The authors consider a binary regression models with linear and monotone parts and with cloglog link function, i.e., the binary outcome (response) $Δ$ depends on the covariates $X$ and $Z$ by $$\text{\bf P}(Δ=0\vert X,Z)=(1-g(Z))^{\exp(β’X)},$$ where $g:[0,+\infty)\to[0,1]$ is an increasing and continuously differentiable unknown function. This model is analogous to the Cox proportional hazards model for current status data and $Λ(z)=-\log(1-g(z))$ plays the role of a cumulative hazard function. The paper is devoted to likelihood ratio tests for $H_0:β=β_0$ and $\tilde H_0:Λ(z_0)=ϑ_0$. It is shown that the LRs are asymptotically pivotal quantities and the asymptotic distribution of LR for $\tilde H_0$ is described in terms of a two-sided Brownian motion with parabolic drift. Asymptotic confidence intervals for $β$ and $Λ(z_0)$ are constructed. Results of simulations and applications to biological data are presented.
Reviewer: R. E. Maiboroda (Kyïv)