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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

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Celebrating the 20th anniversary of the presentation of the paper by Dempster, Laird and Rubin which popularized the EM algorithm, we investigate, after a brief historical account, strategies that aim to make the EM algorithm converge faster while maintaining its simplicity and stability (e.g. automatic monotone convergence in likelihood). First we introduce the idea of a`working parameter' to facilitate the search for ef®cient data augmentation schemes and thus fast EM implementations. Second, summarizing various recent extensions of the EM algorithm, we formulate a general alternating expectation± conditional maximization algorithm AECM that couples¯exible data augmentation schemes with model reduction schemes to achieve ef®cient computations. We illustrate these methods using multivariate t-models with known or unknown degrees of freedom and Poisson models for image reconstruction. We show, through both empirical and theoretical evidence, the potential for a dramatic reduction in computational time with little increase in human effort. We also discuss the intrinsic connection between EM-type algorithms and the Gibbs sampler, and the possibility of using the techniques presented here to speed up the latter. The main conclusion of the paper is that, with the help of statistical considerations, it is possible to construct algorithms that are simple, stable and fast.