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 epanechnikov-like_bump_._pdf [2015/04/09 19:09]nikolaj epanechnikov-like_bump_._pdf [2015/11/10 18:08]nikolaj Both sides previous revision Previous revision 2015/11/10 18:08 nikolaj 2015/11/10 17:58 nikolaj 2015/04/09 19:10 nikolaj 2015/04/09 19:09 nikolaj 2015/04/09 19:08 nikolaj 2015/04/09 19:08 nikolaj 2015/04/09 19:06 nikolaj 2015/04/09 19:05 nikolaj 2015/04/09 18:57 nikolaj 2015/04/09 18:57 nikolaj 2015/04/09 18:56 nikolaj created Next revision Previous revision 2015/11/10 18:08 nikolaj 2015/11/10 17:58 nikolaj 2015/04/09 19:10 nikolaj 2015/04/09 19:09 nikolaj 2015/04/09 19:08 nikolaj 2015/04/09 19:08 nikolaj 2015/04/09 19:06 nikolaj 2015/04/09 19:05 nikolaj 2015/04/09 18:57 nikolaj 2015/04/09 18:57 nikolaj 2015/04/09 18:56 nikolaj created Line 1: Line 1: ===== Epanechnikov-like bump . PDF ===== ===== Epanechnikov-like bump . PDF ===== - ==== Set ==== + ==== Function ​==== | @#55CCEE: context ​    | @#55CCEE: $x_0,​d:​{\mathbb R}$ | | @#55CCEE: context ​    | @#55CCEE: $x_0,​d:​{\mathbb R}$ | - | @#FF9944: definition ​ | @#FF9944: $k_n:​{\mathbb N}\to{\mathbb R}\to{\mathbb R}_{\ge 0}$ | + | @#FF9944: definition ​ | @#FF9944: $k_n:​{\mathbb N}_{\ge 0}\to{\mathbb R}\to{\mathbb R}_{\ge 0}$ | | @#FF9944: definition ​ | @#FF9944: $k_n(x):​=\begin{cases} \dfrac{1}{2d}\left(1+\dfrac{1}{2n}\right)\left(1-\left(\dfrac{x-x_0}{d}\right)^{2n}\right) &​\hspace{.5cm} \mathrm{if}\hspace{.5cm} \vert x \vert\le 1 \\\\ 0 \hspace{.5cm} &​\hspace{.5cm} \mathrm{else} \end{cases}$ | | @#FF9944: definition ​ | @#FF9944: $k_n(x):​=\begin{cases} \dfrac{1}{2d}\left(1+\dfrac{1}{2n}\right)\left(1-\left(\dfrac{x-x_0}{d}\right)^{2n}\right) &​\hspace{.5cm} \mathrm{if}\hspace{.5cm} \vert x \vert\le 1 \\\\ 0 \hspace{.5cm} &​\hspace{.5cm} \mathrm{else} \end{cases}$ | Line 9: Line 9: $\lim_{n\to\infty}k_n$ is the normed $x_0$-centered rectangle of height $\frac{1}{2d}$. $\lim_{n\to\infty}k_n$ is the normed $x_0$-centered rectangle of height $\frac{1}{2d}$. - We can of course linearly combine ​and normalize to gain PDF’s with several such $k$-bumps to obtain new PDF's. + We can of course linearly combine several such $k$-bumps ​and then normalize ​to obtain new PDF's. == Theorems == == Theorems ==