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epanechnikov-like_bump_._pdf [2015/04/09 19:06] nikolaj |
epanechnikov-like_bump_._pdf [2015/11/10 17:58] nikolaj |
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===== Epanechnikov-like bump . PDF ===== | ===== Epanechnikov-like bump . PDF ===== | ||
- | ==== Set ==== | + | ==== Function ==== |
- | | @#55CCEE: context | @#55CCEE: $n:{\mathbb N}$ | | + | |
| @#55CCEE: context | @#55CCEE: $x_0,d:{\mathbb R}$ | | | @#55CCEE: context | @#55CCEE: $x_0,d:{\mathbb R}$ | | ||
- | | @#FF9944: definition | @#FF9944: $k:{\mathbb R}\to{\mathbb R}_{\ge 0}$ | | + | | @#FF9944: definition | @#FF9944: $k_n:{\mathbb N}\to{\mathbb R}\to{\mathbb R}_{\ge 0}$ | |
- | | @#FF9944: definition | @#FF9944: $k(x):=\begin{cases} \frac{1}{2d}\left(1+\frac{1}{2n}\right)\left(1-\left(\frac{x-x_0}{d}\right)^{2n}\right) &\hspace{.5cm} \mathrm{if}\hspace{.5cm} \left|x\right|\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} $ | |
----- | ----- | ||
=== Discussion === | === Discussion === | ||
- | For large $n$, the function converges against the normed 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 == | ||
- | $\int_{x_0-d}^{x_0+d}\left(\dfrac{x}{d}\right)^{2m} k(x)\,{\mathrm d}x=\dfrac{1}{2(n+m)+1}\dfrac{2n+1}{2m+1}$ | + | $\int_{x_0-d}^{x_0+d}\left(\dfrac{x}{d}\right)^{2m} k_n(x)\,{\mathrm d}x=\dfrac{1}{2(n+m)+1}\dfrac{2n+1}{2m+1}$ |
=== Code === | === Code === | ||
<code> | <code> |