The stochastic approximation method for estimation of a distribution function

2016

JJSS

: In this paper we consider the kernel estimators of a distribution function defined by the stochastic approximation algorithm when the observation are contamined by measurement errors. It is well known that this estimators depends heavily on the choice of a smoothing parameter called the bandwidth. We propose a specific second generation plug-in method of the deconvolution kernel distribution estimators defined by the stochastic approximation algorithm. We show that, using the proposed bandwidth selection and the stepsize which minimize the M ISE (Mean Integrated Squared Error), the proposed estimator will be better than the classical one for small sample setting when the error variance is controlled by the noise to signal ratio. We corroborate these theoretical results through simulations and a real dataset.

Section: Assumptions and Main Resultsmentioning

confidence: 96%

Bandwidth Selection in Deconvolution Kernel Distribution Estimators Defined by Stochastic Approximation Method with Laplace Errors

2016

JJSS

“…Now, in order to estimate the optimal bandwidth , we must estimate I 1 and I 2 . We followed the approach of Altman & Leger () and SLAOUI (), which is called the plug‐in estimate, and we use the following kernel estimator of I 1 :

{\hat{I}}_{1} = \frac{Π@ @_{n}}{n} \sum_{i, k = 1}^{n} {@@Π}_{k}^{- 1} γ_{k} b_{k}^{- 1} K_{b} (\frac{X_{i} - X_{k}}{b_{k}}) Y_{i}^{2},

where K b is a kernel and b n is the associated bandwidth.…”

Section: Assumptions and Main Resultsmentioning

confidence: 99%

“…Assumption (A2) on the stepsize and the bandwidth was used in the recursive framework for the estimation of the density function (Mokkadem et al 2009a;Slaoui, 2013Slaoui, , 2014a and for the estimation of the distribution function (Slaoui (2014b)).…”

Section: Assumptions and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Plug‐in bandwidth selector for recursive kernel regression estimators defined by stochastic approximation method

2015

Statistica Neerlandica

“…They concluded that chosen appropriately plug-in and bootstrap selectors both outperform the cross-validation bandwidth, and that neither of the two can be claimed to be better in all cases. Recently, plug-in bandwidth selection method for recursive kernel density estimators defined by stochastic approximation method have been done by Slaoui (2014a) and for recursive kernel distribution estimators have been done by Slaoui (2014b). In this paper, we developed a specific plug-in bandwidth selection method of the semi-recursive kernel estimators of a regression function defined by stochastic approximation method.…”

Section: Introductionmentioning

confidence: 99%

Optimal bandwidth selection for semi-recursive kernel regression estimators

Statistics and Its Interface

2016

In this paper we propose an automatic selection of the bandwidth of the semi-recursive kernel estimators of a regression function defined by the stochastic approximation algorithm. We showed that, using the selected bandwidth and some special stepsizes, the proposed semi-recursive estimators will be very competitive to the nonrecursive one in terms of estimation error but much better in terms of computational costs. We corroborated these theoretical results through simulation study and a real dataset.