject leaves the study before an event occurs, or the study ends before the event has occurred.…”

Nonparametric M-estimation for right censored regression model with stationary ergodic data

“…Observe that, since η i 's are Bernoulli distributed then the processes, described above, do not satisfy the α-mixing condition whereas they are ergodic (see for instance Laïb and Ould Saïd (2000) and the references therein). The first regression model (Model 1) has been used by Wang and Liang (2012) to study the accuracy of the M-estimator of the regression function when data are truncated and satisfy the α-mixing assumption. Model 2 and Model 3 have been used by Khardani el al.…”