Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

The Negative Binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard Normal approximation often does not provide adequate inferences about the data's mean in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's Inequality. We now propose growth estimators of the Negative Binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a Normal-style confidence interval by effectively removing a small, pre--determined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the Normal approximation

Ignition of Fuel Beds by Firebrands

The severity of wildfires around the globe is increasing. At the same time, urban development is expanding outward into areas where severe fires occur. There is an increased risk of home loss to fires in areas where severe fires and urban expansion meet. Ignition of homes or nearby fuel is a significant mechanism of home loss. To date, a general model for ignition has proven elusive due to the limited quantification of parameters that control ignition. Thus, understanding and quantifying the processes and parameters that control ignition is essential to reducing home losses. This work considered the influence of fuel bed properties (i.e., particle size and chemical composition), environmental conditions near the ignition site (i.e., wind speed and direction), and variations of ignition sources (i.e., spacing, count, and energy deposition) on the likelihood of ignition. A combination of experimental and computational approaches were conducted to determine the influence of the parameters studied on ignition. Ignition propensity increased when conduction was favored due to small particle sizes compared to radiation-driven heat transfer to fuel beds with larger particle sizes. Ignition was also sensitive to fuel bed species. Fuels high in lignin were not able to be ignited, and increased wind speed decreased the ignition threshold for some, but not all, materials. The formation of recirculation zones caused by wind decreased the ignition threshold. The propensity of wind orientation with respect to the ignition source influenced the formation of recirculation zones. As the size of the recirculation zone increased, the ignition threshold decreased. In windy conditions, the presence of multiple firebrands had little influence on the ignition threshold. However, at low wind speeds, interactions between additional firebrands significantly influenced ignition. Finally, results from all ignition tests conducted in this work were aggregated, and an ignition model was created. It is anticipated that the insights gained from these studies and the subsequent model may act as a novel framework for predicting ignition

Non-Gaussian Component Analysis using Entropy Methods

Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis which, since its formulation in 2006, has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable $X$ in $n$-dimensional Euclidean space. There is an unknown subspace $\Gamma$ of the $n$-dimensional Euclidean space such that the orthogonal projection of $X$ onto $\Gamma$ is standard multidimensional Gaussian and the orthogonal projection of $X$ onto $\Gamma^{\perp}$, the orthogonal complement of $\Gamma$, is non-Gaussian, in the sense that all its one-dimensional marginals are different from the Gaussian in a certain metric defined in terms of moments. The NGCA problem is to approximate the non-Gaussian subspace $\Gamma^{\perp}$ given samples of $X$. Vectors in $\Gamma^{\perp}$ correspond to `interesting' directions, whereas vectors in $\Gamma$ correspond to the directions where data is very noisy. The most interesting applications of the NGCA model is for the case when the magnitude of the noise is comparable to that of the true signal, a setting in which traditional noise reduction techniques such as PCA don't apply directly. NGCA is also related to dimension reduction and to other data analysis problems such as ICA. NGCA-like problems have been studied in statistics for a long time using techniques such as projection pursuit. We give an algorithm that takes polynomial time in the dimension $n$ and has an inverse polynomial dependence on the error parameter measuring the angle distance between the non-Gaussian subspace and the subspace output by the algorithm. Our algorithm is based on relative entropy as the contrast function and fits under the projection pursuit framework. The techniques we develop for analyzing our algorithm maybe of use for other related problems

Hubble Space Telescope Near-IR Transmission Spectroscopy of the Super-Earth HD 97658b

Recent results from the Kepler mission indicate that super-Earths (planets with masses between 1-10 times that of the Earth) are the most common kind of planet around nearby Sun-like stars. These planets have no direct solar system analogue, and are currently one of the least well-understood classes of extrasolar planets. Many super-Earths have average densities that are consistent with a broad range of bulk compositions, including both water-dominated worlds and rocky planets covered by a thick hydrogen and helium atmosphere. Measurements of the transmission spectra of these planets offer the opportunity to resolve this degeneracy by directly constraining the scale heights and corresponding mean molecular weights of their atmospheres. We present Hubble Space Telescope near-infrared spectroscopy of two transits of the newly discovered transiting super-Earth HD 97658b. We use the Wide Field Camera 3's scanning mode to measure the wavelength-dependent transit depth in thirty individual bandpasses. Our averaged differential transmission spectrum has a median 1 sigma uncertainty of 23 ppm in individual bins, making this the most precise observation of an exoplanetary transmission spectrum obtained with WFC3 to date. Our data are inconsistent with a cloud-free solar metallicity atmosphere at the 10 sigma level. They are consistent at the 0.4 sigma level with a flat line model, as well as effectively flat models corresponding to a metal-rich atmosphere or a solar metallicity atmosphere with a cloud or haze layer located at pressures of 10 mbar or higher.Comment: ApJ in press; revised version includes an updated orbital ephemeris for the plane

