How does bandwidth selection work for low-variance dimensions with KernelMixtureDistribution?

Question

I am puzzled because this toy example (in Mathematica 8)

dist = KernelMixtureDistribution[{{0, 1}, {0.0, 0}, {0, -1}},"Silverman",
                                  MaxMixtureKernels -> All];
DensityPlot[PDF[dist, {x1, x2}], {x1, -3, 3}, {x2, -3, 3}]

produces a density estimate with quite a large extent in x-direction although it has zero standard deviation. I would not have been surprised if that had produced an error because it should be a delta impulse in x-direction and I don't know if that can be represented by Mathematica. But this result does not make sense.

If I change it to

dist = KernelMixtureDistribution[{{0, 1}, {0.01, 0}, {0, -1}},"Silverman",
                                 MaxMixtureKernels -> All];
DensityPlot[PDF[dist, {x1, x2}], {x1, -3, 3}, {x2, -3, 3}]

the result is a very thin distribution as expected. The behaviour is similar for Silverman's and Scott's rules of thumb.

Can someone shed some light on how the standard deviations are interpreted here? Is this a bug maybe ?

Answer 1

The short answer is that when the standard deviation is zero it is treated as 1.

The benefit of this approach is that a single data point can be used to examine a sort of "standard" kernel.

For example:

ContourPlot[Evaluate@
    PDF[KernelMixtureDistribution[{{0, 0}}, Automatic, "Epanechnikov"],{x,y}], 
       {x, -2, 2}, {y, -2, 2}
]

Now whether this is the correct or desired behavior is certainly up for debate.

Keep in mind that automatic bandwidth selection is nice but after some initial exploration one can (and probably should) always set the bandwidth manually in practice based on some intuition and understanding about the data.