On simple wavelet estimators of random signals and their small-sample properties

Abstract

In the paper we suggest certain nonparametric estimators of random signals based on the wavelet transform. We consider stochastic signals embedded in white noise and extractions with wavelet denoizing algorithms utilizing the non-decimated discrete wavelet transform and the idea of wavelet scaling. We evaluate properties of these estimators via extensive computer simulations and partially also analytically. Our wavelet estimators of random signals have clear advantages over parametric maximum likelihood methods as far as computational issues are concerned, while at the same time they can compete with these methods in terms of precision of estimation in small samples. An illustrative example concerning smoothing of survey data is also provided.

Keywords:

• random signals
• signal estimation
• wavelet denoizing
• maximal overlap discrete wavelet transform

MSC codes:

• 60
• 62
• 91

Acknowledgements

The author would also like to thank the anonymous reviewer for valuable comments and suggestions which helped improve the paper.

Notes

1. For other fundamental contributions to this methodology as well as recent developments, see [6,7,8,9] and references therein as well as the collection of papers available at http://www-stat.stanford.edu/donoho/reports.html.

2. In fact, our simulation and empirical studies presented in a companion paper confirm that both methods of signal extraction discussed here can be useful in forecasting.

3. Computer codes are available upon request.

4. It means, in particular, that the estimator of the wavelet variance was unbiased.

5. An alternative strategy consists in performing a sort of backcasting.

6. The MSEs were estimated as: $(1/b)\sum _{i}1/T\sum _t(x_t^{(i)}-{\tilde{x}}_t^{(i)}{)}^2$, where $x_t^{(i)}$ and ${\tilde{x}}_t^{(i)}$ denote the signal and its appropriate estimate obtained in the ith iteration, b stands for the number of replications and T is the sample length used in the evaluation of the estimators.

7. The other results are available upon request.

8. In our computations we used the Matlab® function fmincon run under the following settings: Algorithm = ‘interior-point’, MaxFunEvals = 1000, MaxIter = 1000. All initial values of estimated parameters were set to those of the actual DGPs.

9. In the case of nonstationary processes scaling coefficients were left unchanged.

10. See, e.g. [22] and the discussion about the so-called ARIMA-model-based methodology. The ‘maximum noise’ assumption here corresponds to the canonical decomposition of this approach.

Additional information

Funding

The author acknowledges financial support from the Polish National Science Center (Decision no. DEC-2013/09/B/HS4/02716).