It looks like you have a lower cut-off limit under which the value will always be 1. This isn't suitable for regression as there is a mathematical discontinuity in the trend. It would be better to ignore all samples with Y=1 and fit a regression only on the samples with Y>1.

When applying the model you would then have to decide if that new sample met the criteria for applying Y = 1 or the regression model using the same rules.

If the data is all above the curve you have a computation error. Such is not possible in regression. by the way, This is not a regression problem in the usual sense. Regression mins the SSE. That is not what you have here. Best wishes, DB

Thank you Prof for your reply.

It looks like you have a lower cut-off limit under which the value will always be 1. This isn't suitable for regression as there is a mathematical discontinuity in the trend. It would be better to ignore all samples with Y=1 and fit a regression only on the samples with Y>1.

When applying the model you would then have to decide if that new sample met the criteria for applying Y = 1 or the regression model using the same rules.

Dear Prof Koczkodaj, can you explain more on the method or any link that i can refer to? Thank you Prof

My point of view is the following:

This is a nonlinear regression model Y = f(X) where X is the combination of a nonlinear function (the max function) with a linear transformation of X (x->a_1+a_2x^n). Statistical programs offer a nonlinear function to do the regression (nls in the statistical program R for example), but I doubt that the nondifferentiability of the max would make several complications as mentionned by James (in the answers). Worst case scenario, you may program your self a least square estimation and use a derivative free algorithm such as the Nelder-Mead algorithm.

Use truncated regression

Thank you all for the suggestions. I will try to work with the suggestions given.