We consider a nonparametric model $\mathcal{E}^{n},$ generated by independent observations $X_{i},$ $i=1,...,n,$ with densities $p(x,θ_{i}),$ $i=1,...,n,$ the parameters of which $θ_{i}=f(i/n)\in Θ$ are driven by the values of an unknown function $f:[0,1]\rightarrow Θ$ in a smoothness class. The main result of the paper is that, under regularity assumptions, this model can be approximated, in the sense of the Le Cam deficiency pseudodistance, by a nonparametric Gaussian shift model $Y_{i}=Γ(f(i/n))+\varepsilon _{i},$ where $\varepsilon_{1},...,\varepsilon _{n}$ are i.i.d. standard normal r.v.'s, the function $Γ(θ):Θ\rightarrow \mathrm{R}$ satisfies $Γ^{\prime}(θ)=\sqrt{I(θ)}$ and $I(θ)$ is the Fisher information corresponding to the density $p(x,θ).$