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Table of contents
I Introduction
1 Exoplanet detection
1.1 Methods and discoveries
1.2 The radial-velocity technique
1.2.1 Planetary signatures
1.2.2 Method of measurement and instrumental errors
1.3 Stellar noises
1.3.1 Magnetic activity
1.3.2 Oscillations
1.3.3 Convection
1.3.4 Summary
1.4 Simulations of the convective stellar activity
1.5 Conclusions on the astrophysical context of the thesis
2 Detection of periodic signatures in noise: tools, models and tests
2.1 Hypothesis testing
2.1.1 Design of hypotheses
2.1.2 Statistical tests and decision rule
2.1.3 Frequentist approaches: tests based on likelihood functions
2.1.4 A brief comment on Bayesian approaches
2.2 Periodogram(s)
2.2.1 The classical periodogram
2.2.2 Periodogram of a single sinusoid
2.2.3 Parseval’s identity
2.2.4 Periodogram resolution
2.2.5 Bias and variance of the periodogram
2.2.6 Asymptotic distribution in the case of a pure random process
2.2.7 Asymptotic distribution of P in the case of a periodic signal in noise
2.2.8 A note on some modied periodograms
2.3 Planetary signatures in the periodogram
2.3.1 Signatures in the time domain
2.3.2 Signatures in the Fourier domain
2.3.3 Discussion and signal model
2.4 Statistical tests applicable to the periodogram
2.4.1 Preliminary notations
2.4.2 Classical tests
2.4.3 Tests designed for multiple sinusoids
2.5 Adaptive tests that can be applied to the periodogram
2.5.1 Introduction of the problem
2.5.2 Example: Kolmogorov-Smirnov test
2.5.3 Higher Criticism
2.5.4 Berk-Jones test
2.6 Sinusoid detection in unknown noise statistics: related works
2.6.1 Non parametric approches
2.6.2 Parametric methods
2.7 Conclusions
2.7.1 Main conclusions
2.7.2 This thesis
II Proposed detection method
3 Statistical tests exploiting simulations of the colored noise: theoretical anal- ysis and numerical study
3.1 Statement, assumptions and objectives of the detection problem
3.1.1 Model under both hypotheses
3.1.2 Assumptions and objectives
3.2 Statistical distribution of the proposed standardized periodogram
3.2.1 Standardized periodogram eP
jPL
3.2.2 Statistics of eP
jPL under H0
3.2.3 Statistics of eP
jPL under H1
3.2.4 Mean and variance of eP
jPL under H0
3.3 Statistical tests applied to the standardized periodogram
3.3.1 Test of the maximum periodogram value (TM)
3.3.2 Fisher’s test and its variations (TF , TCh, TF;rob)
3.3.3 Testing the Nth C largest periodogram value (TC)
3.3.4 Adaptive tests (HC, BJ)
3.4 Standardization using parametric estimates of SE
3.4.1 AR parameters estimation
3.4.2 Distribution of ePjbS E;AR
3.5 Numerical studies on synthetic noise processes
3.5.1 Classical tests based on eP jPL
3.5.2 Adaptive tests based on ePjPL
3.5.3 Eect of ignoring noise correlations
3.5.4 Comparison of the proposed standardization with a parametric approach
3.5.5 Application to the design of observational strategies
3.6 Summary and conclusions
4 Extension to the case of uneven sampling
4.1 Eect of uneven sampling on the periodogram
4.1.1 Considered sampling patterns
4.1.2 Classical periodogram and uneven sampling
4.2 Variants of the periodogram
4.2.1 The Lomb-Scargle periodogram
4.2.2 Least-square tting and other related periodograms
4.3 Generalized Extreme Value distribution
4.3.1 Generalities
4.3.2 Parameters estimation
4.3.3 Diagnostic plots
4.4 False alarm rates evaluation
4.4.1 Method of \independent » frequencies
4.4.2 Bootstrap methods
4.4.3 FA evaluation using theory of the extremes
4.4.4 Conclusions
4.5 Proposed bootstrap procedure
4.5.1 Direct bootstrap approach
4.5.2 \Accelerated » bootstrap approach
4.6 Numerical studies on a synthetic noise process
4.6.1 Validation on known processes
4.6.2 \GEV accelerated » algorithm
4.7 Conclusions
Conclusions and perspectives
