parametricsemiparametric

Parametric semiparametric refers to statistical modeling frameworks that combine a finite-dimensional parametric component with an infinite-dimensional nonparametric component. In such models, some aspects of the data-generating process are described by a set of parameters, while other aspects are left unspecified or modeled nonparametrically. The term highlights the hybrid nature of the approach: a parametric core augmented by flexible, nonparametric parts.

Examples include the Cox proportional hazards model, where the regression coefficients are parametric while the baseline hazard function is left unspecified. Partially linear models, where a linear parametric component is coupled with a nonparametric function of covariates, and various semiparametric regression models in biostatistics and econometrics, illustrate the range of structures that can be described in this way.

Estimation approaches often involve partial or profile likelihood methods, estimating equations, or penalized likelihood for the nonparametric part. Nonparametric components can be estimated with kernel methods, splines, or sieve approaches. Semiparametric efficiency theory provides tools such as influence functions and efficiency bounds for the parametric part when the nonparametric component is present, guiding both estimation and inference.

Advantages of parametric semiparametric models include flexibility and interpretability: they allow complex relationships to be modeled without committing to a fully specified parametric form. Challenges include identifiability issues, need for regularity conditions, and computational demands, as well as careful selection of smoothing or basis functions and tuning parameters.