### LSTAT2430 Advanced Nonparametric Statistics: Fourier-based methods

The objective of this course is to develop applications of non-parametric curve estimation methods to two modern fields of statistics: for the one, to the estimation of spectral densities of time series, including multivariate time series, for the other the interpretation of projection-based estimators as (linear and non-linear) smoothers in a general regression or density estimation context.

Students will be able to understand and appreciate finite sample and asymptotic properties of modern curve estimation methods, along the problem of estimating spectral densities of time series (an alternative and compact way to describe the correlation structure in a given time series in an enhanced and interpretable way).

For projection based estimators (e.g. wavelet estimators), the emphasis will be on understanding the link to classical kernel estimation and why non-linear (threshold) based projection estimators can oer interesting
advantages, both in theory and practice. Beyond developing the theoretical background, the numerical performance of the studied methods will be analysed by the students using R.

Contents:

1. Spectral density estimation: Definition, periodogram-based kernel estimators(properties, asymptotics, higher order kernels, multivariate spectral densities, bandwidth selection, time-varying
spectral densities), interpretations.
2. Projection-based estimators: General definition, specific wavelet approach (properties and asymptotics, mainly via simple Haar basis estimators), comparison of linear and non-linear methods (link to kernel estimation, overview on different thresholding methods), examples.

The treated topics will be chosen according to the interest of the students.

Bibliography:

For part 1 (Spectral density estimation):

Brockwell, P. and Davis, R. (2009). Time Series: Theory and Methods. Springer Series in Statistics.
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'Shumway, R. and Stoffer, D. (2011). Time Series and its Applications. Springer.
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Brillinger, D. (1981). Time Series; Data Analysis and Theory. Holden Day.
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For part 2 on Wavelets:

Nason, G.P. (2008). Wavelet Methods in Statistics with R. Springer.

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Jansen, M. (2001). Noise Reduction by Wavelet Thresholding. Springer

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Vidakovic, B. (1999). Statistical Modellng by Wavelets. Wiley.
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Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.B. (1998). Wavelets, Approximation and Statistical Applications. Springer Lecture Notes in Statistics. descriptif-cours-LSTAT2430.pdf