I have a fairly simple question about (multivariate) Kernel Density Estimators, but I somehow don't seem to find the answer anywhere: are these estimators supposed to specify a proper probability density function (integrating to 1.0)?

I am currently implementing a small software package including kernel density estimators, and in particular a product kernel density estimator for the multivariate case, using the formula found here, and Scott's rule of thumb for the bandwidth selection.

The estimator seems to get right the overall shape of the distribution, but the density values are clearly not classical probability densities, as their integration would yield a value far superior to 1. So I am wondering if I made an error in my implementation, or if these values are correct.

I have a follow-up question in case the estimator is not supposed to yield regular probability densities: is there an easy way to convert the estimator function into a standard PDF (i.e. to normalise the function)?

  • $\begingroup$ My understanding is they'd normally be densities, yes. $\endgroup$
    – Glen_b
    Jan 30, 2013 at 6:58
  • 1
    $\begingroup$ This is merely Fubini's Theorem together with the observation that the product of nonnegative values remains nonnegative. $\endgroup$
    – whuber
    Jan 30, 2013 at 16:04

1 Answer 1


After checking the literature more thoroughly, and trying out myself to integrate the kernel densities, it does seem that a KDE yields a proper probability density function.


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