Logspline Deconvolution in Besov Space

ABSTRACT. In this paper we consider logspline density estimation for random variables which are contaminated with random noise. In the logspline density estimation for data without noise, the logarithm of an unknown density function is estimated by a polynomial spline, the unknown parameters of which are given by maximum likelihood. When noise is present, B‐splines and the Fourier inversion formula are used to construct the logspline density estimator of the unknown density function. Rates of convergence are established when the log‐density function is assumed to be in a Besov space. It is shown that convergence rates depend on the smoothness of the density function and the decay rate of the characteristic function of the noise. Simulated data are used to show the finite‐sample performance of inference based on the logspline density estimation.

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