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Let me get right to my problem. I have an experimental distribution and a single-parameter theoretical distribution.

I want to find the value for the best fit theoretical distribution that agrees with my experimental data for the bulk of the distributions (the tails of my distributions differ substantially).

I isolate an equal portion of both distributions and calculate the sum of the squares of the differences between the two distributions for this region. i.e. least squares approach.

R^{2}=Ʃ [ exp_{i}- theo_{i}(x) ]^{2}

I do this for several values that I have chosen for the single-parameter theoretical distribution and obtain a unique parameter value (which I call x=ζ) which results in the minimization of the sum of the squares (exactly what I want). Every other parameter gives a larger value for this sum of squares.

I do not know if this is necessary but I can plot all parameters that I tested vs the R^{2}too. This gives points that combine to form a parabola, ax^{2}+bx+c=R^{2}. I can take the derivative of the parabola curve to obtain the minimum of the parabola which again occurs at ζ. I have attached a pdf of this.

My question is, how do I find a confidence interval for this? I am looking for [itex]\sigma[/itex] in ζ ± [itex]\sigma[/itex]. Do I use the formula for the parabola to find this? Do I use R^{2}?

I have looked through some books and online and been unsuccessful on how to find this [itex]\sigma[/itex] value. Any help is appreciated. A reference book would be of great help too. Thanks!

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# Confidence interval from least squares fitting?

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