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Concept

Statistical significance

Summary

In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; and the p-value of a result, p, is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. The result is statistically significant, by the standards of the study, when p \le \alpha. The significance level for a study is chosen before data collection, and is typically set to 5% or much lower—depending on the field of study. In any experiment or observation that involves drawing a sample from a population, there is always the possibility that an observed effect would have occurred due to sampling error alone. But

Official source

https://en.wikipedia.org/wiki/Statistical_significance

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Ce cours est une introduction à la théorie des probabilités et aux méthodes statistique.

The course covers basic econometric models and methods that are routinely applied to obtain inference results in economic and financial applications.

This course is neither an introduction to the mathematics of statistics nor an introduction to a statistics program such as R. The aim of the course is to understand statistics from its experimental design and to avoid common pitfalls of statistical reasoning. There is space to discuss ongoing work.

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Bayesian vs. frequentist: analysis of statistical schools of thought

Hossein Rahdari, Sui Kai Wong

2011

Multiple testing with test statistics following heavy-tailed distributions

Zhiwen Jiang

In multiple testing problems where the components come from a mixture model of noise and true effect, we seek to first test for the existence of the non-zero components, and then identify the true alternatives under a fixed significance level

\alpha

. Two parameters, namely the fraction of the non-null components

\varepsilon

and the size of the effects

\mu

, characterise the two-point mixture model under the global alternative. When the number of hypotheses

m

goes to infinity, we are interested in an asymptotic framework where the fraction of the non-null components is vanishing, and the true effects need to be sizable to be detected. Donoho and Jin give an explicit form of the asymptotic detectable boundary based on the Gaussian mixture model under the classic calibration of the parameters of the mixture model. We prove the analogous results for the Cauchy mixture distribution as an example heavy-tailed case. This requires a different formulation of the parameters, which reflects the added difficulties. We also propose a multiple testing procedure based on a filtering approach that can discover the true alternatives. Benjamini and Hochberg (BH) compare the observed

p

-values to a linear threshold curve and reject the null hypotheses from the minimum up to the last up-crossing, and prove the false discovery rate (FDR) is controlled. However, there is an intrinsic difference in heavy-tailed settings. Were we to use the BH procedure we would get a highly variable positive false discovery rate (pFDR). In our study we analyse the distribution of the

p

-values and devise a new multiple testing procedure to combine the usual case and the heavy-tailed case based on the empirical properties of the

p

-values. The filtering approach is designed to eliminate most

p

-values that are more likely to be uniform, while preserving most of the true alternatives. Based on the filtered

p

-values, we estimate the mode

\vartheta

and define the rejection region

\mathscr{R}(\vartheta, \delta)=\left[ \vartheta -\delta/2, \vartheta +\delta/2 \right]

such that the most informative

p

-values are included. The length

\delta

is chosen by controlling the data-dependent estimation of FDR at a desired level.

EPFL2021

A high-energy (0.5-3.0 TeV centre of mass) electron-positron Compact Linear Collider (CLIC) is being studied at CERN as a new physics facility. The design study has been optimized for 3 TeV centre-of-mass energy. Intense bunches injected into the main linac must have unprecedentedly small emittances to achieve the design luminosity 1035cm-2s-1 required for the physics experiments. The positron and electron bunch trains will be provided by the CLIC injection complex. This thesis describes an optics design and performance of a positron damping ring developed for producing such ultra-low emittance beam. The linear optics of the CLIC damping ring is optimized by taking into account the combined action of radiation damping, quantum excitation and intrabeam scattering. The required beam emittance is obtained by using a TME (Theoretical Minimum Emittance) lattice with compact arcs and short period wiggler magnets located in dispersion-free regions. The damping ring beam energy is chosen as 2.42 GeV. The lattice features small values of the optical functions, a large number of compact TME cells, and a large number of wiggler magnets. Strong sextupole magnets are needed for the chromatic correction which introduces significant nonlinearities, decreasing the dynamic aperture. The nonlinear optimization of the lattice is described. An appropriate scheme of chromaticity correction is determined that gives reasonable dynamic aperture and zero chromaticity. The nonlinearities induced by the short period wiggler magnets and their influence on the beam dynamics are also studied. In addition, approaches for absorption of synchrotron radiation power produced by the wigglers are discussed. Realistic misalignments of magnets and monitors increase the equilibrium emittance. The sensitivity of the CLIC damping ring to various kinds of alignment errors is studied. Without any correction, fairly small vertical misalignments of the quadrupoles and, in particular, the sextupoles, introduce unacceptable distortions of the closed orbit as well as intolerable spurious vertical dispersion and coupling due to the strong focusing optics of the damping ring. A sophisticated beam-based correction scheme has been developed in order to bring the design target emittances and the dynamic aperture back to the ideal value. The correction using dipolar correctors and several skew quadrupole correctors allows a minimization of the closed-orbit distortion, the cross-talk between vertical and horizontal closed orbits, the residual vertical dispersion and the betatron tune coupling. The small emittance, short bunch length, and high current in the CLIC damping ring could give rise to collective effects which degrade the quality of the extracted beam. A number of possible instabilities and an estimate of their impact on the ring performance are briefly surveyed. The effects considered include fast beam-ion instability, coherent synchrotron radiation, Touschek scattering, intrabeam scattering, resistive-wall wake fields, and electron cloud.

EPFL2006