787,036 research outputs found
Truthful Linear Regression
We consider the problem of fitting a linear model to data held by individuals
who are concerned about their privacy. Incentivizing most players to truthfully
report their data to the analyst constrains our design to mechanisms that
provide a privacy guarantee to the participants; we use differential privacy to
model individuals' privacy losses. This immediately poses a problem, as
differentially private computation of a linear model necessarily produces a
biased estimation, and existing approaches to design mechanisms to elicit data
from privacy-sensitive individuals do not generalize well to biased estimators.
We overcome this challenge through an appropriate design of the computation and
payment scheme.Comment: To appear in Proceedings of the 28th Annual Conference on Learning
Theory (COLT 2015
Bayesian Linear Regression
The paper is concerned with Bayesian analysis under prior-data conflict, i.e. the situation when observed data are rather unexpected under the prior (and the sample size is not large enough to eliminate the influence of the prior). Two approaches for Bayesian linear regression modeling based on conjugate priors are considered in detail, namely the standard approach also described in Fahrmeir, Kneib & Lang (2007) and an alternative adoption of the general construction procedure for exponential family sampling models. We recognize that - in contrast to some standard i.i.d. models like the scaled normal model and the Beta-Binomial / Dirichlet-Multinomial model, where prior-data conflict is completely ignored - the models may show some reaction to prior-data conflict, however in a rather unspecific way. Finally we briefly sketch the extension to a corresponding imprecise probability model, where, by considering sets of prior distributions instead of a single prior, prior-data conflict can be handled in a very appealing and intuitive way
Current status linear regression
We construct -consistent and asymptotically normal estimates for
the finite dimensional regression parameter in the current status linear
regression model, which do not require any smoothing device and are based on
maximum likelihood estimates (MLEs) of the infinite dimensional parameter. We
also construct estimates, again only based on these MLEs, which are arbitrarily
close to efficient estimates, if the generalized Fisher information is finite.
This type of efficiency is also derived under minimal conditions for estimates
based on smooth non-monotone plug-in estimates of the distribution function.
Algorithms for computing the estimates and for selecting the bandwidth of the
smooth estimates with a bootstrap method are provided. The connection with
results in the econometric literature is also pointed out.Comment: 64 pages, 6 figure
Sublinear expectation linear regression
Nonlinear expectation, including sublinear expectation as its special case,
is a new and original framework of probability theory and has potential
applications in some scientific fields, especially in finance risk measure and
management. Under the nonlinear expectation framework, however, the related
statistical models and statistical inferences have not yet been well
established. The goal of this paper is to construct the sublinear expectation
regression and investigate its statistical inference. First, a sublinear
expectation linear regression is defined and its identifiability is given.
Then, based on the representation theorem of sublinear expectation and the
newly defined model, several parameter estimations and model predictions are
suggested, the asymptotic normality of estimations and the mini-max property of
predictions are obtained. Furthermore, new methods are developed to realize
variable selection for high-dimensional model. Finally, simulation studies and
a real-life example are carried out to illustrate the new models and
methodologies. All notions and methodologies developed are essentially
different from classical ones and can be thought of as a foundation for general
nonlinear expectation statistics
Linear Regression Diagnostics
This paper attempts to provide the user of linear multiple regression with a battery of diagnostic tools to determine which, if any, data points have high leverage or influence on the estimation process and how these possibly discrepant data points differ from the patterns set by the majority of the data. The point of view taken is that when diagnostics indicate the presence of anomolous data, the choice is open as to whether these data are in fact unusual and helpful, or possibly harmful and thus in need of modifications or deletion. The methodology developed depends on differences, derivatives, and decompositions of basic regression statistics. There is also a discussion of how these techniques can be used with robust and ridge estimators. An example is given showing the use of diagnostic methods in the estimation of a cross-country savings rate model.
Functional linear regression that's interpretable
Regression models to relate a scalar to a functional predictor are
becoming increasingly common. Work in this area has concentrated on estimating
a coefficient function, , with related to through
. Regions where correspond to places where
there is a relationship between and . Alternatively, points where
indicate no relationship. Hence, for interpretation purposes, it
is desirable for a regression procedure to be capable of producing estimates of
that are exactly zero over regions with no apparent relationship and
have simple structures over the remaining regions. Unfortunately, most fitting
procedures result in an estimate for that is rarely exactly zero and
has unnatural wiggles making the curve hard to interpret. In this article we
introduce a new approach which uses variable selection ideas, applied to
various derivatives of , to produce estimates that are both
interpretable, flexible and accurate. We call our method "Functional Linear
Regression That's Interpretable" (FLiRTI) and demonstrate it on simulated and
real-world data sets. In addition, non-asymptotic theoretical bounds on the
estimation error are presented. The bounds provide strong theoretical
motivation for our approach.Comment: Published in at http://dx.doi.org/10.1214/08-AOS641 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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