Alignment of curves by non-parametric maximum likelihood estimation can be done when the individual transformations of the time axis is modelled by unobserved random shifts. We consider the larger class of models, where the individual time transformations are assumed to be of a parametric form, known up to some individual un-observed random parameters. The nonparametric maximum likelihood approach is used to derive a infinite dimensional score equation, when differentiating with respect to the common shape function. We suggest an algorithm to find the non-parametric maximum likelihood estimator (NPMLE) for the shape function and apply the method to two data examples on feta cheese and crop respectively. We find smooth estimates for the shape functions without choosing any smoothing parameters or kernel function and we estimate realisations of the un-observed transformation parameters that align the curves to satisfy the eye.
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