Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether these data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second, we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods, we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.

Dwarf spheroidal J-factor likelihoods for generalized NFW profiles

ABSTRACT Indirect detection strategies of particle dark matter (DM) in Dwarf spheroidal satellite galaxies (dSphs) typically entail searching for annihilation signals above the astrophysical background. To robustly compare model predictions with the observed fluxes of product particles, most analyses of astrophysical data – which are generally frequentist – rely on estimating the abundance of DM by calculating the so-called J factor. This quantity is usually inferred from the kinematic properties of the stellar population of a dSph using the Jeans equation, commonly by means of Bayesian techniques that entail the presence (and additional systematic uncertainty) of prior choice. Here, extending earlier work, we develop a scheme to derive the profile likelihood for J factors of dwarf spheroidals for models with five or more free parameters. We validate our method on a publicly available simulation suite, released by the Gaia Challenge, finding satisfactory statistical properties for bias and probability coverage. We present the profile likelihood function and maximum likelihood estimates for the J-factor of 10 dSphs. As an illustration, we apply these profile likelihoods to recently published analyses of γ-ray data with the Fermi Large Area Telescope to derive new, consistent upper limits on the DM annihilation cross-section. We do this for a subset of systems, generally referred to as classical dwarfs. The implications of these findings for DM searches are discussed, together with future improvements and extensions of this technique.

Estimating the Area under the ROC Curve with Modified Profile Likelihoods

Receiver operating characteristic (ROC) curves are a frequent tool to study the discriminating ability of a certain characteristic. The area under the ROC curve (AUC) is a widely used measure of statistical accuracy of continuous markers for diagnostic tests, and has the advantage of providing a single summary index of overall performance of the test. Recent studies have shown some critical issues related to traditional point and interval estimates for the AUC, especially for small samples, more complex models, unbalanced samples or values near the boundary of the parameter space, i.e., when the AUC approaches the values 0.5 or 1.Parametric models for the AUC have shown to be powerful when the underlying distributional assumptions are not misspecified. However, in the above circumstances parametric inference may be not accurate, sometimes yielding misleading conclusions. The objective of the paper is to propose an alternative inferential approach based on modified profile likelihoods, which provides more accurate statistical results in any parametric settings, including the above circumstances. The proposed method is illustrated for the binormal model, but can potentially be used in any other complex model and for any other parametric distribution. We report simulation studies to show the improved performance of the proposed approach, when compared to classical first-order likelihood theory. An application to real-life data in a small sample setting is also discussed, to provide practical guidelines.

LIKELIHOOD INFERENCE IN AN AUTOREGRESSION WITH FIXED EFFECTS

We calculate the bias of the profile score for the regression coefficients in a multistratum autoregressive model with stratum-specific intercepts. The bias is free of incidental parameters. Centering the profile score delivers an unbiased estimating equation and, upon integration, an adjusted profile likelihood. A variety of other approaches to constructing modified profile likelihoods are shown to yield equivalent results. However, the global maximizer of the adjusted likelihood lies at infinity for any sample size, and the adjusted profile score has multiple zeros. Consistent parameter estimates are obtained as local maximizers inside or on an ellipsoid centered at the maximum likelihood estimator.