Wavelet-Based Estimation for Univariate Stable Laws

2005

In this paper a method for the parameter estimation of a Weibull distribution is explained. It is based on the estimates of the cumulative probability function performed by a wavelet method. The estimates, obtained by the application of the proposed method, are compared with the results based on other literature estimators. In particular, it is shown that the estimates are stable also when samples show outliers. Such a method could be utilised in mechanical applications (e.g. transmissions, rolling organs, kinematics couples effort subjected) or time series analysis.

Journal of Economic Surveys, 2013

Economists are already familiar with the Discrete Wavelet Transform. However, a body of work using the Continuous Wavelet Transform has also been growing. We provide a selfcontained summary on continuous wavelet tools, such as the Continuous Wavelet Transform, the Cross-Wavelet, the Wavelet Coherency and the Phase-Di¤erence. Furthermore, we generalize the concept of simple coherency to Partial Wavelet Coherency and Multiple Wavelet Coherency, akin to partial and multiple correlations, allowing the researcher to move beyond bivariate analysis. Finally, we describe the Generalized Morse Wavelets, a class of analytic wavelets recently proposed. A user-friendly toolbox, with examples, is attached to this paper.

Statistica Sinica, 2018

In nonparametric regression with errors in the explanatory variable, the regression function is typically assumed to be smooth, and in particular not to have a rapidly changing derivative. Not all data applications have this property. When the property fails, conventional techniques, usually based on kernel methods, have unsatisfactory performance. We suggest an adaptive, wavelet-based approach, founded on the concept of explained sum of squares, and using matrix regularisation to reduce noise. This non-standard technique is used because conventional wavelet methods fail to estimate wavelet coefficients consistently in the presence of measurement error. We assume that the measurement error distribution is known. Our approach enjoys very good performance, especially when the regression function is erratic. Pronounced maxima and minima are recovered more accurately than when using conventional methods that tend to flatten peaks and troughs. We also show that wavelet techniques have advantages when estimating conventional, smooth functions since they require less sophisticated smoothing parameter choice. That problem is particularly challenging in the setting of measurement error. A data example is discussed and a simulation study is presented.

Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2000

This article defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramér (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete non-decimated Haar wavelets and also a real medical time series example.

arXiv (Cornell University), 2015

We present a new framework for the robust estimation of latent time series models which is fairly general and, for example, covers models going from ARMA to state-space models. This approach provides estimators which are (i) consistent and asymptotically normally distributed, (ii) applicable to various classes of time series models, (iii) straightforward to implement and (iv) computationally efficient. The framework is based on the recently developed Generalized Method of Wavelet Moments (GMWM) and a new robust estimator of the wavelet variance. Compared to existing methods, the latter directly estimates the quantity of interest while performing better in finite samples and using milder conditions for its asymptotic properties to hold. Moreover, results are given showing the identifiability of the GMWM for various classes of time series models thereby allowing this method to consistently estimate many models (and combinations thereof) under mild conditions. Hence, not only does this paper provide an alternative estimator which allows to perform wavelet variance analysis when data are contaminated but also a general approach to robustly estimate the parameters of a variety of (latent) time series models. The simulation studies carried out confirm the better performance of the proposed estimators and the usefulness and broadness of the proposed methodology is shown using practical examples from the domains of economics and engineering with sample sizes up to 900,000.

1997

In these notes we present the main uses of wavelets in statistics. Advances in the area are occuring rapidly, with applications in several elds. There are several books about the mathematical aspects of wavelets, like those by Daubechies(1992) and Chui(1992), and others in areas like signal processing and image compression. The content of these notes borrows heavily from recent works, most of them referenced in the bibliography section. We decided to not include the proofs of the theorems, which the interested reader may nd in the original papers. I would like to thank the organizers of the Third International Conference on Statistical Data Analysis Based on the L 1 ?Norm and Related Methods for the invitation to give a tutorial on wavelets. This gave me the opportunity to prepare this text. Comments and suggestions are welcome and can be sent to pam@ime.usp.br. I would like to thank Chang Chiann for several conversations and David Brillinger for the continuous support. A version in portuguese was presented as a short course in the Seventh Time Series and Econometrics School, in Canela, RS, Brazil and is part of a book to be published by the

Quantitative evidence is presented in order to study the performance of a wavelet-based, second order estimator of relevance to mul-tiscaling phenomena such as telecommunications traffic. Special attention is given to the behavior of the confidence intervals of the scaling exponent under Gaussian assumptions with synthetic long-range dependent trajectories, allowing for a straightforward contrast with the theoretical results presented in the bibliography. The bias-variance tradeoff decision behind the choice of the onset of the scaling region is also re-visited, along with its influence on the confidence of the intervals of the exponent.

Nonlinear Processes in Geophysics, 2007

This study proposes and justifies a Bayesian approach to modeling wavelet coefficients and finding statistically significant features in wavelet power spectra. The approach utilizes ideas elaborated in scale-space smoothing methods and wavelet data analysis. We treat each scale of the discrete wavelet decomposition as a sequence of independent random variables and then apply Bayes' rule for constructing the posterior distribution of the smoothed wavelet coefficients. Samples drawn from the posterior are subsequently used for finding the estimate of the true wavelet spectrum at each scale. The method offers two different significance testing procedures for wavelet spectra. A traditional approach assesses the statistical significance against a red noise background. The second procedure tests for homoscedasticity of the wavelet power assessing whether the spectrum derivative significantly differs from zero at each particular point of the spectrum. Case studies with simulated data and climatic time-series prove the method to be a potentially useful tool in data analysis.

Austrian Journal of Statistics, 2014

A robust approach to the estimation of time series models is proposed. Taking froma new estimation method called the Generalized Method of Wavelet Moments (GMWM)which is an indirect method based on the Wavelet Variance (WV), we replace the classicalestimator of the WV with a recently proposed robust M-estimator to obtain a robustversion of the GMWM. The simulation results show that the proposed approach can beconsidered as a valid robust approach to the estimation of time series and state-spacemodels.

Pattern Recognition Letters, 2007