I have a mathematical equation, based on physics, that requires estimating several parameters via nonlinear regression. I have conducted such nonlinear regression estimation with a dataset of 1100 data points. The nonlinear regression output indicates that:
- adjusted R square is high (0.998), AIC and BIC stats were also computed,
- each parameter T statistic value is high (double digit),
- the estimated standard error for each parameter is quite low,
- the maximum intrinsic curvature and maximum parameter effects are lower than the 95% curvature confidence region,
- (standardized) residual plots vs predicted values are fairly randomly distributed,
- (standardized) residual plots vs. each of the predictor variables show well balanced) distributed around the levels of each regressor,
- the Mean Sum of Square and the standard error of regression are quite low relative to the mean value of the observed dependent variable,
- the plot of observed vs predicted values shows a definite straight line pattern with some, but tight, deviations along the span or interval of the dependent variable.
However, the residual histogram shows a bimodal pattern, even though the 2 means are rather close. The distribution exhibits a symmetrical pattern of about 0. At plain sight, and with normality tests - Shapiro-Wilk, Jarque-Bera, D´Agostino, etc. - indicating a Not-normal distribution. So, how or what is the impact or meaning of having a un-normally distributed residual pattern? Does this un-normal residual distribution invalidate the nonlinear regression results?
If I am not mistaken, the assumption of residual normality is important to determine the confidence intervals or errors in the parameter estimates. But according to the curvature statistics, the maximum intrinsic curvature and parameters effects are well under control and below the critical curvature criterion. How should I interpret or take this non-normal residual distribution? Your insights will be greatly appreciated.