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The emails below summarize some tricks and niceties of assuming a chi-square
distribution for two quanties whose errors follow a bivariate Gaussian, 
for example, proper motion measurements along the RA and Dec axes. 
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From:   Frank Masci <fmasci@ipac.caltech.edu>
Subject: Re: more on our (proper) motion measures
Date:   May 23, 2013 1:23:09 PM PDT

Here are some further takeaway points and assumptions that may have not been apparent in John's description below. I think it's a good idea to include a diagnostic related to this in the QA framework.

(1) This tests whether a measured PM (jointly along RA and Dec) significantly differs from zero.

(2) The "chi-square" referred to below is (PMRA/sigPMRA)^2 + (PMDec/sigPMDec)^2.

(3) This definition for chi-square assumes the errors in PMRA and PMDec are independent. As an aside, has anyone plotted PMRA vs PMDec for a dense magnitude bin and tile with ~uniform depth-of-coverage? Is the scatter circular? Given the bu
lk of the population has zero PM, this can tell us if the errors on each axis ar
e correlated.

(4) The Q function as written below assumes the errors follow a bivariate Gaussi
an distribution. A scatter plot of PMRA vs PMDec could also shed light on this.

(5) The Q value from this function gives the probability of obtaining a larger P
M (jointly in RA, Dec) by chance assuming the population is _not_ moving. I.e., 
a tiny Q means it's unlikely the measurements belong to a zero PM population with sigma axes sigPMRA and sigPMDec.

(6) The definition of chi-square above can be interpreted as an effective signal-to-noise ratio: (S/N)^2 = (S/N)_pmra^2 + (S/N)_pmdec^2. I.e., its square-root is the magnitude of a vector whose components are the S/N ratios in PMRA and PMDec.(7) Related to (6): instead of formally computing Q (which depends on the 2-D Gaussian assumption), one can simply key off the effective "joint S/N" or its square where (S/N)^2 = -2*logQ.

Hope this helps.

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From: Frank Masci <fmasci@ipac.caltech.edu>
Date: November 28, 2013 1:29:29 PM PST
To: John Fowler <jwf@ipac.caltech.edu>
Subject: Re: Selecting AllWISE sources to use for AWRef

I think you overlooked the fact that this can be done much simpler. You'll arrive at exactly the same answer if you compute the probability mass in chi-square space since in the end it's the _radial_ motion we're testing for. No optimality is compromised. 

Assuming that the PM model fit "generates" (or plausibly explains) the motion in question, the quantity:

chi^2 = (Dpmra/sigpmra)^2 + (Dpmdec/sigpmdec)^2

will be distributed as chi-square with 2 d.o.f.
where 
Dpmra = PMRA - PMRA_true,
Dpmdec = PMDec - PMDec_true.

The question is: what is the probability that the given PM model fit could have generated the data if indeed the underlying "truth" was instead: PMRA_true=0 and PMDec_true=0?

Your critical chi-square is then:

chi_crit^2 = (PMRA/sigpmra)^2 + (PMDec/sigpmdec)^2; 

The chance probability is then the upper tail of the chi-square distribution: Q = Pr(>chi_crit^2).  Alternatively, you can compute the inverse if you like: P = 1 - Q.

And for a chi-square with 2 degrees of freedom, the upper tail integral of its
p.d.f is simply Q = exp[-0.5*chi_crit^2]. 

Again, you'll get the same answer for the "P" mass if you integrate the bivariate Gaussian directly inside contours, not rectangular regions. Rectangular integration won't catch the full probability mass to decide if a candidate should be accepted or rejected as a PM = 0 source, in a _radial_ sense. 

Regards, Frank