We discuss in this paper the robust equivariant nonparametric regression estimators for ergodic data with the k Nearst Neighbour (kNN) method. We consider a new robust regression estimator when the scale parameter is unknown. The principal aim is to prove the almost complete convergence (with rate) for the proposed estimator. Furthermore, a comparison study based on simulated data is also provided to illustrate the finite sample performances and the usefulness of the kNN approach and to prove the highly sensitive of the kNN approach to the presence of even a small proportion of outliers in the data.
<abstract><p>In this paper, we consider a new method dealing with the problem of estimating the scoring function $ \gamma_a $, with a constant $ a $, in functional space and an unknown scale parameter under a nonparametric robust regression model. Based on the $ k $ Nearest Neighbors ($ k $NN) method, the primary objective is to prove the asymptotic normality aspect in the case of a stationary ergodic process of this estimator. We begin by establishing the almost certain convergence of a conditional distribution estimator. Then, we derive the almost certain convergence (with rate) of the conditional median (scale parameter estimator) and the asymptotic normality of the robust regression function, even when the scale parameter is unknown. Finally, the simulation and real-world data results reveal the consistency and superiority of our theoretical analysis in which the performance of the $ k $NN estimator is comparable to that of the well-known kernel estimator, and it outperforms a nonparametric series (spline) estimator when there are irrelevant regressors.</p></abstract>
The aim of this research was to study a nonparametric estimator of the density and mode function of a scalar response variable given a functional variable, when the observations are i.i.d. This proposed estimator is given by combining Missing At Random (MAR) with the local linear approach. Finally, a comparison study based on simulated data is also provided to illustrate the finite sample performances and the usefulness of the local linear approach with MAR to the presence of even a small proportion of outliers in the data.
<abstract><p>Traditionally, regression problems are examined using univariate characteristics, including the scale function, marginal density, regression error, and regression function. When the correlation between the response and the predictor is reasonably straightforward, these qualities are helpful and instructive. Given the predictor, the response's conditional density provides more specific information regarding the relationship. This study aims to examine a nonparametric estimator of a scalar response variable's function of a density and mode, given a functional variable when the data are spatially dependent. The estimator is then derived and established by combining the local linear and the $ k $ nearest neighbors methods. Next, the suggested estimator's uniform consistency in the number of neighbors (UNN) is proved. Finally, to demonstrate the efficacy and superiority of the acquired results, we applied our new estimator to simulated and real data and compared it to the existing competing estimator.</p></abstract>
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