3,985 research outputs found
On a family of test statistics for discretely observed diffusion processes
We consider parametric hypotheses testing for multidimensional ergodic
diffusion processes observed at discrete time. We propose a family of test
statistics, related to the so called -divergence measures. By taking into
account the quasi-likelihood approach developed for studying the stochastic
differential equations, it is proved that the tests in this family are all
asymptotically distribution free. In other words, our test statistics weakly
converge to the chi squared distribution. Furthermore, our test statistic is
compared with the quasi likelihood ratio test. In the case of contiguous
alternatives, it is also possible to study in detail the power function of the
tests.
Although all the tests in this family are asymptotically equivalent, we show
by Monte Carlo analysis that, in the small sample case, the performance of the
test strictly depends on the choice of the function . Furthermore, in
this framework, the simulations show that there are not uniformly most powerful
tests
Divergences Test Statistics for Discretely Observed Diffusion Processes
In this paper we propose the use of -divergences as test statistics to
verify simple hypotheses about a one-dimensional parametric diffusion process
\de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t, from discrete
observations with , , under the asymptotic scheme , and
. The class of -divergences is wide and includes
several special members like Kullback-Leibler, R\'enyi, power and
-divergences. We derive the asymptotic distribution of the test
statistics based on -divergences. The limiting law takes different forms
depending on the regularity of . These convergence differ from the
classical results for independent and identically distributed random variables.
Numerical analysis is used to show the small sample properties of the test
statistics in terms of estimated level and power of the test
Empirical -distance test statistics for ergodic diffusions
The aim of this paper is to introduce a new type of test statistic for simple
null hypothesis on one-dimensional ergodic diffusion processes sampled at
discrete times. We deal with a quasi-likelihood approach for stochastic
differential equations (i.e. local gaussian approximation of the transition
functions) and define a test statistic by means of the empirical -distance
between quasi-likelihoods. We prove that the introduced test statistic is
asymptotically distribution free; namely it weakly converges to a
random variable. Furthermore, we study the power under local alternatives of
the parametric test. We show by the Monte Carlo analysis that, in the small
sample case, the introduced test seems to perform better than other tests
proposed in literature
Fitting Effective Diffusion Models to Data Associated with a "Glassy Potential": Estimation, Classical Inference Procedures and Some Heuristics
A variety of researchers have successfully obtained the parameters of low
dimensional diffusion models using the data that comes out of atomistic
simulations. This naturally raises a variety of questions about efficient
estimation, goodness-of-fit tests, and confidence interval estimation. The
first part of this article uses maximum likelihood estimation to obtain the
parameters of a diffusion model from a scalar time series. I address numerical
issues associated with attempting to realize asymptotic statistics results with
moderate sample sizes in the presence of exact and approximated transition
densities. Approximate transition densities are used because the analytic
solution of a transition density associated with a parametric diffusion model
is often unknown.I am primarily interested in how well the deterministic
transition density expansions of Ait-Sahalia capture the curvature of the
transition density in (idealized) situations that occur when one carries out
simulations in the presence of a "glassy" interaction potential. Accurate
approximation of the curvature of the transition density is desirable because
it can be used to quantify the goodness-of-fit of the model and to calculate
asymptotic confidence intervals of the estimated parameters. The second part of
this paper contributes a heuristic estimation technique for approximating a
nonlinear diffusion model. A "global" nonlinear model is obtained by taking a
batch of time series and applying simple local models to portions of the data.
I demonstrate the technique on a diffusion model with a known transition
density and on data generated by the Stochastic Simulation Algorithm.Comment: 30 pages 10 figures Submitted to SIAM MMS (typos removed and slightly
shortened
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
Simulation of multivariate diffusion bridge
We propose simple methods for multivariate diffusion bridge simulation, which
plays a fundamental role in simulation-based likelihood and Bayesian inference
for stochastic differential equations. By a novel application of classical
coupling methods, the new approach generalizes a previously proposed simulation
method for one-dimensional bridges to the multi-variate setting. First a method
of simulating approximate, but often very accurate, diffusion bridges is
proposed. These approximate bridges are used as proposal for easily
implementable MCMC algorithms that produce exact diffusion bridges. The new
method is much more generally applicable than previous methods. Another
advantage is that the new method works well for diffusion bridges in long
intervals because the computational complexity of the method is linear in the
length of the interval. In a simulation study the new method performs well, and
its usefulness is illustrated by an application to Bayesian estimation for the
multivariate hyperbolic diffusion model.Comment: arXiv admin note: text overlap with arXiv:1403.176
Markov chain Monte Carlo for exact inference for diffusions
We develop exact Markov chain Monte Carlo methods for discretely-sampled,
directly and indirectly observed diffusions. The qualification "exact" refers
to the fact that the invariant and limiting distribution of the Markov chains
is the posterior distribution of the parameters free of any discretisation
error. The class of processes to which our methods directly apply are those
which can be simulated using the most general to date exact simulation
algorithm. The article introduces various methods to boost the performance of
the basic scheme, including reparametrisations and auxiliary Poisson sampling.
We contrast both theoretically and empirically how this new approach compares
to irreducible high frequency imputation, which is the state-of-the-art
alternative for the class of processes we consider, and we uncover intriguing
connections. All methods discussed in the article are tested on typical
examples.Comment: 23 pages, 6 Figures, 3 Table
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