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epanechnikov-like_bump_._pdf [2015/04/09 19:06] nikolaj |
epanechnikov-like_bump_._pdf [2015/11/10 18:08] 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: $k_n:{\mathbb N}_{\ge 0}\to{\mathbb R}\to{\mathbb R}_{\ge 0}$ | |
| - | | @#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> | ||
