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Let's assume we have a simple linear regression y(x) = a + bx. Is there a way to obtain a probability density function of y for any given x? Would the concept of confidence regions be useful here in any way?

I'm not a big expert in statistics and your help will be very much appreciated

Thank you

edited Jan 22 2010 at 6:36

Pete L. Clark
39.6k●3●102●225

asked Jan 21 2010 at 11:10

Alex
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Regression doesn't mean equality. Indeed, the linear fit can be extremely bad. So, you can't say much at all about one variable given a linear regression with a known variable when the correlation is arbitrary. – Douglas Zare Jan 21 2010 at 14:45

what would change if correlation were 0.95? – Alex Jan 21 2010 at 15:33

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That could happen from one very low probability outlier which is far above the mean for both random variables, so you still couldn't say much of anything. You are pushing on a rope. You need some other tools than the linear regression or correlation to say anything meaningful about either the density function or confidence regions. – Douglas Zare Jan 21 2010 at 17:04

pinus · Accepted Answer · 2010-01-22 01:47:49Z UTC

You need to make your answer more precise. First, the regression should be
y = a + bx + e
where e has some distribution, let's say described by a density g(e)

Now, do you mean:

1) Obtaining the density f(y|x) theoretically?
This one is easy. By the transformation of densities
f(y|x) = 1/(a+bx)*g(y/(a+bx))

2) Estimating the density when you do not know the parameters of the regression model?
This one will depend on what you know and what you don't know, and what data you have at hand, and is a more complex issue, so you need to provide more information.

unknown (google) · Answer 2 · 2010-01-26 17:14:41Z UTC

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This question can be answered, but you need to be more specific:

a) what is the space on which x lies ? (i.e. is x a vector or a matrix) b) how many observations you have ? (what is the dimension of y)

answered Jan 26 2010 at 17:14

unknown (google)
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stat_cook · Answer 3 · 2012-08-16 10:11:40Z UTC

I have a similar problem, but unfortunately I lack the knowledge of the transformation of densities.... :(

I have a linear regression with two variables (which are also a bit correlated unfortunately).

y=a+bx1+cx2+e

I know the distributions for x1, x2, e and also the correlation between x1 and x2.

now I want to calculate the distribution for y analytically (not by simulating) but i have no idea how to proceed. could you please help me?

Distribution function of dependent variable, confidence regions

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