Asymptotic Properties of M-estimators

Functional statistics plays a central role in statistical research. In this study, we focus on conditional models, which can be formulated as follows: let X be a functional random variable and Y a multidimensional random variable. The prediction of Y given X is modeled through a mapping r(.) applied to X.

To approximate Y conditional on X, we construct estimators for functional parameters using the kernel method. Examples include regression, quantile, expectile, and conditional mode estimation.

After constructing an estimator of the functional parameter, the asymptotic analysis proceeds along two main lines:

First, we establish the almost-sure uniform convergence of nonparametric estimators for certain conditional models, specifying the corresponding rates of convergence.

Second, we derive the asymptotic normality of the estimator under standard regularity conditions. We also discuss its application in constructing confidence intervals. Furthermore, we provide an explicit expression for the asymptotic covariance matrix.

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