Selection Of The Order And The Degrees Of Smoothing For Non-parametric Ordered Hidden Markov Models

A hidden Markov model is a mixture model that is based on some parametric familysuch that its mixing process {s_t} is a K-state Markov chain. Under the case of y_t|{s_t=k}-N(α_k+β_kx_t,σ_k²), the problem can be solved from Chopin(2007). But we hote that the mean in the normal distribution is a linear function. This paper solved the problem that the mean is an unknown non-linear function. Firstly, we should fit the unknown non-linear function by spline function; secondly, we need to write the likelihood function of the first t observations y₁,...,y_t;thirdly, we are interestedin the coefficients of the spline function, that isα_k(s)_j,which satisfies the constraint of (?)|α_k(s)j|≤t;at last, we can obtain the maximum likelihood estimation of the coefficients by using Kuhn-Tuckerconditions.