Density deconvolution of a sum variable from noisy data with unknown noise distribution

Abstract Let X 1 , X 2 , …, X ℓ with an integer ℓ ≥ 2 be independent copies of a continuous random variable X with an unknown density f . We study the nonparametric problem of estimating the density f ℓ of the sum variable X 1 + X 2 + ⋯ + X ℓ using a random sample Y 1 , …, Y n from the distribution of Y generated by the model Y = X + ɛ . Here ɛ is a random noise independent of X and distributed with an unknown density g . By assuming the availability of an additional sample from g and applying the ridge regularization method, we propose an estimator for f ℓ which is shown to be consistent with respect to the mean integrated squared error whenever the Fourier transform of g vanishes on a Lebesgue measure zero set. An error estimate is derived when f has a finite smoothness and g is compactly supported. Moreover, our estimator also achieves the same error estimates as in some related works obtained under supersmooth and ordinary smooth g . Some numerical results are provided to illustrate the estimator’s performance with finite samples.