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I was wondering whether there is an equivalent to Kernel Density Estimation to estimate nonparametrically the logarithm of a density. Or if there is any nonparametric method for that. (Taking the logarithm of the kernel density estimate seems suboptimal.)

Could you point towards references dealing with this?

EDIT. With suboptimal I mean that $\log \hat f_h(x)$ has some extra bias with respect to the bias of $\hat f_h(x)$, although it is still asymptotically unbiased. By a Taylor expansion, $$ \log \hat f_h(x) = \log(f(x) + \hat f_h(x) - f(x)) \approx \log f(x) + \frac{\hat f_h(x) - f(x)}{f(x)} - \frac{1}{2}\frac{(\hat f_h(x) - f(x))^2}{f(x)^2}. $$ Then, taking the expectation and pluging-in the bias and MSE expressions for $\hat f_h(x)$: \begin{eqnarray*} \mathbb{E}[\log \hat f_h(x)] &\approx& \log f(x) + \frac{\mu_2(K)f''(x)}{2f(x)} h^2 - \frac{1}{2f(x)^2} \Big(\frac{1}{4}\mu_2(K)^2f''(x)^2h^4 + \frac{R(K)f(x)}{nh}\Big)\\ & = &\log f(x) + O(h^2 + (nh)^{-1}) \end{eqnarray*} which is larger than the bias of $\hat f_h(x)$, $\mathbb{E}[\hat f_h(x)] = f(x) +O(h^2)$.

epsilone
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    "*Taking the logarithm of the kernel density estimate seems suboptimal*" -- what are you trying to optimize, exactly? – Glen_b Nov 10 '15 at 23:19
  • Hi @Glen_b, thanks for your comment! I guess the Mean Integrated Squared Error, or a similar error measure in the log scale that makes more sense. My first concern when I thought about ``suboptimal'' was about the bias as it was larger than the KDE's. I edited the question in that direction. – epsilone Nov 11 '15 at 07:43
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    You could perhaps attempt to correct for the bias. – Glen_b Nov 11 '15 at 08:27

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