194 research outputs found
Plausibility functions and exact frequentist inference
In the frequentist program, inferential methods with exact control on error
rates are a primary focus. The standard approach, however, is to rely on
asymptotic approximations, which may not be suitable. This paper presents a
general framework for the construction of exact frequentist procedures based on
plausibility functions. It is shown that the plausibility function-based tests
and confidence regions have the desired frequentist properties in finite
samples---no large-sample justification needed. An extension of the proposed
method is also given for problems involving nuisance parameters. Examples
demonstrate that the plausibility function-based method is both exact and
efficient in a wide variety of problems.Comment: 21 pages, 5 figures, 3 table
A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets
In nonparametric statistical problems, we wish to find an estimator of an
unknown function f. We can split its error into bias and variance terms;
Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel
estimate, the supremum norm of the variance term is asymptotically distributed
as a Gumbel random variable. In the following, we prove a version of this
result for estimators using compactly-supported wavelets, a popular tool in
nonparametric statistics. Our result relies on an assumption on the nature of
the wavelet, which must be verified by provably-good numerical approximations.
We verify our assumption for Daubechies wavelets and symlets, with N = 6, ...,
20 vanishing moments; larger values of N, and other wavelet bases, are easily
checked, and we conjecture that our assumption holds also in those cases
On Verifiable Sufficient Conditions for Sparse Signal Recovery via Minimization
We propose novel necessary and sufficient conditions for a sensing matrix to
be "-good" - to allow for exact -recovery of sparse signals with
nonzero entries when no measurement noise is present. Then we express the error
bounds for imperfect -recovery (nonzero measurement noise, nearly
-sparse signal, near-optimal solution of the optimization problem yielding
the -recovery) in terms of the characteristics underlying these
conditions. Further, we demonstrate (and this is the principal result of the
paper) that these characteristics, although difficult to evaluate, lead to
verifiable sufficient conditions for exact sparse -recovery and to
efficiently computable upper bounds on those for which a given sensing
matrix is -good. We establish also instructive links between our approach
and the basic concepts of the Compressed Sensing theory, like Restricted
Isometry or Restricted Eigenvalue properties
Asymptotic expansions for renewal measures in the plane
Let P be a distribution in the plane and define the renewal measure R=ΣP *n where * denotes convolution. The main results of this paper are three term asymptotic expansions for R far from the origin. As an application, expansions are obtained for distributions in linear boundary crossing problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47655/1/440_2004_Article_BF00348749.pd
Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds
Let be a random sample from some unknown probability density
defined on a compact homogeneous manifold of dimension . Consider a 'needlet frame' describing a localised
projection onto the space of eigenfunctions of the Laplace operator on with corresponding eigenvalues less than , as constructed in
\cite{GP10}. We prove non-asymptotic concentration inequalities for the uniform
deviations of the linear needlet density estimator obtained from an
empirical estimate of the needlet projection of . We apply these results to construct risk-adaptive
estimators and nonasymptotic confidence bands for the unknown density . The
confidence bands are adaptive over classes of differentiable and
H\"{older}-continuous functions on that attain their H\"{o}lder
exponents.Comment: Probability Theory and Related Fields, to appea
Semiparametric theory and empirical processes in causal inference
In this paper we review important aspects of semiparametric theory and
empirical processes that arise in causal inference problems. We begin with a
brief introduction to the general problem of causal inference, and go on to
discuss estimation and inference for causal effects under semiparametric
models, which allow parts of the data-generating process to be unrestricted if
they are not of particular interest (i.e., nuisance functions). These models
are very useful in causal problems because the outcome process is often complex
and difficult to model, and there may only be information available about the
treatment process (at best). Semiparametric theory gives a framework for
benchmarking efficiency and constructing estimators in such settings. In the
second part of the paper we discuss empirical process theory, which provides
powerful tools for understanding the asymptotic behavior of semiparametric
estimators that depend on flexible nonparametric estimators of nuisance
functions. These tools are crucial for incorporating machine learning and other
modern methods into causal inference analyses. We conclude by examining related
extensions and future directions for work in semiparametric causal inference
Quantum Mechanics from Focusing and Symmetry
A foundation of quantum mechanics based on the concepts of focusing and
symmetry is proposed. Focusing is connected to c-variables - inaccessible
conceptually derived variables; several examples of such variables are given.
The focus is then on a maximal accessible parameter, a function of the common
c-variable. Symmetry is introduced via a group acting on the c-variable. From
this, the Hilbert space is constructed and state vectors and operators are
given a clear interpretation. The Born formula is proved from weak assumptions,
and from this the usual rules of quantum mechanics are derived. Several
paradoxes and other issues of quantum theory are discussed.Comment: 26 page
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