Solved – Difference – Probability-To-Exceed (PTE) and $\chi^2$ distribution

chi-squared-distributionconfidence intervaldistributionsprobability

I would like to understand the difference between the $\chi^{2}$ distribution and the Probability-To-Exceed ?

I have to compare 2 data sets A and B and in the article I am reading, they talk about this PTE :

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I only know the $\chi^{2}$ distribution with $k=2$ degrees of freedom :

$$f(\Delta\chi^{2})=\dfrac{1}{2}\,e^{-\dfrac{\Delta\chi^{2}}{2}}\quad(1)$$

and the relation with confidence level :

$$1-CL={\large\int}_{\Delta\chi^{2}_{CL}}^{+\infty}\,\dfrac{1}{2}\,e^{-\dfrac{\Delta\chi^{2}}{2}}\,d\,\chi^{2}=e^{-\dfrac{\Delta\chi_{CL}^{2}}{2}}\quad(2)$$

I don't know how to do the link with the text above.

In the article, they make appear the integral of gaussian whereas in $(2)$, I can only make appear a simple integration of exponential (I mean, there is no "$\text{erf}$" function appearing unlike into the article).

If someone could tell me the difference between $\chi^{2}$ distribution and $P_{\chi^2}$ (PTE) ?

UPDATE 1: the context is about astrophysics where I have to compare the consistency of 2 data sets (cosmological parameters) . The method is described below :

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Could anyone tell me what's the definition of this Probability-To-Exceed and how to determine it ?

Is it a cumulative function ? How to get the integral of a gaussian in this case (since erf appears) ?

Any help is welcome, regards

Best Answer

Instead of asking what the difference is between the two, it's clearer to ask what the relationship is between 𝜒2 distribution and the Probability-To-Exceed (PTE).

The PTE is the probability of obtaining a higher 𝜒2 than what you actually achieved. 𝜒2 is a measure of how far off your values are from expectation, and a higher value means larger disagreement. A very low PTE means it is very unlikely to get a higher 𝜒2 than what you already have, meaning your values are farther off from expectation than random chance would allow. In the opposite extreme, a very high PTE means it is very likely to get a higher 𝜒2; this is also bad because it usually means you have overestimated the errors on your measurement.

To calculate the PTE, integrate the 𝜒2 distribution up to your value of 𝜒2, and subtract that value from 1. Usually this is done via a look-up table or solved numerically with a computer program, since there is not a closed-form solution.

The quoted text then goes further, wanted to relate this PTE into a "sigma" of a gaussian distribution since that is a more commonly understood metric.

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