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#### Keywords

- hidden variables (2)
- mixture (2)
- nonparametric regression (2)
- EM algorith (1)
- EM algorithm (1)
- conditional quantile (1)
- image denoising (1)
- kernel estimates (1)
- localization (1)
- neural network (1)

We consider data generating mechanisms which can be represented as mixtures of finitely many regression or autoregression models. We propose nonparametric estimators for the functions characterizing the various mixture components based on a local quasi maximum likelihood approach and prove their consistency. We present an EM algorithm for calculating the estimates numerically which is mainly based on iteratively applying common local smoothers and discuss its convergence properties.

We derive some asymptotics for a new approach to curve estimation proposed by Mr'{a}zek et al. cite{MWB06} which combines localization and regularization. This methodology has been considered as the basis of a unified framework covering various different smoothing methods in the analogous two-dimensional problem of image denoising. As a first step for understanding this approach theoretically, we restrict our discussion here to the least-squares distance where we have explicit formulas for the function estimates and where we can derive a rather complete asymptotic theory from known results for the Priestley-Chao curve estimate. In this paper, we consider only the case where the bias dominates the mean-square error. Other situations are dealt with in subsequent papers.

We consider the problem of estimating the conditional quantile of a time series at time \(t\) given observations of the same and perhaps other time series available at time \(t-1\). We discuss sieve estimates which are a nonparametric versions of the Koenker-Bassett regression quantiles and do not require the specification of the innovation law. We prove consistency of those estimates and illustrate their good performance for light- and heavy-tailed distributions of the innovations with a small simulation study. As an economic application, we use the estimates for calculating the value at risk of some stock price series.