Asymptotic theory (statistics)

In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size n may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of n → ∞. In practice, a limit evaluation is considered to be approximately valid for large finite sample sizes too. Overview Most statistical problems begin with a dataset of size n. The asymptotic theory proceeds by assuming that it is possible (in principle) to keep collecting additional data, thus that the sample size grows infinitely, i.e. n → ∞. Under the assumption, many results can be obtained that are unavailable for samples of finite size. An example is the weak law of large numbers. The law states that for a sequence of independent and identically distributed (IID) random variables X1, X2, ..., if one value is drawn from each random variable and

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

Gravitational causality and the self-stress of photons

Giulia Isabella

We study causality in gravitational systems beyond the classical limit. Using on-shell methods, we consider the 1-loop corrections from charged particles to the photon energy-momentum tensor - the self-stress - that controls the quantum interaction between two on-shell photons and one off-shell graviton. The self-stress determines in turn the phase shift and time delay in the scattering of photons against a spectator particle of any spin in the eikonal regime. We show that the sign of the beta-function associated to the running gauge coupling is related to the sign of time delay at small impact parameter. Our results show that, at first post-Minkowskian order, asymptotic causality, where the time delay experienced by any particle must be positive, is respected quantum mechanically. Contrasted with asymptotic causality, we explore a local notion of causality, where the time delay is longer than the one of gravitons, which is seemingly violated by quantum effects.

2022

Optimal solution and asymptotic properties of a stochastic control problem arising in sailboat trajectory optimization

Carlo Ciccarella

We study the optimal strategy for a sailboat to reach an upwind island under the hypothesis that the wind direction fluctuates according to a Brownian motion and the wind speed is constant. The work is motivated by a concrete problem which typically arises during sailing regattas, namely finding the best tacking strategy to reach the upwind buoy as quickly as possible. We assume that there is no loss of time when tacking. We first guess an optimal strategy and then we establish its optimality by using the dynamic programming principle. The Hamilton Jacobi Bellmann equation obtained is a parabolic PDE with Neumann boundary conditions. Since it does not admit a closed form solution, the proof of optimality involves an intricate estimate of derivatives of the value function. We explicitly provide the asymptotic shape of the value function. In order to do so, we prove a result on large time behavior for solutions to time dependent parabolic PDE using a coupling argument. In particular, a boat far from the island approaches the island at

\frac{1}{2} + \frac{\sqrt{2}}{\pi} = 95.02\%

of the boat's speed.

EPFL2017