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I'm doing a multiple nonlinear regression that involves $y=A+B\cdot \text{erf}(\frac{C-x}{\sqrt{2}w})$. Parameters $A$, $B$, and $C$ (the one I really care about), all need to be estimated and $x$ is a vector of around 20 elements (samples). Now, erf is pretty similar to sigmoid (the usual logistic function that people seem to use as a basis). I'm not too familiar with logistic regression, so I'll ask a really open-ended question: do any logistic regression experts see any method or advantage to framing this more as a logistic type regression? I imagine the best thing to do is to stick with least-squares cost function and an L2 (i.e. ridge, tikhonov, etc.) regularizer, my planned course, but don't want to discount the similarity to a whole other field.

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  • $\begingroup$ You refer to this as "multiple nonlinear regression" but you only mention one predictor-variable ($x$). Is $w$ a variable or a parameter? $\endgroup$ – Glen_b -Reinstate Monica Sep 13 '17 at 22:57
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What makes logistic regression "different" from nonlinear least squares (NLS) is not the form of the fitted expectation but the model for the conditional distribution of the response.

I see no benefit in trying to connect a straightforward nonlinear least squares problem (which would still be NLS after adding a $L_2$ penalty in the coefficients) to logistic regression (or another binomial GLM, like probit regression), which it's not especially closely related to nonlinear least squares except in the superficial sense that they have a rough similarity of appearance.

If your response were of a kind that it might reasonably modelled by binomial counts, I could see a reason to start from logistic regression (a GLM) and move from there to a generalized non-linear model (GNM) in order to fit the functional form. We don't have that here.

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