18,608 research outputs found
Perturbation and scaled Cook's distance
Cook's distance [Technometrics 19 (1977) 15-18] is one of the most important
diagnostic tools for detecting influential individual or subsets of
observations in linear regression for cross-sectional data. However, for many
complex data structures (e.g., longitudinal data), no rigorous approach has
been developed to address a fundamental issue: deleting subsets with different
numbers of observations introduces different degrees of perturbation to the
current model fitted to the data, and the magnitude of Cook's distance is
associated with the degree of the perturbation. The aim of this paper is to
address this issue in general parametric models with complex data structures.
We propose a new quantity for measuring the degree of the perturbation
introduced by deleting a subset. We use stochastic ordering to quantify the
stochastic relationship between the degree of the perturbation and the
magnitude of Cook's distance. We develop several scaled Cook's distances to
resolve the comparison of Cook's distance for different subset deletions.
Theoretical and numerical examples are examined to highlight the broad spectrum
of applications of these scaled Cook's distances in a formal influence
analysis.Comment: Published in at http://dx.doi.org/10.1214/12-AOS978 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Perturbation selection and influence measures in local influence analysis
Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133--169] local influence
approach based on normal curvature is an important diagnostic tool for
assessing local influence of minor perturbations to a statistical model.
However, no rigorous approach has been developed to address two fundamental
issues: the selection of an appropriate perturbation and the development of
influence measures for objective functions at a point with a nonzero first
derivative. The aim of this paper is to develop a differential--geometrical
framework of a perturbation model (called the perturbation manifold) and
utilize associated metric tensor and affine curvatures to resolve these issues.
We will show that the metric tensor of the perturbation manifold provides
important information about selecting an appropriate perturbation of a model.
Moreover, we will introduce new influence measures that are applicable to
objective functions at any point. Examples including linear regression models
and linear mixed models are examined to demonstrate the effectiveness of using
new influence measures for the identification of influential observations.Comment: Published in at http://dx.doi.org/10.1214/009053607000000343 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Spectra, vacua and the unitarity of Lovelock gravity in D-dimensional AdS spacetimes
We explicitly confirm the expectation that generic Lovelock gravity in D
dimensions has a unitary massless spin-2 excitation around any one of its
constant curvature vacua just like the cosmological Einstein gravity. The
propagator of the theory reduces to that of Einstein's gravity, but scattering
amplitudes must be computed with an effective Newton's constant which we
provide. Tree-level unitarity imposes a single constraint on the parameters of
the theory yielding a wide range of unitary region. As an example, we
explicitly work out the details of the cubic Lovelock theory.Comment: 9 pages, 2 references adde
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