218 research outputs found
Quantum Monte Carlo simulation
Contemporary scientific studies often rely on the understanding of complex
quantum systems via computer simulation. This paper initiates the statistical
study of quantum simulation and proposes a Monte Carlo method for estimating
analytically intractable quantities. We derive the bias and variance for the
proposed Monte Carlo quantum simulation estimator and establish the asymptotic
theory for the estimator. The theory is used to design a computational scheme
for minimizing the mean square error of the estimator.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS406 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Vast volatility matrix estimation for high-frequency financial data
High-frequency data observed on the prices of financial assets are commonly
modeled by diffusion processes with micro-structure noise, and realized
volatility-based methods are often used to estimate integrated volatility. For
problems involving a large number of assets, the estimation objects we face are
volatility matrices of large size. The existing volatility estimators work well
for a small number of assets but perform poorly when the number of assets is
very large. In fact, they are inconsistent when both the number, , of the
assets and the average sample size, , of the price data on the assets go
to infinity. This paper proposes a new type of estimators for the integrated
volatility matrix and establishes asymptotic theory for the proposed estimators
in the framework that allows both and to approach to infinity. The
theory shows that the proposed estimators achieve high convergence rates under
a sparsity assumption on the integrated volatility matrix. The numerical
studies demonstrate that the proposed estimators perform well for large and
complex price and volatility models. The proposed method is applied to real
high-frequency financial data.Comment: Published in at http://dx.doi.org/10.1214/09-AOS730 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Wavelet modeling of priors on triangles
AbstractParameters in statistical problems often live in a geometry of certain shape. For example, count probabilities in a multinomial distribution belong to a simplex. For these problems, Bayesian analysis needs to model priors satisfying certain constraints imposed by the geometry. This paper investigates modeling of priors on triangles by use of wavelets constructed specifically for triangles. Theoretical analysis and numerical simulations show that our modeling is flexible and is superior to the commonly used Dirichlet prior
Sparse linear discriminant analysis by thresholding for high dimensional data
In many social, economical, biological and medical studies, one objective is
to classify a subject into one of several classes based on a set of variables
observed from the subject. Because the probability distribution of the
variables is usually unknown, the rule of classification is constructed using a
training sample. The well-known linear discriminant analysis (LDA) works well
for the situation where the number of variables used for classification is much
smaller than the training sample size. Because of the advance in technologies,
modern statistical studies often face classification problems with the number
of variables much larger than the sample size, and the LDA may perform poorly.
We explore when and why the LDA has poor performance and propose a sparse LDA
that is asymptotically optimal under some sparsity conditions on the unknown
parameters. For illustration of application, we discuss an example of
classifying human cancer into two classes of leukemia based on a set of 7,129
genes and a training sample of size 72. A simulation is also conducted to check
the performance of the proposed method.Comment: Published in at http://dx.doi.org/10.1214/10-AOS870 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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