Philip B. Stark

Differential rotation and dynamics of the solar interior

Splitting of the suns global oscillation frequencies by large-scale flows can be used to investigate how rotation varies with radius and latitude within the solar interior. The nearly uninterrupted observations by the Global Oscillation Network Group (GONG) yield oscillation power spectra with high duty cycles and high signal-to-noise ratios. Frequency splittings derived from GONG observations confirm that the variation of rotation rate with latitude seen at the surface carries through much of the convection zone, at the base of which is an adjustment layer leading to latitudinally independent rotation at greater depths. A distinctive shear layer just below the surface is discernible at low to mid-latitudes.

Inverse problems as statistics

What mathematicians, scientists, engineers, and statisticians mean by “inverse problem” differs. For a statistician, an inverse problem is an inference or estimation problem. The data are finite in number and contain errors, as they do in classical estimation or inference problems, and the unknown typically is infinite-dimensional, as it is in nonparametric regression. The additional complication in an inverse problem is that the data are only indirectly related to the unknown. Standard statistical concepts, questions, and considerations such as bias, variance, mean-squared error, identifiability, consistency, efficiency, and various forms of optimality apply to inverse problems. This article discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of techniques and results.

Decline of speech understanding and auditory thresholds in the elderly

A group of 29 elderly subjects between 60.0 and 83.7 years of age at the beginning of the study, and whose hearing loss was not greater than moderate, was tested twice, an average of 5.27 years apart. The tests measured pure-tone thresholds, word recognition in quiet, and understanding of speech with various types of distortion (low-pass filtering, time compression) or interference (single speaker, babble noise, reverberation). Performance declined consistently and significantly between the two testing phases. In addition, the variability of speech understanding measures increased significantly between testing phases, though the variability of audiometric measurements did not. A right-ear superiority was observed but this lateral asymmetry did not increase between testing phases. Comparison of the elderly subjects with a group of young subjects with normal hearing shows that the decline of speech understanding measures accelerated significantly relative to the decline in audiometric measures in the seventh to ninth decades of life. On the assumption that speech understanding depends linearly on age and audiometric variables, there is evidence that this linear relationship changes with age, suggesting that not only the accuracy but also the nature of speech understanding evolves with age.

The Solar Acoustic Spectrum and Eigenmode Parameters

The Global Oscillation Network Group (GONG) project estimates the frequencies, amplitudes, and linewidths of more than 250,000 acoustic resonances of the sun from data sets lasting 36 days. The frequency resolution of a single data set is 0.321 microhertz. For frequencies averaged over the azimuthal order m, the median formal error is 0.044 microhertz, and the associated median fractional error is 1.6 × 10−5. For a 3-year data set, the fractional error is expected to be 3 × 10−6. The GONG m-averaged frequency measurements differ from other helioseismic data sets by 0.03 to 0.08 microhertz. The differences arise from a combination of systematic errors, random errors, and possible changes in solar structure.

Global risk of big earthquakes has not recently increased

The recent elevated rate of large earthquakes has fueled concern that the underlying global rate of earthquake activity has increased, which would have important implications for assessments of seismic hazard and our understanding of how faults interact. We examine the timing of large (magnitude M≥7) earthquakes from 1900 to the present, after removing local clustering related to aftershocks. The global rate of M≥8 earthquakes has been at a record high roughly since 2004, but rates have been almost as high before, and the rate of smaller earthquakes is close to its historical average. Some features of the global catalog are improbable in retrospect, but so are some features of most random sequences—if the features are selected after looking at the data. For a variety of magnitude cutoffs and three statistical tests, the global catalog, with local clusters removed, is not distinguishable from a homogeneous Poisson process. Moreover, no plausible physical mechanism predicts real changes in the underlying global rate of large events. Together these facts suggest that the global risk of large earthquakes is no higher today than it has been in the past.

Conservative Statistical Post-Election Audits

There are many sources of error in counting votes: the apparent winner might not be the rightful winner. Hand tallies of the votes in a random sample of precincts can be used to test the hypothesis that a full manual recount would find a different outcome. This paper develops a conservative sequential test based on the vote-counting errors found in a hand tally of a simple or stratified random sample of precincts. The procedure includes a natural escalation: If the hypothesis that the apparent outcome is incorrect is not rejected at stage s, more precincts are audited. Eventually, either the hypothesis is rejected-and the apparent outcome is confirmed-or all precincts have been audited and the true outcome is known. The test uses a priori bounds on the overstatement of the margin that could result from error in each precinct. Such bounds can be derived from the reported counts in each precinct and upper hounds on the number of votes cast in each precinct. The test allows errors in different precincts to be treated differently to reflect voting technology or precinct sizes. It is not optimal, but it is conservative: the chance of erroneously confirming the outcome of a contest if a full manual recount would show a different outcome is no larger than the nominal significance level. The approach also gives a conservative P-value for the hypothesis that a full manual recount would find a different outcome, given the errors found in a fixed size sample. This is illustrated with two contests from November, 2006: the U.S. Senate race in Minnesota and a school hoard race for the Sausalito Marin City School District in California, a small contest in which voters could vote for up to three candidates.

A Gentle Introduction to Risk-Limiting Audits

Risk-limiting audits provide statistical assurance that election outcomes are correct by manually examining portions of the audit trail-paper ballots or voter-verifiable paper records. This article sketches two types of risk-limiting audits, ballot-polling audits and comparison audits, and gives example computations. These audits do not require in-house statistical expertise.

Inference in infinite-dimensional inverse problems - Discretization and duality

Many techniques for solving inverse problems involve approximating the unknown model, a function, by a finite-dimensional “discretization” or parametric representation. The uncertainty in the computed solution is sometimes taken to be the uncertainty within the parametrization; this can result in unwarranted confidence. The theory of conjugate duality can overcome the limitations of discretization within the “strict bounds” formalism, a technique for constructing confidence intervals for functionals of the unknown model incorporating certain types of prior information. The usual computational approach to strict bounds approximates the “primal” problem in a way that the the resulting confidence intervals are at most long enough to have the nominal coverage probability. There is another approach based on “dual” optimization problems that gives confidence intervals with at least the nominal coverage probability. The pair of intervals derived by the two approaches bracket a correct confidence interval. The theory is illustrated with gravimetric, seismic, geomagnetic, and helioseismic problems and a numerical example in seismology.

A SHARPER DISCREPANCY MEASURE FOR POST-ELECTION AUDITS

Post-election audits use the discrepancy between machine counts and a hand tally of votes in a random sample of precincts to infer whether error affected the electoral outcome. The maximum relative overstatement of pairwise margins (MRO) quantifies that discrepancy. The electoral outcome a full hand tally shows must agree with the apparent outcome if the MRO is less than 1. This condition is sharper than previous ones when there are more than two candidates or when voters may vote for more than one candidate. For the 2006 U.S. Senate race in Minnesota, a test using MRO gives a

Reproducing Earth's kernel: Uncertainty of the shape of the core‐mantle boundary from PKP and PcP travel times

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