Networks are a popular tool for representing elements in a system and their interconnectedness. Many observed networks can be viewed as only samples of some true underlying network. Such is frequently the case, for example, in the monitoring and study of massive, online social networks. We study the problem of how to estimate the degree distribution-an object of fundamental interest-of a true underlying network from its sampled network. In particular, we show that this problem can be formulated as an inverse problem. Playing a key role in this formulation is a matrix relating the expectation of our sampled degree distribution to the true underlying degree distribution. Under many network sampling designs, this matrix can be defined entirely in terms of the design and is found to be ill-conditioned. As a result, our inverse problem frequently is ill-posed. Accordingly, we offer a constrained, penalized weighted least-squares approach to solving this problem. A Monte Carlo variant of Stein's unbiased risk estimation (SURE) is used to select the penalization parameter. We explore the behavior of our resulting estimator of network degree distribution in simulation, using a variety of combinations of network models and sampling regimes. In addition, we demonstrate the ability of our method to accurately reconstruct the degree distributions of various sub-communities within online social networks corresponding to Friendster, Orkut and LiveJournal. Overall, our results show that the true degree distributions from both homogeneous and inhomogeneous networks can be recovered with substantially greater accuracy than reflected in the empirical degree distribution resulting from the original sampling.

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"Errors in population forecasts arise from errors in the jump-off population and errors in the predictions of future vital rates. The propagation of these errors through the linear (Leslie) growth model is studied, and prediction intervals for future population are developed. For U.S. national forecasts, the prediction intervals are compared with the U.S. Census Bureau's high-low intervals." In order to assess the accuracy of the predictions of vital rates, the authors "derive the predictions from a parametric statistical model and estimate the extent of model misspecification and errors in parameter estimates. Subjective, expert opinion, so important in real forecasting, is incorporated with the technique of mixed estimation. A robust regression model is used to assess the effects of model misspecification."

Average accuracy of jury verdicts for a set of cases can be studied empirically and systematically even when the correct verdict cannot be known. The key is to obtain a second rating of the verdict, for example, the judge's, as in the recent study of criminal cases in the United States by the National Center for State Courts (NCSC). That study, like the famous Kalven‐Zeisel study, showed only modest judge‐jury agreement. Simple estimates of jury accuracy can be developed from the judge‐jury agreement rate; the judge's verdict is not taken as the gold standard. Although the estimates of accuracy are subject to error, under plausible conditions they tend to overestimate the average accuracy of jury verdicts. The jury verdict was estimated to be accurate in no more than 87 percent of the NCSC cases (which, however, should not be regarded as a representative sample with respect to jury accuracy). More refined estimates, including false conviction and false acquittal rates, are developed with models using stronger assumptions. For example, the conditional probability that the jury incorrectly convicts given that the defendant truly was not guilty (a “Type I error”) was estimated at 0.25, with an estimated standard error (s.e.) of 0.07, the conditional probability that a jury incorrectly acquits given that the defendant truly was guilty (“Type II error”) was estimated at 0.14 (s.e. 0.03), and the difference was estimated at 0.12 (s.e. 0.08). The estimated number of defendants in the NCSC cases who truly are not guilty but are convicted does seem to be smaller than the number who truly are guilty but are acquitted. The conditional probability of a wrongful conviction, given that the defendant was convicted, is estimated at 0.10 (s.e. 0.03).

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Recent large earthquakes that caused great damage in areas predicted to be relatively safe, illustrate the importance of criteria that assess how well earthquake hazard maps used to develop codes for earthquake-resistant construction are actually performing. At present, there is no agreed-upon way of assessing how well a map performed and thus determining whether one map performed better than another. The fractional site exceedance metric implicit in current maps, that during the chosen time interval the predicted ground motion will be exceeded only at a specific fraction of the sites, is useful but permits maps to be nominally successful even if they significantly underpredict or overpredict shaking, or permits them to be nominally unsuccessful but do well in terms of predicting shaking. We explore some possible metrics that better measure the effects of overprediction and underprediction and can be weighted to reflect the two differently and to reflect differences in populations and property at risk. Although no single metric alone fully characterizes map behavior, using several metrics can provide useful insight for comparing and improving hazard maps. For example, both probabilistic and deterministic hazard maps for Italy dramatically overpredict the recorded shaking in a 2200-yr-long historical intensity catalog, illustrating problems in the data (most likely), models, or both.

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