Efficient estimation for a subclass of shape invariant models

TL;DR

This paper introduces an efficient estimation method for a shape invariant model involving unknown periodic functions, phases, and amplitudes, using profile likelihood and Fourier basis, achieving asymptotic efficiency.

Contribution

It proposes a novel asymptotically efficient estimator for the finite-dimensional parameters in the shape invariant model, along with a consistent estimator for the common shape.

Findings

Estimator is asymptotically efficient for phases and amplitudes.

Estimation method yields a consistent, asymptotically linear estimator for the common shape.

Method leverages profile likelihood and Fourier basis for improved accuracy.

Abstract

In this paper, we observe a fixed number of unknown $2\pi$-periodic functions differing from each other by both phases and amplitude. This semiparametric model appears in literature under the name "shape invariant model." While the common shape is unknown, we introduce an asymptotically efficient estimator of the finite-dimensional parameter (phases and amplitude) using the profile likelihood and the Fourier basis. Moreover, this estimation method leads to a consistent and asymptotically linear estimator for the common shape.