An exoplanet transit survey team runs five checks on one target's light curve.

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Error, Chance and the Machine, Ch Tests, intervals and Bayesian inference

Problem

An exoplanet transit survey team runs five checks on one target's light curve. ((a)) A transit-plus-baseline model is fitted to N=16N=16 epochs with p=2p=2 free parameters, giving χ2=21.0\chi^2=21.0. Using Q(χ2;ν)=e−χ2/2∑i=0ν/2−1(χ2/2)i/i!Q(\chi^2;\nu)=e^{-\chi^2/2}\sum_{i=0}^{\nu/2-1}(\chi^2/2)^i/i!, find ν\nu and the p-value. ((b)) Six independent depth measurements (standard deviation σ=1\sigma=1 in the fitted units) test H0:depth=0H_0:\text{depth}=0 against H1:depth=1H_1:\text{depth}=1 at size α=0.05\alpha=0.05 (z0.05=1.645z_{0.05}=1.645). Give the Neyman-Pearson threshold and its power, using Φ(0.804)=0.7893\Phi(0.804)=0.7893. ((c)) Adding a baseline-slope coefficient, free to take either sign and zero under the null, to the light-curve model gives −2ln⁡Λ=2.71-2\ln\Lambda=2.71. Using Φ(1.645)=0.9500\Phi(1.645)=0.9500, find the p-value from Wilks' theorem. ((d)) A follow-up run with negligible background records no secondary-eclipse event. Find the 68 per cent Neyman upper limit on the mean number of such events. ((e)) In injection-recovery tests, the pipeline recovers 18 of 20 injected transits. With a Beta(2,2)\mathrm{Beta}(2,2) prior on the recovery efficiency θ\theta, give the posterior, its mean and its mode, and compare the mode with the maximum-likelihood estimate.

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