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When using a Gaussian kernel to estimate the distribution of a Gaussian-distributed $x$, the bandwidth that minimizes the mean integrated squared error is:

$$h=\left(\frac{4 \hat{\sigma}^5}{3n}\right)^{\frac{1}{5}} $$

where $\hat{\sigma}$ is the estimated standard deviation of $x$ and $n$ is the sample size. I have seen the derivation of this result.

I am aware there are adjustments that use the inter-quartile range rather than $\hat{\sigma}$ and also that use $0.90$ rather than $1.06=\left(\frac{4}{3}\right)^{\frac{1}{5}}$. I am aware that these adjustments are motivated by the fact that $x$ may not be normally distributed and may be skewed.

I have not seen mathematical justification for these adjustments or an explanation of why the adjustments are "optimal". The adjustments feel arbitrary to me. Is there mathematical justification of these?

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  • $\begingroup$ Consider moving this to Cross Validated for a better fit and a greater pool of experts on such topics. $\endgroup$
    – Richard Hardy
    Jun 29 at 8:42
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    $\begingroup$ I asked on stats stack exchange and got nothing back. stats.stackexchange.com/questions/580248/… I'm an economist and know other economists use kernels, so I thought it was worth trying here. I'm not offended if it is deemed to be off topic. $\endgroup$
    – Michael Gmeiner
    Jun 29 at 9:12

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A good discussion, with references for the mathematical justifications, is given in the textbook written by Henderson and Parmeter (2015, Chapter 2.5). See also the technical appendices of the textbook available online at: https://www.the-smooth-operators.com/technical-appendixerratum

Henderson, D. and C. Parmeter, 2015, Applied Nonparametric Econometrics, Cambridge University Press.

To summarize the material exposed there: the bandwith can be calculated optimally in order to minimize either AMSE or AMISE. This yields the expression $$ h_{opt} = g\left(f(x),f^{(2)}(x),R(f^{(2)}),R(k),\kappa_2(k) \right)\hspace{2mm} n^{-1/5} $$ where the proportionality factor $g$ is specific to the objective that is minimized, $f$ denotes the density and $f^{(2)}(x)$ its second derivative, $\kappa_2(k)$ is the second moment of the kernel function $k$, and $R(f) = \int f(v)^2 dv$ represents the "difficulty" of a function $f$. There is no optimal value for this proportionality factor, but it can be calibrated for specific choices of the kernel and density functions. The values you give are obtained for the normal density and gaussian kernel. Further values are given by the authors in their Table 2.3.

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  • $\begingroup$ Thank you for the answer. I did not see an explanation in the freely available technical appendix. Is there a method to obtain an answer that does not involve paying for the textbook? $\endgroup$
    – Michael Gmeiner
    Jul 4 at 12:49
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    $\begingroup$ I have added a short paragraph... but the best alternative is to borrow the textbook at a library. $\endgroup$
    – Bertrand
    Jul 4 at 20:45

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