Explain

Explanation of KDE Variance Bound

The variance of an estimator tells us how much the estimate fluctuates around its average value across different random datasets.

Key Insight from the Derivation:

$\text{var}(\hat{p}(x)) \le \frac{C}{nh^d}$

where $C$ depends on the kernel maximum and the density itself.

1. $1/n$ factor: As we get more data points ($n$ increases), the variance decreases. This is standard for most statistical estimators; more data means more stability.
2. $1/h^d$ factor: As the bandwidth $h$ gets smaller, the variance increases.
   • Think of it this way: if $h$ is very tiny, the density estimate at $x$ depends only on data points falling extremely close to $x$. This is a rare event, so the count will fluctuate wildly (0, 1, or 2 points) between different datasets, leading to high variance.
   • If $h$ is large, we average over a large region, stabilizing the count and reducing variance.

Bias-Variance Tradeoff:

• Part (a) (Bias): Small $h$ reduces bias (less smoothing).
• Part (b) (Variance): Small $h$ increases variance (noisier).

This implies we need to tune $h$ carefully. We want $h \to 0$ as $n \to \infty$ to eliminate bias, but we need $nh^d \to \infty$ to eliminate variance. This means $h$ must shrink, but not too fast relative to the sample size $n$.