36,751 research outputs found
Computational aspects of Bayesian spectral density estimation
Gaussian time-series models are often specified through their spectral
density. Such models present several computational challenges, in particular
because of the non-sparse nature of the covariance matrix. We derive a fast
approximation of the likelihood for such models. We propose to sample from the
approximate posterior (that is, the prior times the approximate likelihood),
and then to recover the exact posterior through importance sampling. We show
that the variance of the importance sampling weights vanishes as the sample
size goes to infinity. We explain why the approximate posterior may typically
multi-modal, and we derive a Sequential Monte Carlo sampler based on an
annealing sequence in order to sample from that target distribution.
Performance of the overall approach is evaluated on simulated and real
datasets. In addition, for one real world dataset, we provide some numerical
evidence that a Bayesian approach to semi-parametric estimation of spectral
density may provide more reasonable results than its Frequentist counter-parts
Bayesian computational methods
In this chapter, we will first present the most standard computational
challenges met in Bayesian Statistics, focussing primarily on mixture
estimation and on model choice issues, and then relate these problems with
computational solutions. Of course, this chapter is only a terse introduction
to the problems and solutions related to Bayesian computations. For more
complete references, see Robert and Casella (2004, 2009), or Marin and Robert
(2007), among others. We also restrain from providing an introduction to
Bayesian Statistics per se and for comprehensive coverage, address the reader
to Robert (2007), (again) among others.Comment: This is a revised version of a chapter written for the Handbook of
Computational Statistics, edited by J. Gentle, W. Hardle and Y. Mori in 2003,
in preparation for the second editio
Bayesian Cointegrated Vector Autoregression models incorporating Alpha-stable noise for inter-day price movements via Approximate Bayesian Computation
We consider a statistical model for pairs of traded assets, based on a
Cointegrated Vector Auto Regression (CVAR) Model. We extend standard CVAR
models to incorporate estimation of model parameters in the presence of price
series level shifts which are not accurately modeled in the standard Gaussian
error correction model (ECM) framework. This involves developing a novel matrix
variate Bayesian CVAR mixture model comprised of Gaussian errors intra-day and
Alpha-stable errors inter-day in the ECM framework. To achieve this we derive a
novel conjugate posterior model for the Scaled Mixtures of Normals (SMiN CVAR)
representation of Alpha-stable inter-day innovations. These results are
generalized to asymmetric models for the innovation noise at inter-day
boundaries allowing for skewed Alpha-stable models.
Our proposed model and sampling methodology is general, incorporating the
current literature on Gaussian models as a special subclass and also allowing
for price series level shifts either at random estimated time points or known a
priori time points. We focus analysis on regularly observed non-Gaussian level
shifts that can have significant effect on estimation performance in
statistical models failing to account for such level shifts, such as at the
close and open of markets. We compare the estimation accuracy of our model and
estimation approach to standard frequentist and Bayesian procedures for CVAR
models when non-Gaussian price series level shifts are present in the
individual series, such as inter-day boundaries. We fit a bi-variate
Alpha-stable model to the inter-day jumps and model the effect of such jumps on
estimation of matrix-variate CVAR model parameters using the likelihood based
Johansen procedure and a Bayesian estimation. We illustrate our model and the
corresponding estimation procedures we develop on both synthetic and actual
data.Comment: 30 page
Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models
Long-Range Dependence (LRD) and heavy-tailed distributions are ubiquitous in natural and socio-economic data. Such data can be self-similar whereby both LRD and heavy-tailed distributions contribute to the self-similarity as measured by the Hurst exponent. Some methods widely used in the physical sciences separately estimate these two parameters, which can lead to estimation bias. Those which do simultaneous estimation are based on frequentist methods such as Whittle’s approximate maximum likelihood estimator. Here we present a new and systematic Bayesian framework for the simultaneous inference of the LRD and heavy-tailed distribution parameters of a parametric ARFIMA model with non-Gaussian innovations. As innovations we use the α-stable and t-distributions which have power law tails. Our algorithm also provides parameter uncertainty estimates. We test our algorithm using synthetic data, and also data from the Geostationary Operational Environmental Satellite system (GOES) solar X-ray time series. These tests show that our algorithm is able to accurately and robustly estimate the LRD and heavy-tailed distribution parameters
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