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Suppose I have iid random variables $X_1,\dots,X_n$ and some parametric function of $\theta,$ $f_{\theta},$ which is linear or nonlinear in $\theta$ (I am more interested in the nonlinear case, but if the linear case has been studied but the nonlinear case has not, I would be interested in the linear case as well).

Is there existing literature on characterizing the distribution of minimizer $\theta_n$ of the sum of squared "pairwise" differences (just including $n^2$ in the denominator to make it look more like an average) $$\theta_n=\arg\min_{\theta}\frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n \left\{f_{\theta}(X_i)-f_{\theta}(X_j)\right\}^2?$$

User0
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    You refer to "some function $f.$" What, then, does the subscript $\theta$ represent? If you mean to ask about a parameterized family of functions, then the extent and nature of that parameterization will be critical to obtaining any results of this nature (just as they are in familiar cases like MLE). In its current form, this question doesn't appear to supply enough details to permit any good answers. – whuber Aug 30 '22 at 21:03
  • Thank you @whuber, updated! – User0 Aug 31 '22 at 13:18
  • The thrust of the question is unclear, though, because this double sum is just twice the variance of the $n$ numbers $f_\theta(X_i).$ Why pose it in such an overly complicated fashion? – whuber Aug 31 '22 at 15:41
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    @whuber, I will try to work out how you got there, but in such a case, you have helped a lot! – User0 Aug 31 '22 at 19:04
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    It's easy once you realize the sum doesn't change when you add a constant to $f_\theta.$ That permits you to assume the mean of the $f_\theta(X_i)$ is zero and that the mean of their squares is the variance. Expand the quadratic, separate the terms, and you're done. – whuber Aug 31 '22 at 19:37
  • If you want anonymity, I think you must change the name on your account. You can always change it back later. I do not know of any way to change or hide the name on only one post. – kjetil b halvorsen Mar 03 '24 at 17:03

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