ABSTRACT
As we indicated in Section I.1, our concern in this chapter will be with methods for inference about parameters in non- and semiparametric models whose qualitative large sample behaviour is like that of the corresponding procedures for regular parametric models. In the i.i.d. case, estimates converge at rate nhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429153822/6c2e9a20-ec71-4535-a7cd-56f78736ac55/content/in9_u001.tif"/> to Gaussian distributions, and the behaviour of tests is governed by that of Gaussian processes. Section 9.1 introduces theory for estimates based on maximizing modified and empirical likelihoods in important semiparametric models such as linear models with stochastic covariates, biased sampling models, Cox proportional hazard models with censoring, and independent component analysis models. Section 9.2 and 9.3 deal, respectively, with asymptotic normality and efficiency for estimates. Section 9.4 similarly deals with asymptotic inference for tests in semiparametric models. We need to use the machinery of Chapter 7 to extend the techniques of Chapter 5 and 6. In Section 9.5 we show how the powerful notions of contiguity and local asymptotic normality can clarify power and information bound calculations.
