A note on the properties of some time varying bilinear models

In this note, a sufficient condition is given for the existence and uniqueness of a stable causal solution for bilinear time series with time-varying coefficients; also some conditions for invertibility and the optimal prediction procedure are given. The notions of controllability, observability and minimality are discussed.Time-varying bilinear models Quasi-stationary processes Causality Invertibility Controllability Observability Minimality

Testing variances in wavelet regression models

In this paper we develop an asymptotically locally optimal partial score test for testing the suitability of a homoscedastic wavelet model against a general heteroscedastic wavelet model. As the construction of the partial score test requires a consistent estimate for the nuisance parameter, namely the constant variance estimate under the null hypothesis, we conduct a comprehensive investigation in order to choose its best possible estimate among some competitors. The size and power performances of the partial score test are reported for testing for heteroscedasticity in a time series of finite length.Daubechies wavelet Gasser-Muller estimator Haar wavelet Partial score test Weighted least squares

On exact minimax wavelet designs obtained by simulated annealing

We construct minimax robust designs for estimating wavelet regression models. Such models arise from approximating an unknown nonparametric response by a wavelet expansion. The designs are robust against errors in such an approximation, and against heteroscedasticity. We aim for exact, rather than approximate, designs; this is facilitated by our use of simulated annealing. The relative simplicity of annealing allows for a much more complete treatment of some hitherto intractable problems initially addressed in Oyet and Wiens (J. Nonparametric Stat. 12 (2000) 837). Thus, we are able to exhibit integer-valued designs for estimating higher order wavelet approximations of nonparametric curves. The exact designs constructed for multiwavelet approximations of various orders are found to be symmetric and periodic, as anticipated in Oyet and Wiens (J. Nonparametric Stat. 12 (2000) 837). We also construct integer-valued designs based on the Daubechies wavelet system with a wavelet number of 5.Daubechies wavelet Heteroscedastic Multiwavelet Nonparametric regression Rounding Weighted least squares