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We propose a method for setting limits that avoids excluding parameter values for which the sensitivity falls below a specified threshold. These power-constrained limits (PCL) address the issue that motivated the widely used CLs procedure, but do so in a way that makes more transparent the properties of the statistical test to which each value of the parameter is subjected. A case of particular interest is for upper limits on parameters that are proportional to the cross section of a process whose existence is not yet established. The basic idea of the power constraint can easily be applied, however, to other types of limits.
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In counting experiments, one can set an upper limit on the rate of a Poisson process based on a count of the number of events observed due to the process. In some experiments, one makes several counts of the number of events, using different instruments, different event detection algorithms, or observations over multiple time intervals. We demonstrate how to generalize the classical frequentist upper limit calculation to the case where multiple counts of events are made over one or more time intervals using several (not necessarily independent) procedures. We show how different choices of the rank ordering of possible outcomes in the space of counts correspond to applying different levels of significance to the various measurements. We propose an ordering that is matched to the sensitivity of the different measurement procedures and show that in typical cases it gives stronger upper limits than other choices. As an example, we show how this method can be applied to searches for gravitational-wave bursts, where multiple burst-detection algorithms analyse the same data set, and demonstrate how a single combined upper limit can be set on the gravitational-wave burst rate.
The question of exclusion region construction in new phenomenon searches has been causing considerable discussions for many years and yet no clear mathematical definition of the problem has been stated so far. In this paper we formulate the problem in mathematical terms and propose a solution to the problem within the framework of statistical tests. The proposed solution avoids problems of the currently used procedures.
Time series of observables measured from complex systems do often exhibit non-normal statistics, their statistical distributions (PDFs) are not gaussian and often skewed, with roughly exponential tails. Departure from gaussianity is related to the intermittent development of large-scale coherent structures. The existence of these structures is rooted into the nonlinear dynamical equations obeyed by each system, therefore it is expected that some prior knowledge or guessing of these equations is needed if one wishes to infer the corresponding PDF; conversely, the empirical knowledge of the PDF does provide information about the underlying dynamics. In this work we suggest that it is not always necessary. We show how, under some assumptions, a formal evolution equation for the PDF $p(x)$ can be written down, corresponding to the progressive accumulation of measurements of the generic observable $x$. The limiting solution to this equation is computed analytically, and shown to interpolate between some of the most common distributions, Gamma, Beta and Gaussian PDFs. The control parameter is just the ratio between the rms of the fluctuations and the range of allowed values. Thus, no information about the dynamics is required.
Measurements are inseparable from inference, where the estimation of signals of interest from other observations is called an indirect measurement. While a variety of measurement limits have been defined by the physical constraint on each setup, the fundamental limit of an indirect measurement is essentially the limit of inference. Here, we propose the concept of statistical limits on indirect measurement: the bounds of distinction between signals and noise and between a signal and another signal. By developing the asymptotic theory of Bayesian regression, we investigate the phenomenology of a typical indirect measurement and demonstrate the existence of these limits. Based on the connection between inference and statistical physics, we also provide a unified interpretation in which these limits emerge from phase transitions of inference. Our results could pave the way for novel experimental design, enabling assess to the required quality of observations according to the assumed ground truth before the concerned indirect measurement is actually performed.
Many time series produced by complex systems are empirically found to follow power-law distributions with different exponents $alpha$. By permuting the independently drawn samples from a power-law distribution, we present non-trivial bounds on the memory strength (1st-order autocorrelation) as a function of $alpha$, which are markedly different from the ordinary $pm 1$ bounds for Gaussian or uniform distributions. When $1 < alpha leq 3$, as $alpha$ grows bigger, the upper bound increases from 0 to +1 while the lower bound remains 0; when $alpha > 3$, the upper bound remains +1 while the lower bound descends below 0. Theoretical bounds agree well with numerical simulations. Based on the posts on Twitter, ratings of MovieLens, calling records of the mobile operator Orange, and browsing behavior of Taobao, we find that empirical power-law distributed data produced by human activities obey such constraints. The present findings explain some observed constraints in bursty time series and scale-free networks, and challenge the validity of measures like autocorrelation and assortativity coefficient in heterogeneous systems.
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