Supplement paper to 'Nonparametric estimation in hidden Markov models'
Thierry Dumont, Sylvain Le Corff · 2012
This document is a supplementary material to the article “Nonparametric estimation using partially observed Markov chains”. It provides additional proofs of some technical results given in the original paper. Section 1 recalls the model, the definitions and the assumptions used in the paper. Section 2 provides proofs of some results stated in the paper and Section 3 gives details on the algorithm used to perform the Expectation-Maximization based estimation. 1 Model and definitions In this section, we recall the model and the assumptions given in Dumont and Le Corff [2012]. Comments on these assumptions can be found in [Dumont and Le Corff, 2012, Section 2]. Let ℓ and m be positive integers and K be a subset of R m. The main statistical problem considered in this paper is the estimation of an unknown target function f ⋆ : K → R ℓ when observing a process {Yk} k∈N such that for any k ≥ 0, Yk belongs to R ℓ and satisfies Yk def = f⋆(Xk) + ɛk. {ɛk} k∈N is assumed to be an i.i.d Gaussian process with common distribution N (0, σ 2 Iℓ), Iℓ being the identity matrix of size ℓ and σ 2 a fixed positive parameter. Denote by ϕ the probability distribution of ɛ0, i.e. ∀z ∈ R ℓ, ϕ(z) def