Many thanks to those who responded, and apologies for taking some time to
post a summary.
1)=========
Nick Cox and Martin Hazelton both pointed out that a simple way to achieve
this would be to transform the data, smooth with a symmetric kernel, and
back transform.
Nick added that a very good reason for this is to avoid smearing some of a
distribution beyond its natural support, e.g. when a variable is necessarily
positive.
He also comments that: "The transformation and back transformation procedure
is mentioned briefly by Silverman (1986, pp.27-30), although his worked
example (p.28) is not very encouraging. Good expositions are given by Wand
and Jones (1995, pp.43-45), Simonoff (1996, pp.61-64) and Bowman and
Azzalini (1997, pp.14-16). The underlying principle is that for a continuous
monotone transformation t(x), the densities f(x) and f(t(x)) are related by
f(x) = f(t(x)) |dt/dx|."
The references are:
Bowman, A.W. and Azzalini, A. 1997. Applied smoothing techniques for
data analysis: the kernel approach with S-Plus applications. Oxford:
Oxford University Press.
Silverman, B.W. 1986. Density estimation for statistics and data analysis.
London: Chapman and Hall.
Simonoff, J.S. 1996. Smoothing methods in statistics. New York: Springer.
Wand, M.P. and Jones, M.C. 1995. Kernel smoothing. London: Chapman and Hall.
(Arthur Pewsey also suggested this last reference -- but see (3) below).
2)=========
Alun Pope points out that this issue arises in discontinuity detection and
in forecasting.
In the first case he suggests
1. Inge Koch and Alun Pope. Asymptotic bias and variance of a kernel-based
estimator for the location of a discontinuity. Nonparametric Statistics, 8
(1997) 45-64.
In this case the kernel is the derivative of a symmetric kernel with one
maximum so is exactly antisymmetric.
For the forecasting case he suggests
2. I. Gijbels, A Pope and M. Wand. Understanding exponential smoothing via
kernel regression. J Royal Statistical Society, Series B, 61 (1999) 39-50.
In this case the kernel is "one-sided" in an obvious way.
3)=========
However, it may be that anyone inspired by the above to rush ahead and try
out asymmetric kernels on their existing problems should be cautious.
Chris Jones -- in answer to my question as to whether the notion had been
studied -- replied:
"Not really, because in general use at least, there are theoretically
essentially no advantages to be had from their use. At least, not yet ....."
4)=========
Alun Pope very reasonably asked what the application was that I had in mind.
It is this:
I am trying to estimate the distribution of rates of profit; I have
accounting data on companies (lots of them, since it is essentially the
Companies House database), but what I would like is the distribution across
the total capital employed by the corporate sector.
It is relatively easy to apply a suitable scheme of weighted random sampling
from the companies data, but to take the results as genuinely being the
estimate sought involves the assumption that each £ of capital employed by a
firm achieves the same rate of return.
If I am not mistaken, smoothing the resulting distribution with any of the
usual symmetric kernels effectively assumes that all firms have
identically-shaped symmetric *internal* distributions of rates of return,
albeit with differing scales.
This seems rather unlikely, at least for the larger ones.
Instead, an economically plausible assumption would be that the internal
distribution approximates the economy-wide distribution, which is of course
unknown (but likely to be skewed on the basis of both theory and the
evidence -- as far as it goes-- of my current weighted sampling).
Hence my interest in skewed kernels, which would seem to allow one to
implement this latter assumption. (Actually, one might reason that small
companies have roughly symmetrical distributions, with the internal
distributions becoming more skewed the larger the company -- but this raises
the question of applying kernels of differing forms to different
observations....)
A further feature of my problem makes it difficult to implement the idea of
transforming the data, smoothing, and back-transforming: a small but
important minority of firms have negative profit rates.
For the obvious transformations one could, if one knew the lower bound of
the distribution, think of adding the appropriate quantity to each
observation before applying any transformation -- but part of the aim of the
work is precisely to estimate a reasonable lower bound for rates of
return....
Julian Wells
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