Bootstrap confidence sets under model misspecification

A multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and a possible model misspecification. Theoretical results justify the bootstrap validity for a small or moderate sample size and allow to control the impact of the parameter dimension $p$: the bootstrap approximation works if $p^3/n$ is small. The main result about bootstrap validity continues to apply even if the underlying parametric model is misspecified under the so-called small modelling bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modelling bias. We illustrate the results with numerical examples for misspecified linear and logistic regressions.Comment: Published at http://dx.doi.org/10.1214/15-AOS1355 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simultaneous likelihood-based bootstrap confidence sets for a large number of models

The paper studies a problem of constructing simultaneous likelihood-based confidence sets. We consider a simultaneous multiplier bootstrap procedure for estimating the quantiles of the joint distribution of the likelihood ratio statistics, and for adjusting the confidence level for multiplicity. Theoretical results state the bootstrap validity in the following setting: the sample size $n$ is fixed, the maximal parameter dimension $p_{\textrm{max}}$ and the number of considered parametric models $K$ are s.t. $(\log K)^{12}p_{\max}^{3}/n$ is small. We also consider the situation when the parametric models are misspecified. If the models' misspecification is significant, then the bootstrap critical values exceed the true ones and the simultaneous bootstrap confidence set becomes conservative. Numerical experiments for local constant and local quadratic regressions illustrate the theoretical results

Spatial distribution of Far infrared emission in spiral galaxies II. Heating sources and gas-to-dust ratio

We study the radial distribution of the temperature of the warm dust and gas-to-dust mass ratios in a sample of 22 spiral galaxies. The heating capabilities of the diffuse interstellar radiation field (ISRF), based on Desert et al. model, are investigated in 13 of the sample galaxies. In general, the temperature of the warm dust decreases away from the center, reaches a minimum value at the mid-disk and increases again in the outer parts of galaxies. Heating a mixture of small and big grains by the ISRF is able to explain the observed behavior qualitatively. However, ultraviolet photons from recent star formation events are necessary for a detailed matching of the warm dust temperature profiles. Very small grains contribute typically more than 50% to the observed flux at 60 micron beyond half the disk radius in galaxies. Optical depth profiles, derived from the observed 60 micron and warm dust temperature profiles, peak at or close to the galactic center. In 13 of the galaxies, where dust temperature profiles are modeled, we obtain gas-to-dust mass ratio profiles, after correction for the contaminating effects of very small grains. The gas-to-dust mass ratio decreases by a factor of 8 from the center to the optical isophotal radius, where the value approaches the local galactic value. We demonstrate that the observed steep gradient is a result of the over-estimation of the molecular mass, and can be flattened out to within a factor of 2, if the molecular hydrogen mass (H2) is recomputed assuming a metallicity dependent conversion factor from CO intensity to H2 column density. The flattened radial profiles indicate a global gas-to-dust ratio of around 300, which is within a factor of two of the local galactic value.Comment: Uses aas2pp4.sty and epsfig.sty, 27 pages. To appear in Astronomical Journal, September 199

Spatial distribution of far infrared emission in spiral galaxies I. Relation with radio continuum emission

We use high resolution IRAS and 20 cm radio continuum (RC) images of a sample of 22 spiral galaxies to study the correlation between the far infra-red (FIR) and RC emissions within the galactic disks. A combination of exponential and gaussian profiles rather than a single exponential profile is found to be a better representation of the observed intensity profiles in the two bands. The gaussian component, which we show is not due to the effects of limited beam-resolution, contains more than 60% of the total flux in majority of the galaxies. The dominance of the gaussian component suggests that the nuclear star forming regions and the bulge stars are more important contributors to the emission in the two bands, rather than the outer exponential stellar disks. The RC profile is flatter compared to the FIR profile, resulting in a decrease of their ratio, Q60, away from the center. However, the Q60 increases in the extreme outer parts, where the dispersion in the FIR and RC correlation is also higher than in the central regions. The global Q60 and its dispersion match those in the inner parts of the galaxies. These results imply that the observed tight correlation in the global quantities reflects processes in the inner regions only where OB stars and the associated Type II supernovae control the FIR and RC emission. In the outer parts heating of very small dust grains by the old disk stars provides a secondary component in the FIR emission, without associated RC emission. The edge-on galaxy NGC3079 shows extended FIR and RC emissions along its minor axis, probably associated with the nuclear starburst activity.Comment: Uses aas2pp4.sty and epsfig.sty, 21 pages. Figure 1 is spread over 22 pages and is available at http://www.inaoep.mx/~ydm/preprint.html To appear in Astronomical Journal, September 199

Spatially Resolved Star Formation History Along the Disk of M82 Using Multi-Band Photometric Data

We present the results on the star formation history and extinction in the disk of M82 over spatial scales of 10" (~180 pc). Multi-band photometric data covering from the far ultraviolet to the near infrared bands were fitted to a grid of synthetic spectral energy distributions. We obtained distribution functions of age and extinction for each of the 117 apertures analyzed, taking into account observational errors through Monte-Carlo simulations. These distribution functions were fitted with gaussian functions to obtain the mean ages and extinctions along with errors on them. The analyzed zones include the high surface brightness complexes defined by O'Connell & Mangano (1978). We found that these complexes share the same star formation history and extinction as the field stellar populations in the disk. There is an indication that the stellar populations are marginally older at the outer disk (450 Myr at ~3 kpc) as compared to the inner disk (100 Myr at 0.5 kpc). For the nuclear regions (radius less than 500 pc), we obtained an age of less than 10 Myr. The results obtained in this work are consistent with the idea that the 0.5-3 kpc part of the disk of M82 formed around 90% of the stellar mass in a star-forming episode that started around 450 Myr ago lasting for about 350 Myr. We found that field stars are the major contributors to the flux over the spatial scales analyzed in this study, with stellar cluster contribution being 7% in the nucleus and 0.7% in the disk.Comment: 19 pages, 14 figures. Accepted for publication in The Astrophysical Journa