Examples of moderate deviation principle for diffusion processes

Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family $$S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH(X_s)ds, \ t\to\infty$$ for an ergodic diffusion process $X_t$ under $0.5<\kappa<1$ and appropriate $H$. We mean a decomposition with ``corrector'': $$\frac{1}{t^\kappa}\int_0^tH(X_s)ds={\rm corrector}+\frac{1}{t^\kappa}\underbrace{M_t}_{\rm martingale}.$$ and show that, as in the CLT analysis, the corrector is negligible but in the MD scale, and the main contribution in the MD brings the family ``$\frac{1}{t^\kappa}M_t, t\to\infty.$'' Starting from Bayer and Freidlin, \cite{BF}, and finishing by Wu's papers \cite{Wu1}-\cite{WuH}, in the MD study Laplace's transform dominates. In the paper, we replace the Laplace technique by one, admitting to give the conditions, providing the MD, in terms of ``drift-diffusion'' parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named ``Examples ...''

Moderate deviations for particle filtering

Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and (Y_t) are the observations. In order to solve the so-called filtering problem, one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations (Y_1,...,Y_t). The particle filtering method gives an approximation of the law L(X_t|Y_1,...,Y_t) by an empirical measure \frac{1}{n}\sum_1^n\delta_{x_{i,t}}. In this paper we establish the moderate deviation principle for the empirical mean \frac{1}{n}\sum_1^n\psi(x_{i,t}) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.Comment: Published at http://dx.doi.org/10.1214/105051604000000657 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Off-line detection of multiple change points with the Filtered Derivative with p-Value method

This paper deals with off-line detection of change points for time series of independent observations, when the number of change points is unknown. We propose a sequential analysis like method with linear time and memory complexity. Our method is based at first step, on Filtered Derivative method which detects the right change points but also false ones. We improve Filtered Derivative method by adding a second step in which we compute the p-values associated to each potential change points. Then we eliminate as false alarms the points which have p-value smaller than a given critical level. Next, our method is compared with the Penalized Least Square Criterion procedure on simulated data sets. Eventually, we apply Filtered Derivative with p-Value method to segmentation of heartbeat time series, and detection of change points in the average daily volume of financial time series

Fast change point analysis on the Hurst index of piecewise fractional Brownian motion

In this presentation, we introduce a new method for change point analysis on the Hurst index for a piecewise fractional Brownian motion. We first set the model and the statistical problem. The proposed method is a transposition of the FDpV (Filtered Derivative with p-value) method introduced for the detection of change points on the mean in Bertrand et al. (2011) to the case of changes on the Hurst index. The underlying statistics of the FDpV technology is a new statistic estimator for Hurst index, so-called Increment Bernoulli Statistic (IBS). Both FDpV and IBS are methods with linear time and memory complexity, with respect to the size of the series. Thus the resulting method for change point analysis on Hurst index reaches also a linear complexity

Convergence of adaptive mixtures of importance sampling schemes

In the design of efficient simulation algorithms, one is often beset with a poor choice of proposal distributions. Although the performance of a given simulation kernel can clarify a posteriori how adequate this kernel is for the problem at hand, a permanent on-line modification of kernels causes concerns about the validity of the resulting algorithm. While the issue is most often intractable for MCMC algorithms, the equivalent version for importance sampling algorithms can be validated quite precisely. We derive sufficient convergence conditions for adaptive mixtures of population Monte Carlo algorithms and show that Rao--Blackwellized versions asymptotically achieve an optimum in terms of a Kullback divergence criterion, while more rudimentary versions do not benefit from repeated updating.Comment: Published at http://dx.doi.org/10.1214/009053606000001154 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

How long should arthroscopic clavicular resection be in acromioclavicular arthropathy? A radiological-clinical study (with computed tomography) of 18 cases at a mean 4 years’ follow-up

AbstractIntroductionEndoscopic clavicular resection is a common procedure, but few studies have analyzed predictive factors for outcome.Hypotheses1) Computed tomography (CT) of clavicular resection is reproductible; 2) Functional outcome correlates with resection length; 3) Other factors also influence outcome.Material and methodsPatients operated on between 2005 and 2010 were called back to establish functional scores (Constant, Simple Shoulder Test [SST], satisfaction) and undergo low-dose bilateral comparative computed tomography (CT) centered on the acromioclavicular joints. The assessment criteria were resection edge parallelism and resection length, measured using OsiriX® software. Radiological and clinical data were correlated.Results18 out of 21 patients (85%: 3 female, 15 male) were assessed. Mean age at surgery was 49 years (range, 40–62 yrs); mean follow-up was 4.2 years (1.6–7.2 yrs). Mean Constant score rose from 57.7 (25–85) to 70.2 (30–96); mean postoperative SST was 9.3 (3–12). 11 patients had very good and 4 poor results. CT resection length was reproducible, with intraclass, intra- and interobserver correlation coefficients >95%. There was no significant correlation between articular resection length on CT and functional scores (P=0.2). Functional scores were negatively influenced by an occupational pathologic context (P<0.01) and by associated tendinopathy.Discussion and conclusionLow-dose CT enabled reproducible analysis of clavicular resection. The hypothesized correlation between resection length and functional result was not confirmed. Work accidents and occupational disease emerged as risk factors.Level of evidenceSingle-center retrospective analytic cohort study. Level 4, guideline grade C

Entrapment and traumatic neuropathies of the elbow and hand: An imaging approach

AbstractUltrasound and magnetic resonance imaging currently offer a detailed analysis of the peripheral nerves. Compressive and traumatic nerve injuries are the two main indications for imaging investigation of nerves with several publications describing the indications, technique and diagnostic capabilities of imaging signs. Investigation of entrapment neuropathies has three main goals, which are to confirm neuronal distress, search for the cause of nerve compression and exclude a differential diagnosis on the entire nerve. For traumatic nerve injuries, imaging, predominantly ultrasound, occasionally provides essential information for management including the type of nerve lesion, its exact site and local extension

Fast change point analysis on the Hurst index of piecewise fractional Brownian motion

International audienceIn this presentation, we introduce a new method for change point analysis on the Hurst index for a piecewise fractional Brownian motion. We first set the model and the statistical problem. The proposed method is a transposition of the FDpV (Filtered Derivative with p-value) method introduced for the detection of change points on the mean in Bertrand et al. (2011) to the case of changes on the Hurst index. The underlying statistics of the FDpV technology is a new statistic estimator for Hurst index, so-called Increment Bernoulli Statistic (IBS). Both FDpV and IBS are methods with linear time and memory complexity, with respect to the size of the series. Thus the resulting method for change point analysis on Hurst index reaches also a linear complexity

Adaptive Importance Sampling in General Mixture Classes

In this paper, we propose an adaptive algorithm that iteratively updates both the weights and component parameters of a mixture importance sampling density so as to optimise the importance sampling performances, as measured by an entropy criterion. The method is shown to be applicable to a wide class of importance sampling densities, which includes in particular mixtures of multivariate Student t distributions. The performances of the proposed scheme are studied on both artificial and real examples, highlighting in particular the benefit of a novel Rao-Blackwellisation device which can be easily incorporated in the updating scheme.Comment: Removed misleading comment in Section