Generalized Linear Models (GLMs) may be nonlinear function of explanatory variables. Even though we use word "Linear" on the ground that explanatory variable are linear in the parameters. In other words because the covariates affect the distribution of response variable only through linear combination of explanatory variables in terms of regression parameters in creating the so called linear predictor. I did not understand the meaning of linear in the parameters or linear combination of explanatory variables in terms of regression parameters.
Let's say you're estimating claim frequency using a very simple model with two parameters, alpha and beta. The following models are all linear in the parameters alpha and beta: \(y = \alpha x + \beta\) \(y = \alpha x^2 + \beta\) \(y = \alpha log x + \beta\) But the following models are NOT linear in the parameters: \(y = \alpha^2 x + \beta\) \(y = x log \alpha + \beta\) In other words it's the parameters that need to be linear, even though the x needn't be. "Explanatory variables" are the things that affect the estimated claim frequency, eg age, gender, etc. The explanatory variable in the model above is x.
Thanks but still have little doubt regarding the fact that can we use word linear in such a way. I though word linear can be used only for function which can be plotted as straight line. sorry if it very basic question. Once again thank for detail reply.
