A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

Precambrian non-marine stromatolites in alluvial fan deposits, the Copper Harbor Conglomerate, upper Michigan

Laminated cryptalgal carbonates occur in the Precambrian Copper Harbor Conglomerate of northern Michigan, which was deposited in the Keweenawan Trough, an aborted proto-oceanic rift. This unit is composed of three major facies deposited by braided streams on a large alluvial-fan complex. Coarse clastics were deposited in braided channels, predominantly as longitudinal bars, whereas cross-bedded sandstones were deposited by migrating dunes or linguoid bars. Fine-grained overbank deposits accumulated in abandoned channels. Gypsum moulds and carbonate-filled cracks suggest an arid climate during deposition. Stromatolites interstratified with these clastic facies occur as laterally linked drapes over cobbles, as laterally linked contorted beds in mudstone, as oncolites, and as poorly developed mats in coarse sandstones. Stromatolites also are interbedded with oolitic beds and intraclastic conglomerates. Stromatolitic microstructure consists of alternating detrital and carbonate laminae, and open-space structures. Radial-fibrous calcite fans are superimposed on the laminae. The laminae are interpreted as algal in origin, whereas the origin of the radial fibrous calcite is problematic. The stromatolites are inferred to have grown in lakes which occupied abandoned channels on the fan surface. Standing water on a permeable alluvial fan in an arid climate requires a high water table maintained by high precipitation, or local elevation of the water table, possibly due to the close proximity of a lake. Occurrence of stromatolites in the upper part of the Copper Harbor Conglomerate near the base of the lacustrine Nonesuch Shale suggests that these depositional sites may have been near the Nonesuch Lake.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72022/1/j.1365-3091.1983.tb00713.x.pd

Distribution of Major Health Risks: Findings from the Global Burden of Disease Study

BACKGROUND: Most analyses of risks to health focus on the total burden of their aggregate effects. The distribution of risk-factor-attributable disease burden, for example by age or exposure level, can inform the selection and targeting of specific interventions and programs, and increase cost-effectiveness. METHODS AND FINDINGS: For 26 selected risk factors, expert working groups conducted comprehensive reviews of data on risk-factor exposure and hazard for 14 epidemiological subregions of the world, by age and sex. Age-sex-subregion-population attributable fractions were estimated and applied to the mortality and burden of disease estimates from the World Health Organization Global Burden of Disease database. Where possible, exposure levels were assessed as continuous measures, or as multiple categories. The proportion of risk-factor-attributable burden in different population subgroups, defined by age, sex, and exposure level, was estimated. For major cardiovascular risk factors (blood pressure, cholesterol, tobacco use, fruit and vegetable intake, body mass index, and physical inactivity) 43%–61% of attributable disease burden occurred between the ages of 15 and 59 y, and 87% of alcohol-attributable burden occurred in this age group. Most of the disease burden for continuous risks occurred in those with only moderately raised levels, not among those with levels above commonly used cut-points, such as those with hypertension or obesity. Of all disease burden attributable to being underweight during childhood, 55% occurred among children 1–3 standard deviations below the reference population median, and the remainder occurred among severely malnourished children, who were three or more standard deviations below median. CONCLUSIONS: Many major global risks are widely spread in a population, rather than restricted to a minority. Population-based strategies that seek to shift the whole distribution of risk factors often have the potential to produce substantial reductions in disease burden