I'm down the rabbit hole on a least squares spline approximation problem I'm working on. I've solved the least squares piece, as well as calculating a statistic called "Q1" (the Durbin-Watson statistic) which is supposed to be used to calculate the adequacy of an approximation to determine when further refinement of knot positions is no longer needed.
The issue with using Q1 is that calculating it's exact distribution is not feasible. A solution, in the paper I'm working from's own words:
...there are two limiting distributions for Q1 , both only depending on N and m, between which the actual distribution is guaranteed to lie. These limiting ‘upper’ and ‘lower’ distributions, which we denote as fU(Q1) and fL(Q1), can be approximated by beta distributions and are easily computed.
I'm stuck on what I believe is simply a gap in my knowledge. For the algorithm I'm coding my criteria is supposed to be:
In a separate paper on this same topic I've found instructions for calculating the parameters of Q1's beta distribution. These instructions are included at the bottom of my question, but assuming I've done the math correctly for calculating the values p and q which will be used as parameters for B(p,q), I'm at a loss for how to use this information to test for the above criteria. Am I supposed to simply plug Q1/4 into the formula (12) listed below? Or do I calculate the mean and stddev and use that to test for the above 5% criteria? (How would I do that?)
It's a bit tricky for me to trial and error this based on the numbers I get when I try different approaches since I'm not 100% sure what values to expect at what points, and if an error lies in my calculations of the application of the theory.
Thanks for any hints!
Directions for calculating Q1's beta distribution parameters:
