In this paper, we study the differentiability of implicitly defined functions which we encounter in the profile likelihood estimation of parameters in semi-parametric models. Scott and Wild (Biometrika 84 (1997) 57–71; J. Statist. Plann. Inference 96 (2001) 3–27) and Murphy and van der Vaart (J. Amer. Statist. Assoc. 95 (2000) 449–485) developed methodologies that can avoid dealing with such implicitly defined functions by parametrizing parameters in the profile likelihood and using an approximate least favorable submodel in semi-parametric models. Our result shows applicability of an alternative approach presented in Hirose (Ann. Inst. Statist. Math. 63 (2011) 1247–1275) which uses the direct expansion of the profile likelihood.
"On differentiability of implicitly defined function in semi-parametric profile likelihood estimation." Bernoulli 22 (1) 589 - 614, February 2016. https://doi.org/10.3150/14-BEJ669