The problem addressed in both these papers [1] [2] is taking the output of a simulation and fitting to a target distribution (in this case an empirical estimate derived from historical observation of a real system under possibly different conditions and with great uncertainty when extrapolating to extremes).

( [not an answer] Reduce Dimensions

Question whether you really need to estimate so many potentially biased

things? Are they just "nuisance parameters"?

When you say the distribution should be "unbiased" .. why not consider

instead what statistic you really are trying to estimate and simply unbias that? )

A. State-Space Transforms (direct or via copulae)

both papers essentially define the non-linear map of a 2-d state-space to be separable into 2 1-d non-linear transforms, note that this already changes the correlation value and conditional dependence structure, but there is nothing stopping you defining an arbitrary "warp" in higher dimensions. As long as it is a small warp you should not be in danger of introducing a major difference in statistics.

B. Adaptive Rejection / Metropolis–Hastings / Gibbs sampling

the desired frequency of samples is known, you can choose to randomly reject those that appear over-sampled in order to better fit the desired profile, this brute-force approach becomes expensive in high dimensions, hence Metropolis-Hastings algorithm or Gibbs sampling are generally used to synthesise some "random extra similar" samples cheaply (Markov chain Monte Carlo). see [3]

C. Weighted Monte-Carlo

Rather than adding samples you can weight them, in probability theory you can look at that as dP/dQ a change of measure. This will complicate calculating the statistics, particularly if you try numerical integration (like the trapezium rule) rather than the naive approach as the interpolations of weights are not simple and the noise they might add can matter. In Financial maths we can happily set up the weights to be variance reducing though and thus accelerate Monte-Carlo convergence.

D. Model Mixtures (input Parameters)

to avoid purely statistical techniques you can look to calibrate a range of input parameters to your model itself and use a combination of those to provide real samples but under a range of scenarios. This is very expensive particularly if your model is slow, but in Financial applications it has the benefit of interpretation as uncertainty about the input parameters and admitting analysis of Baysian updating of that knowledge. Note that you could also just be defining a new model which happens to be a mixture of samples produced with varying parameters.

Calibration. All three approaches require you to estimate a kernel from a finite sample. The simplest statistical way to look at "good-ness" of fit is moment matching .. the most apparently sophisticated is "maximum a posteriori probability (MAP)" - but beware of that approach as information/entropy relies on a good understanding of the uncertainty itself .. not much use with unknown unknowns!

see also http://en.wikipedia.org/wiki/Estimation_theory

[1] H. Hoffmann, T. Rath, "Meteorologically consistent bias correction of climate time series for agricultural models" (Theoretical & Applied Climatology Oct2012, Vol. 110 Issue 1/2, p129)

[2] C. Piani, J. O. Haerter, "Two dimensional bias correction of temperature and precipitation copulas in climate models"

[3] D. Raje and R. Krishnan "Bayesian parameter uncertainty modeling in a macroscale hydrologic model and its impact on Indian river basin hydrology under climate change" (Water Resour. Res., doi:10.1029/2011WR011123, in press)

Dear Holger,

it is already long time ago ...

I am working on the same topic these days. Thank you for giving me the link to your paper, which I did not know yet. For the other paper (PH2012), however, they do not use a copula for a multivariate BC. The rather correct P for the different quantiles of T using a QM approach. The copula is simply used to assess the performance of the bias corrected results.

Cheers, Patrick