On a small sample adjustment for the profile score function in semiparametric smoothing models

We consider the profile score function in models with smooth and parametric components. If local respectively weighted likelihood estimation is used for fitting the smooth component, the resulting profile likelihood estimate for the parametric component is asymptotically efficient as shown in T. A. Severini and W. H. Wong (1992, Ann. Statist.20, 1768–1802). However, as in solely parametric models the profile score function is not unbiased. We propose a small sample bias adjustment which results by extending the correction suggested in P. McCullagh and R. Tibshirani (1990, J. Roy. Statist. Soc. Ser. B52, 325–344) to the framework of semiparametric models