The likelihood ratio test (LRT) and the related F test, 1 popularized in astrophysics by Eadie et al. (1971), Bevington (1969), Lampton, Margon, and Bowyer (1976), Cash (1979), andAvni et al. (1978) do not (even asymptotically) adhere to their nominal χ 2 and F distributions in many statistical tests common in astrophysics, thereby casting many marginal line or source detections and non-detections into doubt. Although the above references illustrate the many legitimate uses of these statistics, in some important cases it can be impossible to compute the correct false positive rate. For example, it has become common practice to use the LRT or the F test for detecting a line in a spectral model or a source above background despite the lack of certain required regularity conditions. (These applications were not originally suggested by Cash (1979) or by Bevington (1969)). In these and other settings that involve testing a hypothesis that is on the boundary of the parameter space, contrary to common practice, the nominal χ 2 distribution for the LRT or the F distribution for the F test should not be used. In this paper, we characterize an important class of problems where the LRT and the F test fail and illustrate this non-standard behavior. We briefly sketch several possible acceptable alternatives, focusing on Bayesian posterior predictive probability-values. We present this method in some detail, as it is a simple, robust, and intuitive approach. This alternative method is illustrated using the gamma-ray burst of May 8, 1997 (GRB 970508) to investigate the presence of an Fe K emission line during the initial phase of the observation.There are many legitimate uses of the LRT and the F test in astrophysics, and even when these tests are inappropriate, there remain several statistical alternatives (e.g., judicious use of error bars and Bayes factors). Nevertheless, there are numerous cases of the inappropriate use of the LRT and similar tests in the literature, bringing substantive scientific results into question. * The authors gratefully acknowledge funding for this project partially provided by NSF grants DMS-97-05157 and DMS-01-04129, and by NASA Contract NAS8-39073 (CXC).1 The F test for an additional term in a model, as defined in Bevington (1969) on pp. 208-209, is the ratio 6 A probability-value or p-value is the probability of observing a value of the test statistic (such as χ 2 ) as extreme or more extreme than the value actually observed given that the null model holds (e.g. χ 2 30 ≥ 2.0) Small p-values are taken as evidence against the null model; i.e., p-values are used to calibrate tests. Posterior predictive p-values are a Bayesian analogue; see Section 4.2.3 usually contaminated with background counts, degraded by instrument response, and altered by the effective area of the instrument and interstellar absorption. Thus, we model the observed counts in a detector channel l as independent Poisson 7 random variables with expectation

Wavelets are scaleable, oscillatory functions that deviate from zero only within a limited spatial regime and have average value zero, and thus may be used to simultaneously characterize the shape, location, and strength of astronomical sources. But in addition to their use as source characterizers, wavelet functions are rapidly gaining currency within the source detection field. Wavelet-based source detection involves the correlation of scaled wavelet functions with binned, two-dimensional image data. If the chosen wavelet function exhibits the property of vanishing moments, significantly nonzero correlation coefficients will be observed only where there are high-order variations in the data; e.g., they will be observed in the vicinity of sources. Source pixels are identified by comparing each correlation coefficient with its probability sampling distribution, which is a function of the (estimated or a priori-known) background amplitude.In this paper, we describe the mission-independent, wavelet-based source detection algorithm WAVDETECT, part of the freely available Chandra Interactive Analysis of Observations (CIAO) software package. Our algorithm uses the Marr, or "Mexican Hat" wavelet function, but may be adapted for use with other wavelet functions. Aspects of our algorithm include: (1) the computation of local, exposure-corrected normalized (i.e. flat-fielded) background maps; (2) the correction for exposure variations within the field-of-view (due to, e.g., telescope support ribs or the edge of the field); (3) its applicability within the low-counts regime, as it does not require a minimum number of background counts per pixel for the accurate computation of source detection thresholds; (4) the generation of a source list in a manner that does not depend upon a detailed knowledge of the point spread function (PSF) shape; and (5) error analysis. These features make our algorithm considerably more general than previous methods developed for the analysis of X-ray image data, especially in the low count regime. We demonstrate the robustness of WAVDETECT by applying it to an image from an idealized detector with a spatially invariant Gaussian PSF and an exposure map similar to that of the Einstein IPC; to Pleiades Cluster data collected by the ROSAT PSPC; and to simulated Chandra ACIS-I image of the Lockman Hole region. 4 The CIAO software package may downloaded from http://asc.harvard.edu/ciao/. WAVDETECT is composed of WTRANSFORM, a source detector, and WRECON, a source list generator; these programs may be run separately.5 While it can operate to a limited extent if nothing at all is known about the PSF, our algorithm is most effective if characteristic PSF sizes, e.g. the radii of circles containing 50% of the encircled energy for different off-axis angles, are computable.12 These maps are later combined into a single map used in the calculation of source properties. See §3.2.1.13 If one does not provide an exposure map, a flat one is assumed, to account for the edge of the FOV.

