Statistical test for two Maxwell-Boltzmann distributions

Suppose we have two processes A and B, and N subjects, x_i.
A(x) is distributed as a Maxwell-Boltzmann distribution with scale parameter alpha_a; B(x) is distributed as a Maxwell-Boltzmann distribution with scale parameter alpha_b.

Our experiment measures A(x_i) and B(x_i) for each subject.

Is there a standard test for testing the null hypothesis: alpha_a = alpha_b ?

Note 1: Ideally, because the subjects under each process are the same subjects, this would be a paired test.
Note 2: The reason we are testing that specific null hypothesis is that the scale parameter corresponds to a meaningful error in our process, and we are trying to compare the error from A to the error from B.

Thank you,

Eli Gibson


TS Contributor
Would you like to elaborate more on what assumption you made on the \( x_i \) ?

How did the processes \( A(x_i), B(x_i) \) related to \( x_i \)?
And their dependence structure?

First, some context:
x_i is a known point in 3D space. The processes behind A and B are essentially methods for estimating x_i (we can call them C and D). C and D have approximately isotropic Gaussian error in 3D with standard deviations sigma_C and sigma_D. We are interested in the magnitude of the error (i.e. A(x_i)=||C(x_i)-x_i|| and B(x_i)=||D(x_i)-x_i|| ), which should have a Maxwell-Boltzmann distribution. The scaling parameter alpha_C can be related to the standard deviation of the error, C(x_i)-x_i, and thus relates to the probability of having an error magnitude of less than a given threshold.

We are assuming that each x_i is an independent sample, but its actual value is not inherently meaningful. Rather the important value is the estimation error C(x_i)-x_i. Some x_i are harder to estimate, regardless of the method, so A(x_i) and B(x_i) are somewhat correlated.

Interestingly C(x_i)-x_i and D(x_i)-x_i are not linearly correlated, because the error can be in any direction.

Does that answer your questions? I am not a statistician by background, so I may have misinterpreted your questions.

Thank you,