Douglass et al assume that observed b_o is the true trend, and in this case one-sample t-test can be performed. With large n_2, t-distribution approaches rapidly normal d., and the 95 % confidence interval for the trend would be

Length of this interval approaches zero as n_2 gets larger, but there is nothing special about that. That’s what should happen under this assumption.

Let’s take two samples of size and , compute sample means and variances, , and , . If these are from same distribution, the statistic

(where ), follows a Student’s t with dof. If , this simplifies to

This test is more SE like than SD. But when you turn this to as prediction interval for one future sample (), you’ll get

and I think test based on this or y in the above is more or less the same thing. Proof is left to the reader, though This is more SD-like test, specially if is large.

In Santer17, the variances seem to be different for those two samples, so the situation gets more complicated. And it is a bit unclear to me how sample deviations are obtained in H2 case.

Did someone actually say that weather with AR(1) and lag 1 autocorrelation of 0.9 is reasonable?

I get that impression from Santer17. Confusing stuff, just some time ago someone told us that The conclusion is inescapable, that global temperature cannot be adequately modeled as a linear trend plus AR(1) process.

The interesting point is that AR(1) with p of about 0.9 is suddenly accepted for ‘climate noise’.

Did someone actually say that weather with AR(1) and lag 1 autocorrelation of 0.9 is reasonable?

If you were to run AR(1) simulations with this lag 1 autocorrelation, the autocorrelation for observataions of GMST since GMST would be highly unlikely. Highly.. Heck, if “weahter noise” is AR(1) with autocorrelation of 0.728, there’s only a 1.7% chance of getting lag 1 autocorrelations as low as we’ve gotten since 2001.

You can read a bit about the analysis here.

The autocorrealtions are higher during periods when volcanos like Pinatubo and El Chicon are going off. Otherwise… well… The observational evidence suggests it’s lower.

I think beaker was sucked through a wormhole into a parallel universe around the time he posited that Santer et. al. contained a typo. He might be having some trouble posting from there.

The interesting point is that AR(1) with p of about 0.9 is suddenly accepted for ‘climate noise’. Mann & Lees 96 paper told us that such value would be unphysical. The problem is, you need high p to keep AGW running even if the temperatures go down. On the other hand, you need low p to be able to say ‘one cannot simulate the evolution of the climate over last 30 years without including in the simulations mankind’s influence on sulfate aerosols and greenhouse gases.’