Ground-based Transit Spectroscopy of the Hot-Jupiter WASP-19b in the Near-infrared

We present ground-based measurements of the transmission and emission spectra of the hot-Jupiter WASP-19b in nine spectroscopic channels from 1.25 to 2.35 μm. The measurements are based on the combined analysis of time-series spectroscopy obtained during two complete transits and two complete secondary eclipses of the planet. The observations were performed with the MMIRS instrument on the Magellan II telescope using the technique of multi-object spectroscopy with wide slits. We compare the transmission and emission data to theoretical models to constrain the composition and thermal structure of the planet's atmosphere. Our measured transmission spectrum exhibits a scatter that corresponds to 1.3 scale heights of the planet's atmosphere, which is consistent with the size of spectral features predicted by theoretical models for a clear atmosphere. We detect the secondary eclipses of the planet at significances ranging from 2.2σ to 14.4σ. The secondary eclipse depths, and the significances of the detections increase toward longer wavelengths. Our measured emission spectrum is consistent with a 2250 K effectively isothermal one-dimensional model for the planet's dayside atmosphere. This model also matches previously published photometric measurements from the Spitzer Space Telescope and ground-based telescopes. These results demonstrate the important role that ground-based observations using multi-object spectroscopy can play in constraining the properties of exoplanet atmospheres, and they also emphasize the need for high-precision measurements based on observations of multiple transits and eclipses

A Precise Water Abundance Measurement for the Hot Jupiter WASP-43b

The water abundance in a planetary atmosphere provides a key constraint on the planet's primordial origins because water ice is expected to play an important role in the core accretion model of planet formation. However, the water content of the Solar System giant planets is not well known because water is sequestered in clouds deep in their atmospheres. By contrast, short-period exoplanets have such high temperatures that their atmospheres have water in the gas phase, making it possible to measure the water abundance for these objects. We present a precise determination of the water abundance in the atmosphere of the 2 $M_\mathrm{Jup}$ short-period exoplanet WASP-43b based on thermal emission and transmission spectroscopy measurements obtained with the Hubble Space Telescope. We find the water content is consistent with the value expected in a solar composition gas at planetary temperatures (0.4-3.5x solar at 1 $\sigma$ confidence). The metallicity of WASP-43b's atmosphere suggested by this result extends the trend observed in the Solar System of lower metal enrichment for higher planet masses.Comment: Accepted to ApJL; this version contains three supplemental figures that are not included in the published paper. See also our companion paper "Thermal structure of an exoplanet atmosphere from phase-resolved emission spectroscopy" by Stevenson et a

Clouds in the atmosphere of the super-Earth exoplanet GJ 1214b

Recent surveys have revealed that planets intermediate in size between Earth and Neptune (‘super-Earths’) are among the most common planets in the Galaxy. Atmospheric studies are the next step towards developing a comprehensive understanding of this new class of object. Much effort has been focused on using transmission spectroscopy to characterize the atmosphere of the super-Earth archetype GJ 1214b, but previous observations did not have sufficient precision to distinguish between two interpretations for the atmosphere. The planet’s atmosphere could be dominated by relatively heavy molecules, such as water (for example, a 100 per cent water vapour composition), or it could contain high-altitude clouds that obscure its lower layers. Here we report a measurement of the transmission spectrum of GJ 1214b at near-infrared wavelengths that definitively resolves this ambiguity. The data, obtained with the Hubble Space Telescope, are sufficiently precise to detect absorption features from a high mean-molecular-mass atmosphere. The observed spectrum, however, is featureless. We rule out cloud-free atmospheric models with compositions dominated by water, methane, carbon monoxide, nitrogen or carbon dioxide at greater than 5σ confidence. The planet’s atmosphere must contain clouds to be consistent with the data