On semiparametric generalized linear models with nonparametric canonical link

Since the seminal paper by Nelder and Wedderburn (1972), generalized linear models (GLMs) have become a popular option for building regression models. Although GLMs offer a rich family of distributions and link functions, modellers often encounter the perplexing challenge of finding a suitable combination of response distribution and link function that yields a satisfactory fit to the data. The semiparametric GLM is a useful and powerful alternative as it offers flexibility in the response distribution , but it still requires specifications of a link function and its maximum likelihood computation is difficult. In this paper, we propose a novel extension of the semiparametric GLM in which the link function is required to be canonical, but its functional form is left unspecified. That is, the link function is unknown (apart from the canonical requirement) and is determined nonparametrically from the data.