18,255 research outputs found
Geometric Inference on Kernel Density Estimates
We show that geometric inference of a point cloud can be calculated by
examining its kernel density estimate with a Gaussian kernel. This allows one
to consider kernel density estimates, which are robust to spatial noise,
subsampling, and approximate computation in comparison to raw point sets. This
is achieved by examining the sublevel sets of the kernel distance, which
isomorphically map to superlevel sets of the kernel density estimate. We prove
new properties about the kernel distance, demonstrating stability results and
allowing it to inherit reconstruction results from recent advances in
distance-based topological reconstruction. Moreover, we provide an algorithm to
estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure
Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation
Approximate Bayesian computation has emerged as a standard computational tool
when dealing with the increasingly common scenario of completely intractable
likelihood functions in Bayesian inference. We show that many common Markov
chain Monte Carlo kernels used to facilitate inference in this setting can fail
to be variance bounding, and hence geometrically ergodic, which can have
consequences for the reliability of estimates in practice. This phenomenon is
typically independent of the choice of tolerance in the approximation. We then
prove that a recently introduced Markov kernel in this setting can inherit
variance bounding and geometric ergodicity from its intractable
Metropolis--Hastings counterpart, under reasonably weak and manageable
conditions. We show that the computational cost of this alternative kernel is
bounded whenever the prior is proper, and present indicative results on an
example where spectral gaps and asymptotic variances can be computed, as well
as an example involving inference for a partially and discretely observed,
time-homogeneous, pure jump Markov process. We also supply two general
theorems, one of which provides a simple sufficient condition for lack of
variance bounding for reversible kernels and the other provides a positive
result concerning inheritance of variance bounding and geometric ergodicity for
mixtures of reversible kernels.Comment: 26 pages, 10 figure
A selective overview of nonparametric methods in financial econometrics
This paper gives a brief overview on the nonparametric techniques that are
useful for financial econometric problems. The problems include estimation and
inferences of instantaneous returns and volatility functions of
time-homogeneous and time-dependent diffusion processes, and estimation of
transition densities and state price densities. We first briefly describe the
problems and then outline main techniques and main results. Some useful
probabilistic aspects of diffusion processes are also briefly summarized to
facilitate our presentation and applications.Comment: 32 pages include 7 figure
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