6,479 research outputs found
Limits and Confidence Intervals in the Presence of Nuisance Parameters
We study the frequentist properties of confidence intervals computed by the
method known to statisticians as the Profile Likelihood. It is seen that the
coverage of these intervals is surprisingly good over a wide range of possible
parameter values for important classes of problems, in particular whenever
there are additional nuisance parameters with statistical or systematic errors.
Programs are available for calculating these intervals.Comment: 6 figure
The Consistency of Partial Observability for PDEs
In this paper, a new definition of observability is introduced for PDEs. It
is a quantitative measure of partial observability. The quantity is proved to
be consistent if approximated using well posed approximation schemes. A first
order approximation of an unobservability index using empirical gramian is
introduced. For linear systems with full state observability, the empirical
gramian is equivalent to the observability gramian in control theory. The
consistency of the defined observability is exemplified using a Burgers'
equation.Comment: 28 pages, 3 figure
Estimation of the Distribution of Random Parameters in Discrete Time Abstract Parabolic Systems with Unbounded Input and Output: Approximation and Convergence
A finite dimensional abstract approximation and convergence theory is
developed for estimation of the distribution of random parameters in infinite
dimensional discrete time linear systems with dynamics described by regularly
dissipative operators and involving, in general, unbounded input and output
operators. By taking expectations, the system is re-cast as an equivalent
abstract parabolic system in a Gelfand triple of Bochner spaces wherein the
random parameters become new space-like variables. Estimating their
distribution is now analogous to estimating a spatially varying coefficient in
a standard deterministic parabolic system. The estimation problems are
approximated by a sequence of finite dimensional problems. Convergence is
established using a state space-varying version of the Trotter-Kato semigroup
approximation theorem. Numerical results for a number of examples involving the
estimation of exponential families of densities for random parameters in a
diffusion equation with boundary input and output are presented and discussed
Estimating the Distribution of Random Parameters in a Diffusion Equation Forward Model for a Transdermal Alcohol Biosensor
We estimate the distribution of random parameters in a distributed parameter
model with unbounded input and output for the transdermal transport of ethanol
in humans. The model takes the form of a diffusion equation with the input
being the blood alcohol concentration and the output being the transdermal
alcohol concentration. Our approach is based on the idea of reformulating the
underlying dynamical system in such a way that the random parameters are now
treated as additional space variables. When the distribution to be estimated is
assumed to be defined in terms of a joint density, estimating the distribution
is equivalent to estimating the diffusivity in a multi-dimensional diffusion
equation and thus well-established finite dimensional approximation schemes,
functional analytic based convergence arguments, optimization techniques, and
computational methods may all be employed. We use our technique to estimate a
bivariate normal distribution based on data for multiple drinking episodes from
a single subject.Comment: 10 page
Asymptotic equivalence and adaptive estimation for robust nonparametric regression
Asymptotic equivalence theory developed in the literature so far are only for
bounded loss functions. This limits the potential applications of the theory
because many commonly used loss functions in statistical inference are
unbounded. In this paper we develop asymptotic equivalence results for robust
nonparametric regression with unbounded loss functions. The results imply that
all the Gaussian nonparametric regression procedures can be robustified in a
unified way. A key step in our equivalence argument is to bin the data and then
take the median of each bin. The asymptotic equivalence results have
significant practical implications. To illustrate the general principles of the
equivalence argument we consider two important nonparametric inference
problems: robust estimation of the regression function and the estimation of a
quadratic functional. In both cases easily implementable procedures are
constructed and are shown to enjoy simultaneously a high degree of robustness
and adaptivity. Other problems such as construction of confidence sets and
nonparametric hypothesis testing can be handled in a similar fashion.Comment: Published in at http://dx.doi.org/10.1214/08-AOS681 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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