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I have x and y related in the way $x = a + \frac{1}{b+y}$. x is measured. I am interested in y. a and b are known. y must be $>=0$ from physics, x is also always $>0$. Solving for y: $y = \frac{1}{x-a}-b$.

Measurements of x are about normal distributed around some value (1 experiment - several values). However I'm interested in y. For values close to a things get messy. But in many cases I have a number of points $<a$ and don't know how to deal with them.

Can you give me pointers what I can do to estimate y in these cases?

Extra information - maybe it helps with suggesting a solution.

My aim would be to to calculate y for each experiment with some kind of information about accuracy. In addition I need to do some kind of summary statistic over all my measurements of y.

  • x is a slope - calculated by linear regression, practically it is in the range from 0 to 1, typically close to 0.5 (with sd < 1%, but I did not use this so far)
  • a and b are assumed as constants calculated from empirical values ( however the values by different researchers disagree by 20% - life sciences )
  • both a and b are >0, also a < 1
  • I would assume the distribution of y is similar to a lognormal distribution.
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bdecaf
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    If you know that y must be greater than zero, then you can put a lower bound on x (and justify this appropriately), such that y will never be non-positive. That is some lower bound below which you justify it as measurement error, since it's not possible. – user2974951 Sep 13 '18 at 06:44
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    I wonder putting the measurement error term like $x_{measured}=x_{true }+e$ could be helpful in this situation. I hope someone will answer this post. – KDG Sep 13 '18 at 07:42
  • @user2974951 unfortunately this is only half the problem. I was already thinking along making a+eps a lower bound for x - but then I will end up with lots of items valued 1/eps. – bdecaf Sep 13 '18 at 09:31
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    Since $a$ and $b$ are known, this is not primarily an estimation problem (as you have tagged it): it appears to be one of prediction, if I read your post correctly. (It is possible you will need to estimate the error variance, but that is a "nuisance parameter" that does not appear of direct interest.) The solution depends on what you know or assume about $y$ and the error variance. For instance, is $y$ a random variable with a known distribution? – whuber Sep 13 '18 at 15:19
  • @whuber I see - sorry my statistical vocubaulary is rusty. I will retag. I also gave some extra information in case it helps suggest an approach. – bdecaf Sep 13 '18 at 15:44
  • Thank you, but now I'm totally mystified, because your question begins with a statement of its solution! Since you state you know $a$ and $b,$ then for any given $y$ you predict $x$ by plugging the values $a,b,y$ into the formula for it. The accuracy information comes from analyzing the measurement system. However, there is much that is mysterious, because evidently you do not know $y:$ you estimate it with regression (on data you don't describe). I have a very strong feeling that you haven't articulated the problem you actually face and the lack of detail might hurt you. – whuber Sep 13 '18 at 15:46
  • @whuber - thanks for correcting me - when I wrote I had x and y mixed up. let me correct. That's what I get for using SE on the go. – bdecaf Sep 13 '18 at 16:04

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