Recursive Kernel Density Estimation and Optimal Bandwidth Selection Under α: Mixing Data

Nonparametric relative recursive regression

“…It is clear, that the use of the proposed estimator (4) can improve considerably the computational cost. non-recursive correspond to the estimator (5) using the proposed plug-in bandwidth selection (17), Recursive 1 correspond to the estimator (4) using the proposed plug-in bandwidth selection (14) and the stepsize Figure 1: Qualitative comparison between three kernel relative regression estimators; non-recursive correspond to the estimator (5) using the proposed plug-in bandwidth selection (17), Recursive 1 correspond to the estimator (4) using the proposed plug-in bandwidth selection (14) and the stepsize (γn) = n − , Recursive 2 correspond to the estimator (4) using the proposed plug-in bandwidth selection (14) and the stepsize (γn) = hn / n k= h k using model 1; µ (X) = . + x, X ∼ U ( , ) and ε ∼ N ( , ).…”

“…Table 2 shows that the recursive estimator with the choice (γn) = hn / n k= h k outperforms the nonrecursive estimator and the recursive one with the choice (γn) = n − : the empirical levels of the intervals I ,n are greater than those of I ,n . Figure 2: Qualitative comparison between three kernel relative regression estimators; non-recursive correspond to the estimator (5) using the proposed plug-in bandwidth selection (17), Recursive 1 correspond to the estimator (4) using the proposed plug-in bandwidth selection (14) and the stepsize (γn) = n − , Recursive 2 correspond to the estimator (4) using the proposed plug-in bandwidth selection (14) and the stepsize (γn) = hn / n k= h k using model 2; µ (X) = log . + (x − . )…”

“…In conclusion, the proposed estimators allowed us to obtain a good results. A future research direction would be to extend our ndings to the α-mixing framework see [17]. Another direction is to investigate the relative regression estimation based on the transformation of the data, in the case when the response variable are not positive see [44].…”

“…: The empirical levels (# {r (x) ∈ I i,n } /N, with N = ) of three models using three estimators; non-recursive correspond to the estimator (5) using the proposed plug-in bandwidth selection (17), Recursive 1 correspond to the estimator (4) using the proposed plug-in bandwidth selection (14) and the stepsize (γn) = n − , Recursive 2 correspond to the estimator (4) using the proposed plug-in bandwidth selection (14) and the stepsize (γn) = hn / n k= h k . vergence rate O n − /(d+ ) .…”

“…vergence rate O n − /(d+ ) . The recursive estimators using the plug-in bandwidth selection developed in the subsection 3.1 (see, (14)) are then compared with the non-recursive one proposed by [15] using the plug-in bandwidth selection developed in the subsection 3.2 (see, (17)). We showed that, using some particularly choice, the proposed estimators can give in some situation a better results compared to the non-recursive approach in terms of estimation error.…”