A transform to change skew without affecting kurtosis? I am curious if there is a transform which alters the skew of a random variable without affecting the kurtosis. This would be analogous to how an affine transform of a RV affects the mean and variance, but not the skew and kurtosis (partly because the skew and kurtosis are defined to be invariant to changes in scale). Is this a known problem?
 A: My answer is the beginnings of a total hack, but I am not aware of any established way to do what you ask.  
My first step would be to rank order your dataset you can find the proportional position in your dataset and then transform it to a normal distribution, this method was used in Reynolds & Hewitt, 1996.  See sample R code below in PROCMiracle.
Once the distribution is normal, then the problem has been turned on its head - a matter of adjusting kurtosis but not skew.  A google search suggested that one could follow the procedures of John & Draper, 1980 to adjust the kurtosis but not the skew - but I could not replicate that result.  
My attempts to develop a crude spreading/narrowing function that takes the input (normalized) value and adds or subtracts a value from it proportional to the position of the variable on the normal scale does result in a monotonic adjustment, but in practice tends to create a bimodal distribution though one that has the desired skewness and kurtosis values.
I realize this is not a complete answer, but I thought it might provide a step in the right direction.
PROCMiracle <- function(datasource,normalrank="BLOM")
  {
     switch(normalrank,
      "BLOM" = {
                  rmod <- -3/8
                  nmod <- 1/4
                },
      "TUKEY" = {
                  rmod <- -1/3
                  nmod <- 1/3
                },
      "VW" ={
                  rmod <- 0
                  nmod <- 1
            },
      "NONE" = {
                  rmod <- 0
                  nmod <- 0
                }
    )
    print("This may be doing something strange with NA values!  Beware!")
    return(scale(qnorm((rank(datasource)+rmod)/(length(datasource)+nmod))))
  }

A: Another possible interesting technique has come to mind, though this doesn't quite answer the question, is to transform a sample to have a fixed sample L-skew and sample L-kurtosis (as well as a fixed mean and L-scale). These four constraints are linear in the order statistics. To keep the transform monotonic on a sample of $n$ observations would then require another $n-1$ equations. This could then be posed as a quadratic optimization problem: minimize the $\ell_2$ norm between the sample order statistics and the transformed version subject to the given constraints. This is a kind of wacky approach, though. In the original question, I was looking for something more basic and fundamental. I also was implicitly looking for a technique which could be applied to individual observations, independent of having an entire cohort of samples.
A: I would rather model this data set using a leptokurtic distribution instead of using data-transformations.  I like the sinh-arcsinh distribution from Jones and Pewsey (2009), Biometrika.
