TEST 01
Setup
UPDATE 333
Assume
- \(n\) sample data points \(X_i\), where \(i\in\{0,1,\dots, (n-1)\}\),
- from an underlying probability density \(f(x)\) (and cumulative distribution function \(F(x)\)).
\[\begin{aligned}
\int k(x)\,dx&=1\\
k(-x)&=k(x)\\
\int x^2\,k(x)\,dx&=1
\end{aligned}\]
MORE TEXT
\[\operatorname{MISE} (h) = \mathbb{E}\!\left\{\, \int \left[\hat{f}_h(x) - f(x)\right]^2 \, dx \right\}\]
| Issue | Solution / mitigation | Notes |
|---|---|---|
| 1. KDE doesn’t characterise the distribution (because it requires that we store the original samples). | Resample the smoothed distribution. | Need the relevant C# classto distinguish whether the data is original sample or smoothed. |
| 2. KDE distorts the variance, which is a key risk measure. | Explicitly correct for the increase in variance – see below. | Implicit assumption that this does not distort the distribution. I think we are assuming at least symmetry (which would not be true e.g. for log-normals). |
| 3. KDE tails are asymptotically \(\exp(-(x/h)^2)\), i.e. the tail depends on \(h\), not the actual tail shape. | Not sure – some thoughts set out below. | It is dangerous to make assumptions about the tails (including whether the tails on either side are similar. |