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I wanted to cluster one-dimentional data with kernel density estimation. I tried to count the optimal bandwidth with Silverman's rule of thumb and also using cross-validation. However, the obtained bandwidths are large, and KDE returns only one cluster of data. Is there a method that will force to estimate the bandwidth which will cluster the data?

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You can set the bandwidth to whatever value you like. You could start at the bandwidth suggested by Silverman's rule, decrease it until you find the number of clusters you think there should be, and make clear when you're reporting your results that that's what you did.

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