155 research outputs found
Combining kernel estimators in the uniform deconvolution problem
We construct a density estimator and an estimator of the distribution
function in the uniform deconvolution model. The estimators are based on
inversion formulas and kernel estimators of the density of the observations and
its derivative. Asymptotic normality and the asymptotic biases are derived
Nonparametric volatility density estimation for discrete time models
We consider discrete time models for asset prices with a stationary
volatility process. We aim at estimating the multivariate density of this
process at a set of consecutive time instants. A Fourier type deconvolution
kernel density estimator based on the logarithm of the squared process is
proposed to estimate the volatility density. Expansions of the bias and bounds
on the variance are derived
Nonparametric methods for volatility density estimation
Stochastic volatility modelling of financial processes has become
increasingly popular. The proposed models usually contain a stationary
volatility process. We will motivate and review several nonparametric methods
for estimation of the density of the volatility process. Both models based on
discretely sampled continuous time processes and discrete time models will be
discussed.
The key insight for the analysis is a transformation of the volatility
density estimation problem to a deconvolution model for which standard methods
exist. Three type of nonparametric density estimators are reviewed: the
Fourier-type deconvolution kernel density estimator, a wavelet deconvolution
density estimator and a penalized projection estimator. The performance of
these estimators will be compared. Key words: stochastic volatility models,
deconvolution, density estimation, kernel estimator, wavelets, minimum contrast
estimation, mixin
Deconvolution for an atomic distribution
Let be i.i.d. observations, where and
and are independent. Assume that unobservable 's are distributed
as a random variable where and are independent, has a
Bernoulli distribution with probability of zero equal to and has a
distribution function with density Furthermore, let the random
variables have the standard normal distribution and let Based
on a sample we consider the problem of estimation of the
density and the probability We propose a kernel type deconvolution
estimator for and derive its asymptotic normality at a fixed point. A
consistent estimator for is given as well. Our results demonstrate that our
estimator behaves very much like the kernel type deconvolution estimator in the
classical deconvolution problem.Comment: Published in at http://dx.doi.org/10.1214/07-EJS121 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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