The Chandra Source Catalog (CSC) is a general purpose virtual X-ray astrophysics facility that provides access to a carefully selected set of generally useful quantities for individual X-ray sources, and is designed to satisfy the needs of a broad-based group of scientists, including those who may be less familiar with astronomical data analysis in the X-ray regime. The first release of the CSC includes information about 94,676 distinct X-ray sources detected in a subset of public Advanced CCD Imaging Spectrometer imaging observations from roughly the first eight years of the Chandra mission. This release of the catalog includes point and compact sources with observed spatial extents 30. The catalog (1) provides access to the best estimates of the X-ray source properties for detected sources, with good scientific fidelity, and directly supports scientific analysis using the individual source data; (2) facilitates analysis of a wide range of statistical properties for classes of X-ray sources; and (3) provides efficient access to calibrated observational data and ancillary data products for individual X-ray sources, so that users can perform detailed further analysis using existing tools. The catalog includes real X-ray sources detected with flux estimates that are at least 3 times their estimated 1σ uncertainties in at least one energy band, while maintaining the number of spurious sources at a level of 1 false source per field for a 100 ks observation. For each detected source, the CSC provides commonly tabulated quantities, including source position, extent, multi-band fluxes, hardness ratios, and variability statistics, derived from the observations in which the source is detected. In addition to these traditional catalog elements, for each X-ray source the CSC includes an extensive set of file-based data products that can be manipulated interactively, including source images, event lists, light curves, and spectra from each observation in which a source is detected.

A commonly used measure to summarize the nature of a photon spectrum is the so-called hardness ratio, which compares the numbers of counts observed in different passbands. The hardness ratio is especially useful to distinguish between and categorize weak sources as a proxy for detailed spectral fitting. However, in this regime classical methods of error propagation fail, and the estimates of spectral hardness become unreliable. Here we develop a rigorous statistical treatment of hardness ratios that properly deals with detected photons as independent Poisson random variables and correctly deals with the non-Gaussian nature of the error propagation. The method is Bayesian in nature and thus can be generalized to carry out a multitude of source-populationYbased analyses. We verify our method with simulation studies and compare it with the classical method. We apply this method to real-world examples, such as the identification of candidate quiescent low-mass X-ray binaries in globular clusters and tracking the time evolution of a flare on a low-mass star.

Extreme Ultraviolet Explorer Deep Survey observations of cool stars (spectral type F to M) have been used to investigate the distribution of coronal flare rates in energy and its relation to activity indicators and rotation parameters. Cumulative and differential flare rate distributions were constructed and fitted with different methods. Power laws are found to approximately describe the distributions. A trend toward flatter distributions for latertype stars is suggested in our sample. Assuming that the power laws continue below the detection limit, we have estimated that the superposition of flares with radiated energies of about 10 29 − 10 31 ergs could explain the observed radiative power loss of these coronae, while the detected flares are contributing only ≈ 10 %. While the power-law index is not correlated with rotation parameters (rotation period, projected rotational velocity, Rossby number) and only marginally with the X-ray luminosity, the flare occurrence rate is correlated with all of them. The occurrence rate of flares with energies larger than 10 32 ergs is found to be proportional to the average total stellar X-ray luminosity. Thus, energetic flares occur more often in X-ray bright stars than in X-ray faint stars. The normalized occurrence rate of flares with energies larger than 10 32 ergs increases with increasing L X /L bol and stays constant for saturated stars. A similar saturation is found below a critical Rossby number. The findings are discussed in terms of simple statistical flare models in an attempt to explain the previously observed trend for higher average coronal temperatures in more active stars. It is concluded that flares can contribute a significant amount of energy to coronal heating in active stars.

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We investigate the EUV and X-ray flare rate distribution in radiated energy of the late-type active star AD Leo. Occurrence rates of solar flares have previously been found to be distributed in energy according to a power law, dN/dE ∝ E −α , with a power-law index α in the range 1.5−2.6. If α ≥ 2, then an extrapolation of the flare distribution to low flare energies may be sufficient to heat the complete observable X-ray/EUV corona.We have obtained long observations of AD Leo with the EUVE and Bep-poSAX satellites. Numerous flares have been detected, ranging over almost two orders of magnitude in their radiated energy. We compare the observed light curves with light curves synthesized from model flares that are distributed in energy according to a power law with selectable index α. Two methods are applied, the first comparing flux distributions of the binned data, and the second using the distributions of photon arrival time differences in the unbinned data (for EUVE). Subsets of the light curves are tested individually, and the quiescent flux has optionally been treated as a superposition of flares from the same flare distribution. We find acceptable α values between 2.0−2.5 for the EUVE DS and the BeppoSAX LECS data. Some variation is found depending on whether or not a strong and long-lasting flare occurring in the EUVE data is included. The BeppoSAX MECS data indicate a somewhat shallower energy distribution (smaller α) than the simultaneously observed LECS data, which is attributed to the harder range of sensitivity of the MECS detector and the increasing peak temperatures of flares with increasing total (radiative) energy. The results suggest that flares can play an important role in the energy release of this active corona. We discuss caveats related to time variability, total energy, and multiple power-law distributions. Studying the limiting case of a corona that is entirely heated by a population of flares, we derive an expression for the time-averaged coronal differential emission measure distribution (DEM) that can be used as a diagnostic for the flare energy distribution. The shape of the analytical DEM agrees with previously published DEMs from observations of active stars.