Local Asymptotic Normality of Ranks and Covariates in Transformation Models

In this paper we exhibit an important but rather complicated example to which the Le Cam-Yang methods may, after some work, be applied. Semiparametric transformation models arise quite naturally in survival analysis. Specifically, let (Z1, T1), (Z2, T2), . . . , (Zn, Tn) be independent and identically distributed with Z, a vector of covariates, having distribution H, and T real. We suppose there exists an unknown strictly monotone transformation a0 : R → R such that, given Z = z, a0(T ) is distributed with distribution function F (·, z, θ) where {F (·, z, θ) : θ ∈ Θ ⊂ Rd} is a regular parametric model. That is, the F (·, z, θ) are dominated by μ with densities f(·, z, θ) and the map θ → √ f(·, z, θ) is Hellinger differentiable. As is usual in these models we take μ to be Lebesgue measure and a0 > 0. The most important special cases of these models are the regression models where θ = (η, ν), and defining the distribution of T given Z structurally, a0(T ) = η + ν′Z + 2,