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Suppose that I have a curve $y(x|\theta)$, where $\theta$ are a set of parameters. My data set is tables $y_{ij}$ and $x_{ij}$. I fit the curve for each row $i$, and obtain the set of parameters $\hat\theta_i^k$ that best fit to this row, where $k$ is the parameter index.

Next I obtain the $n_k\times n_k$ correlation matrix for the parameter estimates $\hat\theta^k$ across rows $i$. What does high or low correlation mean between, say, $\hat\theta^k$ and $\hat\theta^l$?

@RichardHardy suggested to start with a linear case.

Aksakal
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  • Would simplifying this to line fitting still be relevant? Then there would be a bunch of regressions and the question would be, what the correlation between the estimated slopes $\hat\beta_k$ and $\hat\beta_l$ means (I think many of the $\theta$s should have hats in your post, too). Even in this simplified setup, I am not sure what the answer is (besides some basic algebraic or geometric translations of the fact). A further simplification would be to assume the regressors are uncorrelated. But even there I do not immediately see any interesting properties of the question or a possible answer. – Richard Hardy May 09 '18 at 18:28
  • @RichardHardy, maybe the linear case can provide some intuition too, we can try; though I'm dealing with a nonlinear curve. – Aksakal May 09 '18 at 18:31

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