Bandwidth Selection for Recursive Kernel Density Estimators Defined by Stochastic Approximation Method

Slaoui, Yousri

doi:10.1155/2014/739640

Cited by 44 publications

(40 citation statements)

References 21 publications

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“…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%

“…We followed the same steps as in Slaoui (), and we showed that in order to minimize the MISE of

{\overset{I}{false}}_{1}

, the pilot bandwidth ( b n ) must equal

(0.68 {\{\frac{R (K_{b})}{μ_{2}^{2} (K_{b})} \frac{\int_{R} E [Y^{4} | X = x] f^{2} (x) dx}{{(\int_{R} E [Y^{2} | X = x] f^{(2)} (x) f (x) dx)}^{2}}\}}^{1 / 5} n^{- 2 / 5}) .

Moreover, we showed that

\begin{matrix} italicBias ([], {\overset{I}{false}}_{1}) & ≃ 0.57 \times normalΘ {(K_{b})}^{1 / 2} I_{1}^{2 / 5} {((), \int_{double-struckR} double-struckE ([], Y^{2} | X = x) f^{(2)} (x) f (x) dx)}^{1 / 5} n^{- 4 / 5} \\ + o ((), n^{- 4 / 5}) \end{matrix}

and

…”

Section: Assumptions and Main Resultsmentioning

confidence: 99%

“…We followed the same steps as in Slaoui () and we showed that in order to minimize the MISE of

{\overset{I}{false}}_{2}

, the pilot bandwidth

((), b_{n}^{'})

must equal

(0.97 {\{\frac{R {(K_{b^{'}}^{(2)})}^{4} \int_{R} m^{2} (x) f (x) dx}{μ_{2}^{2} (K_{b^{'}}) {(\int_{R} m^{(4)} (x) m^{(2)} (x) f (x) dx)}^{2}}\}}^{1 / 14} n^{- 3 / 14}),

Moreover, we showed that

\begin{matrix} italicBias ([], {\overset{I}{false}}_{2}) & ≃ 1.33 \times normalΓ {((), K_{b^{'}})}^{1 / 2} {((), \int_{double-struckR} m^{2} (x) f (x) dx)}^{1 / 7} \\ \times ((), \int_{double-struckR} m^{(4)} (x) m^{(2)} (x) f (x) dx) \end{matrix}

…”

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%

“…Now, in order to estimate the optimal bandwidth (12), we must estimate I 1 and I 2 . We followed the approach of Altman and Leger (1995) and Slaoui (2014aSlaoui ( , 2014b, which is called the plug-in estimate, and we use the following kernel estimator of I 1 :…”

Section: Ifmentioning

confidence: 99%

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