I am trying to follow this paper to estimate the density for a heavy-tailed distributions using the champernowne transformation.
However, I do not understand the final step to transform the kernel density estimate of the transformed data back to the untransformed data set.
An outline of the procedure is below:
Firstly, the data, X, is transformed:
Where T() is a modified Champernowne CDF. The parameter alpha, M and c have already been estimated.
Then a Kernel Density Estimate, with a Gaussian kernel is done on the transformed data. However, the data must lie in the interval (0,1), so we only take the that part of the estimated density and then divide by the integral of that part of the density.
The final step, which I don't understand is the formula below. What does the denominator mean?
I understand that the numerator is the estimate of the transformed data set.
I can also see the transformered data set in the denominator, T(), but what is T'?
The authors of the paper then write the following expression for the density estimator of the untransformed dataset: