This paper is also the originator of the Markov Chain Monte Carlo methods developed in the following chapters. The potential of these two simultaneous innovations has been discovered much latter by statisticians (Hastings 1970; Geman and Geman 1984) than by of physicists (see also Kirkpatrick et al. 1983). 5.5.5 ] PROBLEMS 211

High-frequency financial data are not only discretely sampled in time but the time separating successive observations is often random. We analyze the consequences of this dual feature of the data when estimating a continuous-time model. In particular, we measure the additional effects of the randomness of the sampling intervals over and beyond those due to the discreteness of the data. We also examine the effect of simply ignoring the sampling randomness. We find that in many situations the randomness of the sampling has a larger impact than the discreteness of the data.

This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...

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This paper analyzes the performance of Bayesian object recognition algorithms in the context of deformable templates. Rigid CAD surface models represent the underlying targets; low-dimensional matrix Lie groups (rotation and translation) extend them to the particular instance of pose and position. For a target ff, I ff represents its templates and sI ff is the target template at the pose/location denoted by the parameter s. The remote sensors observing the objects are modeled by the projective transformation T , that is, T sI ff is the signature of target ff at pose s when viewed by the sensor T . The observations I D are modeled as a random fields with mean T sI ff . In a Bayesian approach, object recognition and pose estimation are basically optimizations for a given cost function related to the posterior. Recognition performance is analyzed through probability of error: given a target ff 0 at pose s 0 what is the probability of it being recognized as ff 1 . Asymptotic ex...

Convergence rates of Markov chains have been widely studied in recent years. In particular, quantitative bounds on convergence rates have been studied in various forms by Meyn and Tweedie [Ann. Appl.

A subthreshold signal is transmitted through a channel and may be detected when some noise – with known structure and proportional to some level – is added to the data. There is an optimal noise level, called stochastic resonance, that corresponds to the highest Fisher information in the problem of estimation of the signal. As noise we consider an ergodic diffusion process and the asymptotic is considered as time goes to infinity. We propose consistent estimators of the subthreshold signal and we solve further a problem of hypotheses testing. We also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples.

This article develops a direct filtration-based maximum likelihood methodology for estimating the parameters and realizations of latent affine processes. Filtration is conducted in the transform space of characteristic functions, using a version of Bayes ’ rule for recursively updating the joint characteristic function of latent variables and the data conditional upon past data. An application to daily stock market returns over 1953-96 reveals substantial divergences from EMM-based estimates; in particular, more substantial and time-varying jump risk. The implications for pricing stock index options are examined. 3 “The Lion in Affrik and the Bear in Sarmatia are Fierce, but Translated into a Contrary Heaven, are of less Strength and Courage.” Jacob Ziegler; translated by Richard Eden (1555) While models proposing time-varying volatility of asset returns have been around for thirty years, it has proven extraordinarily difficult to estimate the parameters of the underlying volatility process,

We consider a Bayesian approach to multiple hypothesis testing. A hierarchical prior model is based on imposing a prior distribution π(k) on the number of hypotheses arising from alternatives (false nulls). We then apply the maximum a posteriori (MAP) rule to find the most likely configuration of null and alternative hypotheses. The resulting MAP procedure and its closely related step-up and step-down versions compare ordered Bayes factors of individual hypotheses with a sequence of critical values depending on the prior. We discuss the relations between the proposed MAP procedure and the existing frequentist and Bayesian counterparts. A more detailed analysis is given for the normal data, where we show, in particular, that by choosing a specific π(k), the MAP procedure can mimic several known familywise error (FWE) and false discovery rate (FDR) controlling procedures. The performance of MAP procedures is illustrated on a simulated example. AMS (2000) subject classification. Primary 62F15, 62F03.

We consider a log-concave density f in R m satisfying certain weak conditions, particularly on the Hessian matrix of \Gamma log f . For such a density, we prove tail exactness of the multivariate saddlepoint approximation. The proof is based on a local limit theorem for the exponential family generated by f . However, the result refers not to asymptotic behaviour under repeated sampling, but to a limiting property at the boundary of the domain of f . Our approach does not apply any complex analysis, but relies totally on convex analysis and exponential models arguments. Running headline: Multivariate saddlepoint approximations. AMS 1991 Subject Classifications: primary: 62E17, 62E20, 62F11, 62H10 secondary: 60E10, 60F05 Keywords: asymptotic normality, convex analysis, exponential models, local limit theorem, moment generating function, Legendre transform, saddlepoint approximation. Department of Mathematical Sciences, Aarhus University, DK-8000 Aarhus C, Denmark, email: atsoebn@mi...