What did Silverman (1981) mean by 'critical bandwidth'? In the selection of a bandwidth for a Kernel Density Estimator, critical bandwidth according to my understanding is:
"For every integer k, where 1<k<n, we can find the minimum width h(k) such that the kernel density estimate has at most k maxima. Silverman calls these h(k) values “critical widths.”
I don't intuitively understand this concept. Any help would be appreciated.
Thank you!
 A: If you have a really wide bandwidth, you'll get one peak in your KDE. If you reduce it a bit, its still one peak. Keep reducing it until you get to the switchover point of adding a second peak. That bandwidth is $h(1)$. 
Now make it smaller still, until you get to the switchover between two peaks and three. That's $h(2)$.
And so forth. 

At any bandwidth between $h(i-1)$ and $h(i)$ you will have $i$ peaks in your KDE.
Silverman wanted a name for that set of $h$-values; he called them critical bandwidths.
This comes up in his test for multimodality, for example.
A: I hate animations in Web pages, but this question begs for an animated answer:

These are KDEs for a set of three values (near -2.5, 0.5, and 2.5).  Their bandwidths continually vary, growing from small to large.  Watch as three peaks become two and ultimately one.

A KDE puts a pile of "probability" at each data point.  As the bandwidth widens, the pile "slumps."  When you start with tiny bandwidths, each data value contributes its own discrete pile.  As the bandwidths grow, the piles slump and merge and accumulate on top of each other (the thick blue line), ultimately becoming one single pile.  Along the way, the maxima change discontinuously from the starting value of $n$ (assuming the kernel has a single maximum, which is almost always the case) to $1.$  The critical width for $k$ maxima is the first (smallest) width that reduces the KDE to a curve with no more than $k$ maxima.
