1. Winsorisation for estimates of change
Daniel Lewis

2. Overview The outlier problem Winsorisation for estimates of level
Winsorisation for estimates of change
Implementing change Winsorisation
Evaluation of methods for Winsorising change
Conclusions

3. The outlier problem Estimate population parameters from sample data:
Assumes sample data represent non-sample data
Outliers present in sample can unduly influence estimates
Unusual weighted values, wi yi
Outlier theory treats ‘representative’ outliers
Winsorisation modifies the values of outliers

4. Winsorisation for level estimates
Choose cut-off ki to minimise MSE
Clarify that this is one-sided type two Winsorisation.

5. Cut-offs for level Winsorisation
Minimising MSE gives:
B is the bias introduced by Winsorisation
This is estimated by:
L is the negative of the bias – estimated by algorithm using past survey data. Ideally multiple periods
Issue of how best to calculate L

6. Winsorisation for estimates of change
Focus on absolute difference: Y2 – Y1.
Currently Winsorise separately (one-sided) and take difference
Tends to work well as Winsorisation bias ‘cancels out’ to some degree
Can we improve by Winsorising change directly?
Derived and tested two possible methods

7. Method 1 for Winsorising change
Derive cut-offs for two adjacent level estimates that minimise the MSE of the difference:
Assume single cut-off ki for a unit in both periods to assist in developing theory
Justified by high overlap between samples and fact that L-values are usually kept constant

8. Method 1 change cut-offs
Cut-off that minimises MSE under these conditions is given by:

9. Method 2 for Winsorising change
Winsorise the difference directly:
Winsorise three terms separately
Need to Winsorise

10. Method 2 continued… Re-write first term as:
Second term here should be relatively insignificant
Defining we have:
Then Winsorise first term with w2i as weight, keep second term unaltered and Winsorise non-common terms as usual

11. How to Winsorise Δi
Δi can be positive or negative, so we need to define:
kUi is cut-off for unusually large positive changes
kLi is cut-off for unusually large negative changes
How should we derive cut-offs?
Investigated different approaches for two-sided Winsorisation
Talk about Kokic-Smith method, Winsorising symmetrically about 0 and ABS approach.

12. Winsorising Δi – approach implemented
Treat cut-offs kLi and kUi separately
Minimise MSE relative to kLi for negative cut-off and relative to kUi for positive cut-off
Not optimal overall, but identifies and adjusts a sensible number of outliers and reduces MSE
Leads to the following cut-offs:

13. Evaluations Two methods of evaluation:
MSE estimates using full Monthly Production Inquiry sample returns
Simulation, sub-sampling from Monthly Production Inquiry sample
Range of options for adjusting y1 and y2 when identify outlier in change
Only going to look at Winsorising change results.

14. Winsorising change (selected results)
Survey estimated results (RMSE (‘000s)):
Simulation results (RMSE (‘000s)):
Very different message from the two evaluations!
No Wins
Wins Change 1
Wins Change 2
Wins Level
234
215
181
No Wins
Wins Change 1
Wins Change 2
Wins Level
832
714
1008
572

15. Conclusions
Winsorising change directly appears too volatile to offer improvements on Winsorising component level estimates and taking the difference
Accuracy of MSE estimates from survey data does not appear sufficient to judge which Winsorisation strategies are most efficient