4,100 research outputs found
ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz
For independent and in the inequality , we give sharp
lower bounds for unimodal distributions having finite variance, and sharp upper
bounds assuming symmetric densities bounded by a finite constant. The lower
bounds depend on a result of Dubins about extreme points and the upper bounds
depend on a symmetric rearrangement theorem of F. Riesz. The inequality was
motivated by medical imaging: find bounds on the area under the Receiver
Operating Characteristic curve (ROC).Comment: Published in at http://dx.doi.org/10.1214/08-AAP536 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the behavior of Bayesian credible intervals for some restricted parameter space problems
For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as
Roe and Woodroofe (2000) investigate 100( HPD credible sets
associated with priors obtained as the truncation of noninformative priors onto
the restricted parameter space. Namely, they establish the attractive lower
bound of for the frequentist coverage probability
of these procedures. In this work, we establish that the lower bound of
is applicable for a substantially more general
setting with underlying distributional symmetry, and obtain various other
properties. The derivations are unified and are driven by the choice of a right
Haar invariant prior. Investigations of non-symmetric models are carried out
and similar results are obtained. Namely, (i) we show that the lower bound
still applies for certain types of asymmetry (or
skewness), and (ii) we extend results obtained by Zhang and Woodroofe (2002)
for estimating the scale parameter of a Fisher distribution; which arises in
estimating the ratio of variance components in a one-way balanced random
effects ANOVA. Finally, various examples illustrating the wide scope of
applications are expanded upon. Examples include estimating parameters in
location models and location-scale models, estimating scale parameters in scale
models, estimating linear combinations of location parameters such as
differences, estimating ratios of scale parameters, and problems with
non-independent observations.Comment: Published at http://dx.doi.org/10.1214/074921706000000635 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
A distributionally robust perspective on uncertainty quantification and chance constrained programming
The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones
Joint Mixability of Elliptical Distributions and Related Families
In this paper, we further develop the theory of complete mixability and joint
mixability for some distribution families. We generalize a result of
R\"uschendorf and Uckelmann (2002) related to complete mixability of continuous
distribution function having a symmetric and unimodal density. Two different
proofs to a result of Wang and Wang (2016) which related to the joint
mixability of elliptical distributions with the same characteristic generator
are present. We solve the Open Problem 7 in Wang (2015) by constructing a
bimodal-symmetric distribution. The joint mixability of slash-elliptical
distributions and skew-elliptical distributions is studied and the extension to
multivariate distributions is also investigated.Comment: 15page
On the maximum bias functions of MM-estimates and constrained M-estimates of regression
We derive the maximum bias functions of the MM-estimates and the constrained
M-estimates or CM-estimates of regression and compare them to the maximum bias
functions of the S-estimates and the -estimates of regression. In these
comparisons, the CM-estimates tend to exhibit the most favorable
bias-robustness properties. Also, under the Gaussian model, it is shown how one
can construct a CM-estimate which has a smaller maximum bias function than a
given S-estimate, that is, the resulting CM-estimate dominates the S-estimate
in terms of maxbias and, at the same time, is considerably more efficient.Comment: Published at http://dx.doi.org/10.1214/009053606000000975 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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