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Inference for semiparametric models: Some questions and an answer. (With comments). (English) Zbl 0997.62028

Summary: Non- and semi-parametric models have flourished during the last twenty five years. They have been applied widely and their theoretical properties have been studied extensively. We briefly review some of their development and list a few questions that we would like to see addressed. We develop an answer to one of these questions by formulating a ‘calculus’ similar to that of the i.i.d. case that enables us to analyze the efficiency of procedures in general semiparametric models when a nonparametric model has been defined. Our approach is illustrated by applying it to regression models, counting process models in survival analysis and submodels of Markov chains, which traditionally require elaborate special arguments. In the examples, the calculus lets us easily read off the structure of efficient estimators and check if a candidate estimator is efficient.

MSC:

62G08 Nonparametric regression and quantile regression
62F12 Asymptotic properties of parametric estimators
62G20 Asymptotic properties of nonparametric inference
62N02 Estimation in survival analysis and censored data
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