No I’m not posting about Blue’s Clues but rather the Gauss-Markov theorem. Blue stands for Best Linear Unbiased Estimator (as almost all of you know). This is an important Theorem for linear regressions. It is my contention that our ranking system is a type of linear regression. Or more accurately a not so linear regression (with regard to the factors used in the ranking system). The weights used in a ranking system are just coefficients in a linear regression equation.
Isn’t BLUE what we want in a ranking system. Note I did not say PEST (that would be me). I know there is no perfect estimator. But we want the best.
How good can we do with our present ranking system estimator? Pretty darn good. But could a few easy changes make it better? This is actually a question. But a very easy (possible) answer comes to mind.
Note that the “L” in BLUE stands for linear. Percentile is a linear scale but certainly it is not linear with regard to the factor or function used in the ranking system. There is no reason to think that a change from the 50th percentile to the 55th percentile causes the same change in a factor (or function) as a change from the 95th percentile to the 100th percentile. In fact, you will almost never be able to convert a factor to its percentile using a linear function. Sometimes we might want to convert to a log scale but to percentile? Probably never really. Clearly percentile cannot always lead to BLUE. In fact it is probably never the best scale to use in order to get the Best Linear Unbiased Estimator and may be far from the best scale to use many times.
Is this why some great ideas from finance fail in ranking systems? I’m guessing so.
One commonly used remedy is to use standard deviation from the mean as the X axis or independent variable(s). It is a linear conversion from the factor to the number of standard deviations from the mean. Standard deviation from the mean is a scale that is pretty easy to understand even if you are not real familiar with the factor itself. Could we be able to convert from percentile to standard deviation then back? Would this be hard? Would it actually be desirable? In the end we still would need to be able to convert to rank so we could pick the top 5 or 10 or 20 highest ranking stocks.
This is not a feature request because so many of you know much more about this that I do. You probably already know whether this is a good idea or not. I would love to know too.
Thank you.
Jim