On B W Silverman Explanation of KDE:
It is stated that from the definition of a probability density, if the random variable X has density f, then
$f(x) = \lim_{h\to 0} \dfrac{1}{2h}\Pr(x-h < X \le x+h) $
And that a natural naive density estimator is
$f(x) = \dfrac{1}{2hn}[\text{no. of } X_1,..,X_n \text{ falling in }(x-h, x+h)] $
Where is the $\frac{1}{2}$ coming from?
If I have 2 values from a sample of 10 on the $(x-h,x+h)$ range $\Pr(x-h < X \le x+h) = \frac{2}{10}$ how does multiplying by $\frac{1}{2h}$ gives me the density?
The probability $2/10$ is the area under the density from $x-h$ to $x+h$, which has length $2h$ and has height $f(x)$ when approximating the area by a rectangle, thus
$$ \frac{2}{10} = f(x) \, 2h $$