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Density estimation of a mixture distribution with unknown point-mass and normal error
Density estimation of a mixture distribution with unknown point-mass and normal error
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Date
2021
Authors
Dang Duc Trong
Nguyen Hoang Thanh
Nguyen Dang Minh
Nguyen Nhu Lan
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Abstract
We consider the model Y = X + ξ where Y is observable, ξ is a noise random variable
with density fξ , X has an unknown mixed density such that P(X = Xc ) = 1 − p,
P(X = a) = p with Xc being continuous and p ∈ (0, 1), a ∈ R. Typically, in the last
decade, the model has been widely considered in a number of papers for the case of
fully known quantities a, fξ . In this paper, we relax the assumptions and consider the
parametric error ξ ∼ σN(0, 1) with an unknown σ > 0. From i.i.d. copies Y1, . . . , Ym of
Y we will estimate (σ, p, a, fXc ) where fXc is the density of Xc . We also find the lower
bound of convergence rate and verify the minimax property of established estimators.
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Keywords
Deconvolution,
Mixture distribution,
Inversion problems,
Nonparametric estimation