Comparing Continuous Probability Distributions: Finding Significance

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To compare continuous probability distributions, one approach involves using regression analysis with two variables representing the true and alternative distributions. The goal is to test the joint hypothesis that the intercept (a) equals 0 and the slope (b) equals 1, indicating a strong relationship. To assess significance when a is close to 0 and b is close to 1, a two-tailed hypothesis test can be employed, allowing for a range of values around these parameters. This method provides a more nuanced understanding of the relationship between the distributions. Overall, utilizing regression and hypothesis testing can effectively evaluate the significance of the comparison.
jjstuart79
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Hi,
I was searching the forum about comparing continuous probability distributions and came across this post back in 2005.

"You could make two variables X(t) = value of the "true" disrtibution (expensive simulation) at point t and Y(t) = value of the alternative dist. (practical simulation) at point t. Then run the regression Y(t) = a + b X(t) for as many t's as you can (or like), then show that the joint hypothesis "(a = 0) AND (b = 1)" is highly statistically significant."

My question is about the last sentence. What would be the best way to check to see if a = 0 and b =1? I know I could count how many times that is exactly true, but what if a = close to 0 and b = close to 1? I would like a way for that to count for some significance as well.

I appreciate any help.
 
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I think I found the answer. A two-tailed hypothesis test should work.
 
The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...

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