33,654 research outputs found
Quantile tomography: using quantiles with multivariate data
The use of quantiles to obtain insights about multivariate data is addressed.
It is argued that incisive insights can be obtained by considering directional
quantiles, the quantiles of projections. Directional quantile envelopes are
proposed as a way to condense this kind of information; it is demonstrated that
they are essentially halfspace (Tukey) depth levels sets, coinciding for
elliptic distributions (in particular multivariate normal) with density
contours. Relevant questions concerning their indexing, the possibility of the
reverse retrieval of directional quantile information, invariance with respect
to affine transformations, and approximation/asymptotic properties are studied.
It is argued that the analysis in terms of directional quantiles and their
envelopes offers a straightforward probabilistic interpretation and thus
conveys a concrete quantitative meaning; the directional definition can be
adapted to elaborate frameworks, like estimation of extreme quantiles and
directional quantile regression, the regression of depth contours on
covariates. The latter facilitates the construction of multivariate growth
charts---the question that motivated all the development
Quantile regression for mixed models with an application to examine blood pressure trends in China
Cardiometabolic diseases have substantially increased in China in the past 20
years and blood pressure is a primary modifiable risk factor. Using data from
the China Health and Nutrition Survey, we examine blood pressure trends in
China from 1991 to 2009, with a concentration on age cohorts and urbanicity.
Very large values of blood pressure are of interest, so we model the
conditional quantile functions of systolic and diastolic blood pressure. This
allows the covariate effects in the middle of the distribution to vary from
those in the upper tail, the focal point of our analysis. We join the
distributions of systolic and diastolic blood pressure using a copula, which
permits the relationships between the covariates and the two responses to share
information and enables probabilistic statements about systolic and diastolic
blood pressure jointly. Our copula maintains the marginal distributions of the
group quantile effects while accounting for within-subject dependence, enabling
inference at the population and subject levels. Our population-level regression
effects change across quantile level, year and blood pressure type, providing a
rich environment for inference. To our knowledge, this is the first quantile
function model to explicitly model within-subject autocorrelation and is the
first quantile function approach that simultaneously models multivariate
conditional response. We find that the association between high blood pressure
and living in an urban area has evolved from positive to negative, with the
strongest changes occurring in the upper tail. The increase in urbanization
over the last twenty years coupled with the transition from the positive
association between urbanization and blood pressure in earlier years to a more
uniform association with urbanization suggests increasing blood pressure over
time throughout China, even in less urbanized areas. Our methods are available
in the R package BSquare.Comment: Published at http://dx.doi.org/10.1214/15-AOAS841 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Improving Point and Interval Estimates of Monotone Functions by Rearrangement
Suppose that a target function is monotonic, namely, weakly increasing, and
an available original estimate of this target function is not weakly
increasing. Rearrangements, univariate and multivariate, transform the original
estimate to a monotonic estimate that always lies closer in common metrics to
the target function. Furthermore, suppose an original simultaneous confidence
interval, which covers the target function with probability at least
, is defined by an upper and lower end-point functions that are not
weakly increasing. Then the rearranged confidence interval, defined by the
rearranged upper and lower end-point functions, is shorter in length in common
norms than the original interval and also covers the target function with
probability at least . We demonstrate the utility of the improved
point and interval estimates with an age-height growth chart example.Comment: 24 pages, 4 figures, 3 table
On a new NBUE property in multivariate sense: an application
Since multivariate lifetime data frequently occur in applications, various properties of multivariate distributions have been previously considered to model and describe the main concepts of aging commonly considered in the univariate setting. The generalization of univariate aging notions to the multivariate case involves, among other factors, appropriate definitions of multivariate quantiles and related notions, which are able to correctly describe the intrinsic characteristics of the concepts of aging that should be generalized, and which provide useful tools in the applications. A new multivariate version of the well-known New Better than Used in Expectation univariate aging notion is provided, by means of the concepts of the upper corrected orthant and multivariate excess-wealth function. Some of its properties are described, with particular attention paid to those that can be useful in the analysis of real data sets. Finally, through an example it is illustrated how the new multivariate aging notion influences the final results in the analysis of data on tumor growth from the Comprehensive Cohort Study performed by the German Breast Cancer Study Grou
Local bilinear multiple-output quantile/depth regression
A new quantile regression concept, based on a directional version of Koenker
and Bassett's traditional single-output one, has been introduced in [Ann.
Statist. (2010) 38 635-669] for multiple-output location/linear regression
problems. The polyhedral contours provided by the empirical counterpart of that
concept, however, cannot adapt to unknown nonlinear and/or heteroskedastic
dependencies. This paper therefore introduces local constant and local linear
(actually, bilinear) versions of those contours, which both allow to
asymptotically recover the conditional halfspace depth contours that completely
characterize the response's conditional distributions. Bahadur representation
and asymptotic normality results are established. Illustrations are provided
both on simulated and real data.Comment: Published at http://dx.doi.org/10.3150/14-BEJ610 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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