The five main statistical problems are:

- Problem 1. The inhomogeneous reference problem
- Neighboring stations are typically used as reference. Homogenization methods should take into account that this reference is also inhomogeneous
- Problem 2. The multiple breakpoint problem
- A longer climate series will typically contain more than one break. Methods designed to take this into account are more accurate as ad-hoc solutions based single breakpoint methods
- Problem 3. Computing uncertainties
- We do know about the remaining uncertainties of homogenized data in general, but need methods to estimate the uncertainties for a specific dataset or station
- Problem 4. Correction as model selection problem
- We need objective selection methods for the best correction model to be used
- Problem 5. Deterministic or stochastic corrections?
- Current correction methods are deterministic. A stochastic approach would be more elegant

## Problem 4. Correction as model selection problem

The number of degrees of freedom (DOF) of the various correction methods varies widely. From just one degree of freedom for annual corrections of the means, to 12 degrees of freedom for monthly correction of the means, to 120 for decile corrections (for the higher order moment method (HOM) for daily data, Della-Marta & Wanner, 2006) applied to every month, to a large number of DOF for quantile or percentile matching.What is the best correction method depends on the characteristics of the inhomogeneity. For a calibration problem just the annual mean would be sufficient, for a serious exposure problem (e.g. insolation of the instrument) a seasonal cycle in the monthly corrections may be expected and the full distribution of the daily temperatures may need to be adjusted.

The best correction method also depends on the reference. Whether the variables of a certain correction model can be reliably estimated depends on how well-correlated the neighboring reference stations are.

Currently climatologists choose their correction method mainly subjectively. For precipitation annual correction are typically applied and for temperature monthly correction are typical. The HOME benchmarking study showed these are good choices. For example, an experimental contribution correcting precipitation on a monthly scale had a larger error as the same method applied on the annual scale because the data did not allow for an accurate estimation of 12 monthly correction constants.

One correction method is typically applied to the entire regional network, while the optimal correction method will depend on the characteristics of each individual break and on the quality of the reference. These will vary from station to station and from break to break. Especially in global studies, the number of stations in a region and thus the signal to noise ratio varies widely and one fixed choice is likely suboptimal. Studying which correction method is optimal for every break is much work for manual methods, instead we should work on automatic correction methods that objectively select the optimal correction method, e.g., using an information criterion. As far as I know, no one works on this yet.