fixed design - as well as its natural generalization to deterministic covariates - is often used in real-life applications when time is involved in the data set. This is the case for instance in Jirak et al. (2014) where the authors studied the monthly sunspot observations and the annual best running times of 1500 meters. Besides, deterministic design is met accross a number of papers in regression models, see for instance Brown and Low (1996), Meister and Reiß (2013) and the references within. The case of random covariates is obviously the most relevant and appears in essence in many applications in boundary models, among other, in insurance and financial risk modelling when analyzing the optimality of portfolios (see Markowitz (1952) for the seminal contribution).

The remaining part of the manuscript is organized as follows. In section 2 the model is explained, while in section 3 the estimation procedure is described. In section 4 we show consistency of the transformation parameter estimator. In section 5 we present simulation results. The proofs for the random covariate case are given in the appendix, while supplementary material contains proofs for the fixed design case and some additional figures and simulation results.

Model
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The 