Well, I tryed out the new technique on the Solar System. At first I was dismayed and bummed out that the Solar System didn't even come close to satisfying the 95% confidence level. Now, I'm merely perplexed. It turns out that the probability distributions based on the law of increasing differences behave oppositely compared to the distributions based on the pure random spacing null hypothesis. That is, as you add planets, the former more closely resemble exponential progressions and the median R-squared scores actually improve, whereas for the latter, the median and 95% critical values decline as one adds planets.
Thus, the R-squared for the Solar System, 0.9933--figure includes Mercury and Ceres, but excludes Pluto for the a priori reasons (1) that we know enough about Pluto to know that it is Kuiper Belt object that has somehow made its way into the inner Solar System, and (2) Pluto is in any case locked in an orbital resonance with Neptune, and therefore does not constitute an independent sample--is less than that for 55 Cancri (0.9975); but on the purely random null hypothesis, they had the same level of significance (0.1%). I had expected that under the law of increasing differences null hypothesis, the situation would be similar.
However, the Solar System's R-squared fell out at about the 87th percentile according to the probability distribution I generated based on 9 planets, no empty slots, rather than close to the expected 95th percentile. But as you can see from the attached charts, the first clearly shows how sensitive the median R-squared is to increasing planets. The second chart shows that the 95% critical value is not as sensitive, but the important point is that it doesn't decline when more planets are added. (The arrow at the far left is the observed R-squared of the Solar System. The number of trials for each distribution was 10 million.)
So now I'm coming around to Hayes and Tremaine's (1998) way of thinking: that the law of increasing differences is an unfair null hypothesis to test against Titius-Bode laws.
I'm open to suggestions at this point.
Meanwhile, back to the drawing board. . . .
