Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSimple simulation of diffusion bridges with application to likelihood inference for diffusions
151
0
0.0
(
0
)
Added by
Mogens Bladt
Publication date
2014
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
With a view to statistical inference for discretely observed diffusion models, we propose simple methods of simulating diffusion bridges, approximately and exactly. Diffusion bridge simulation plays a fundamental role in likelihood and Bayesian inference for diffusion processes. First a simple method of simulating approximate diffusion bridges is proposed and studied. Then these approximate bridges are used as proposal for an easily implemented Metropolis-Hastings algorithm that produces exact diffusion bridges. The new method utilizes time-reversibility properties of one-dimensional diffusions and is applicable to all one-dimensional diffusion processes with finite speed-measure. One advantage of the new approach is that simple simulation methods like the Milstein scheme can be applied to bridge simulation. Another advantage over previous bridge simulation methods is that the proposed method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. For $rho$-mixing diffusions the approximate method is shown to be particularly accurate for long time intervals. In a simulation study, we investigate the accuracy and efficiency of the approximate method and compare it to exact simulation methods. In the study, our method provides a very good approximation to the distribution of a diffusion bridge for bridges that are likely to occur in applications to statistical inference. To illustrate the usefulness of the new method, we present an EM-algorithm for a discretely observed diffusion process.
rate research
Read More
The asymptotic variance of the maximum likelihood estimate is proved to decrease when the maximization is restricted to a subspace that contains the true parameter value. Maximum likelihood estimation allows a systematic fitting of covariance models to the sample, which is important in data assimilation. The hierarchical maximum likelihood approach is applied to the spectral diagonal covariance model with different parameterizations of eigenvalue decay, and to the sparse inverse covariance model with specified parameter values on different sets of nonzero entries. It is shown computationally that using smaller sets of parameters can decrease the sampling noise in high dimension substantially.
We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e. in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space-dependent and depend on multiple unknown parameters. It is known that classical estimators, such as Maximum Likelihood and Quadratic Variation of the Path Estimators, fail to obtain reasonable estimates for parameters in the effective dynamics when based on observations of the underlying multiscale diffusion. We propose a novel algorithm for estimating both the drift and diffusion coefficients in the effective dynamics based on a semi-parametric framework. We demonstrate by means of extensive numerical simulations of a number of selected examples that the algorithm performs well when applied to data from a multiscale diffusion. These examples also illustrate that the algorithm can be used effectively to obtain accurate and unbiased estimates.
An approximate maximum likelihood method of estimation of diffusion parameters $(vartheta,sigma)$ based on discrete observations of a diffusion $X$ along fixed time-interval $[0,T]$ and Euler approximation of integrals is analyzed. We assume that $X$ satisfies a SDE of form $dX_t =mu (X_t ,vartheta ), dt+sqrt{sigma} b(X_t ), dW_t$, with non-random initial condition. SDE is nonlinear in $vartheta$ generally. Based on assumption that maximum likelihood estimator $hat{vartheta}_T$ of the drift parameter based on continuous observation of a path over $[0,T]$ exists we prove that measurable estimator $(hat{vartheta}_{n,T},hat{sigma}_{n,T})$ of the parameters obtained from discrete observations of $X$ along $[0,T]$ by maximization of the approximate log-likelihood function exists, $hat{sigma}_{n,T}$ being consistent and asymptotically normal, and $hat{vartheta}_{n,T}-hat{vartheta}_T$ tends to zero with rate $sqrt{delta}_{n,T}$ in probability when $delta_{n,T} =max_{0leq i<n}(t_{i+1}-t_i )$ tends to zero with $T$ fixed. The same holds in case of an ergodic diffusion when $T$ goes to infinity in a way that $Tdelta_n$ goes to zero with equidistant sampling, and we applied these to show consistency and asymptotical normality of $hat{vartheta}_{n,T}$, $hat{sigma}_{n,T}$ and asymptotic efficiency of $hat{vartheta}_{n,T}$ in this case.
We deal with a general class of extreme-value regression models introduced by Barreto- Souza and Vasconcellos (2011). Our goal is to derive an adjusted likelihood ratio statistic that is approximately distributed as c{hi}2 with a high degree of accuracy. Although the adjusted statistic requires more computational effort than its unadjusted counterpart, it is shown that the adjustment term has a simple compact form that can be easily implemented in standard statistical software. Further, we compare the finite sample performance of the three classical tests (likelihood ratio, Wald, and score), the gradient test that has been recently proposed by Terrell (2002), and the adjusted likelihood ratio test obtained in this paper. Our simulations favor the latter. Applications of our results are presented. Key words: Extreme-value regression; Gradient test; Gumbel distribution; Likelihood ratio test; Nonlinear models; Score test; Small-sample adjustments; Wald test.
For high-dimensional inference problems, statisticians have a number of competing interests. On the one hand, procedures should provide accurate estimation, reliable structure learning, and valid uncertainty quantification. On the other hand, procedures should be computationally efficient and able to scale to very high dimensions. In this note, I show that a very simple data-dependent measure can achieve all of these desirable properties simultaneously, along with some robustness to the error distribution, in sparse sequence models.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
