# Methods to Find the Best Bandwidth for Kernel Density Estimation

I have one dimensional data. All data points are larger than 0. The median and mean are about 10 and 25 respectively. The distribution appears to be lognorm but with really high frequency around the median and fat tail, so lognorm does not fit well. Then I am thinking to use Kernel Density Estimation to describe the data. I tried different ways to find the best bandwidth. (Reference: https://jakevdp.github.io/blog/2013/12/01/kernel-density-estimation/)

R (reference rules)

bw.SJ(data)
bw.nrd(data)
bw.nrd0(data)
bw.ucv(data)


All results are too small (smaller than 0.2) and the graph shows too many bumps, which makes it difficult to analyze.

Python sklearn (cross validation)

grid = GridSearchCV(KernelDensity(), {'bandwidth': np.linspace(0.1, 1.0, 30)}, cv=20)


I ended up testing values of bandwidths from 0.01 to 50 and the best one was 20. Since 20 is too large, the graph is almost flat and does not fit the data at all.

Do you have any ideas why these methods do not work well with my data? Could you tell me other methods to find better bandwidths?

• Use least squares leave One Out Cross validation – Repmat Aug 14 '16 at 10:48
• Thank you for your comment. Could you tell me how to implement it using Python? – Nickel Aug 14 '16 at 13:16
• Sorry I don't know python, but here is something which looks useful jakevdp.github.io/blog/2013/12/01/kernel-density-estimation – Repmat Aug 14 '16 at 14:19
• Can you post a link to your data - then things become more concrete ... :) – wolfies Aug 14 '16 at 17:12

3. If you believe your data is unimodal, you may want to compare the fit of the KDE with that of a log concave estimator. These are implemented in the R packages logcondens and LogConcDEAD.
• Do you mean KDE can work well with both unimodal data and multimodal data, but logcondens and LogConcDEAD are applicable only to unimodal data? – Nickel Aug 14 '16 at 13:23