Numeracy—the ability to understand and use numeric information—is linked to good decision-making. Several problems exist with current numeracy measures, however. Depending on the participant sample, some existing measures are too easy or too hard; also, established measures often contain items well-known to participants. The current article aimed to develop new numeric understanding measures (NUMs) including a 1-item (1-NUM), 4-item (4-NUM), and 4-item adaptive measure (A-NUM). In a calibration study, 2 participant samples (n = 226 and 264 from Amazon’s Mechanical Turk [MTurk]) each responded to half of 84 novel numeracy items. We calibrated items using 2-parameter logistic item response theory (IRT) models. Based on item parameters, we developed the 3 new numeracy measures. In a subsequent validation study, 600 MTurk participants completed the new numeracy measures, the adaptive Berlin Numeracy Test, and the Weller Rasch-Based Numeracy Test, in randomized order. To establish predictive and convergent validities, participants also completed judgment and decision tasks, Raven’s progressive matrices, a vocabulary test, and demographics. Confirmatory factor analyses suggested that the 1-NUM, 4-NUM, and A-NUM load onto the same factor as existing measures. The NUM scales also showed similar association patterns to subjective numeracy and cognitive ability measures as established measures. Finally, they effectively predicted classic numeracy effects. In fact, based on power analyses, the A-NUM and 4-NUM appeared to confer more power to detect effects than existing measures. Thus, using IRT, we developed 3 brief numeracy measures, using novel items and without sacrificing construct scope. The measures can be downloaded as Qualtrics files (https://osf.io/pcegz/).

Initially motivated by a practical issue in target detection vialaser vibrometry, we are interested in the problem of periodicsignal detection in a Gaussian fixed design regression framework.Assuming that the signal belongs to some periodic Sobolev ball andthat the variance of the noise is known, we first consider theproblem from a minimax point of view: we evaluate the so-calledminimax separation rate which corresponds to the minimall 2-distance between the signal and zero so that the detection ispossible with prescribed probabilities of error. Then, we propose atesting procedure which is available when the variance of the noiseis unknown and which does not use any prior information about thesmoothness degree or the period of the signal. We prove that it isadaptive in the sense that it achieves, up to a possible logarithmicfactor, the minimax separation rate over various periodic Sobolevballs simultaneously. The originality of our approach as compared torelated works on the topic of signal detection is that our testingprocedure is sensitive to the periodicity assumption on the signal.A simulation study is performed in order to evaluate the effect ofthis prior assumption on the power of the test. We do observe thegains that we could expect from the theory. At last, we turn to theapplication to target detection by laser vibrometry that we had inview.

We propose a test of a qualitative hypothesis on the mean of a n-Gaussianvector. The testing procedure is available when the variance of theobservations is unknown and does not depend on any prior information onthe alternative. The properties of the test are non-asymptotic. Fortesting positivity or monotonicity, weestablish separation rates with respect to the Euclidean distance, oversubsets of $\mathbb{R}^{n}$ which are related to Hölderian balls in functionalspaces. We provide a simulation study in order to evaluate theprocedure when the purpose is to test monotonicity in a functionalregression model and to check the robustness of the procedure tonon-Gaussian errors.