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Shrinkage Estimation in Multilevel Normal Models
This review traces the evolution of theory that started when Charles Stein in
1955 [In Proc. 3rd Berkeley Sympos. Math. Statist. Probab. I (1956) 197--206,
Univ. California Press] showed that using each separate sample mean from
Normal populations to estimate its own population mean can be
improved upon uniformly for every possible . The
dominating estimators, referred to here as being "Model-I minimax," can be
found by shrinking the sample means toward any constant vector. Admissible
minimax shrinkage estimators were derived by Stein and others as posterior
means based on a random effects model, "Model-II" here, wherein the
values have their own distributions. Section 2 centers on Figure 2, which
organizes a wide class of priors on the unknown Level-II hyperparameters that
have been proved to yield admissible Model-I minimax shrinkage estimators in
the "equal variance case." Putting a flat prior on the Level-II variance is
unique in this class for its scale-invariance and for its conjugacy, and it
induces Stein's harmonic prior (SHP) on .Comment: Published in at http://dx.doi.org/10.1214/11-STS363 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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Number of Pages: 6Integrative BiologyGeological Science
Examining and contrasting the cognitive activities engaged in undergraduate research experiences and lab courses
While the positive outcomes of undergraduate research experiences (UREs) have
been extensively categorized, the mechanisms for those outcomes are less
understood. Through lightly structured focus group interviews, we have
extracted the cognitive tasks that students identify as engaging in during
their UREs. We also use their many comparative statements about their
coursework, especially lab courses, to evaluate their experimental
physics-related cognitive tasks in those environments. We find there are a
number of cognitive tasks consistently encountered in physics UREs that are
present in most experimental research. These are seldom encountered in lab or
lecture courses, with some notable exceptions. Having time to reflect and fix
or revise, and having a sense of autonomy, were both repeatedly cited as key
enablers of the benefits of UREs. We also identify tasks encountered in actual
experimental research that are not encountered in UREs. We use these findings
to identify opportunities for better integration of the cognitive tasks in UREs
and lab courses, as well as discussing the barriers that exist. This work
responds to extensive calls for science education to better develop students'
scientific skills and practices, as well as calls to expose more students to
scientific research.Comment: 11 pages, 3 figure
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