Volume 6 Issue 6 | Bernoulli

Bernoulli

VOL. 6 · NO. 6 | December 2000

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Front Matter

Table of contents, Bernoulli Journal, vol. 6, no. 6 (2000)

Bernoulli 6 (6), (December 2000) Open Access

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Editorial board, Bernoulli Journal, vol. 6, no. 6 (2000)

Bernoulli 6 (6), (December 2000) Open Access

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Articles

Favourite sites, favourite values and jump sizes for random walk and Brownian motion

Endre Csáki,, Pál Révész, Zhan Shi

Bernoulli 6 (6), 951-975, (December 2000) Open Access

KEYWORDS: Brownian motion, favourite site, Local time, Random walk

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A large-deviation principle for random evolution equations

Mohamed Mellouk

Bernoulli 6 (6), 977-999, (December 2000) Open Access

KEYWORDS: Hölder spaces, large-deviation principle, random evolution equations, relative compactness

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Comparison of stochastic Volterra equations

Guillermo Ferreyra, Padamanbhan Sundar

Bernoulli 6 (6), 1001-1006, (December 2000)
KEYWORDS: Semimartingales, Stochastic differential equation, Stochastic Volterra equation

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A pathwise comparison theorem for a class of one-dimensional stochastic Volterra equations driven by continuous semimartingales is proved under suitable conditions. The result is applied to equations appearing in applications.

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A note on hyperbolic von Mises distributions

Jean-Claude Gruet

Bernoulli 6 (6), 1007-1020, (December 2000)
KEYWORDS: Brownian motion, hyperbolic von Mises distribution

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We define hyperbolic von Mises distributions in any integral dimension as exit distributions of hyperbolic Brownian motion (H^(α)_t,t≥0) with drift outside hyperbolic balls centred on the starting point H₀. Bidimensional unwrapped hyperbolic distributions are also considered.

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A large-deviation principle for Dirichlet posteriors

Ayalvadi J. Ganesh, Neil O'connell

Bernoulli 6 (6), 1021-1034, (December 2000)
KEYWORDS: asymptotics, Bayesian nonparametrics, Dirichlet process, large deviations

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Let X_k be a sequence of independent and identically distributed random variables taking values in a compact metric space Ω, and consider the problem of estimating the law of X₁ in a Bayesian framework. A conjugate family of priors for nonparametric Bayesian inference is the Dirichlet process priors popularized by Ferguson. We prove that if the prior distribution is Dirichlet, then the sequence of posterior distributions satisfies a large-deviation principle, and give an explicit expression for the rate function. As an application, we obtain an asymptotic formula for the predictive probability of ruin in the classical gambler's ruin problem.

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On reduction of finite-sample variance by extended Latin hypercube sampling

Nobuaki Hoshino, Akimichi Takemura

Bernoulli 6 (6), 1035-1050, (December 2000)
KEYWORDS: computer experiments, Monte Carlo simulation, numerical integration, ortho\-gonal arrays, variance reduction

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McKay, Conover and Beckman introduced Latin hypercube sampling (LHS) for reducing the variance of Monte Carlo simulations. LHS is a method for stratifying a univariate margin. We consider an extension of LHS to stratify an m-variate margin with orthogonal arrays, after Owen and Tang. We define extended Latin hypercube sampling of strength m (henceforth denoted by ELHS(m)), such that ELHS(1) reduces to LHS. Using the results obtained by Owen, we first derive an explicit formula for the finite-sample variance of ELHS(m). Based on this formula, we give a sufficient condition for variance reduction by ELHS(m), generalizing similar results from McKay et al. for m=1. Actually, our sufficient condition for m=1 contains the sufficient condition of McKay et al. and thus strengthens their result.

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Stochastic volatility models as hidden Markov models and statistical applications

Valentine Genon-Catalot, Thierry Jeantheau, Catherine Larédo

Bernoulli 6 (6), 1051-1079, (December 2000)
KEYWORDS: Diffusion processes, Discrete-time observations, Hidden Markov models, Mixing, Parametric inference, stochastic volatility

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This paper deals with the fixed sampling interval case for stochastic volatility models. We consider a two-dimensional diffusion process (Y_t, V_t), where only (Y_t) is observed at n discrete times with regular sampling interval ͉. The unobserved coordinate (V_t) is ergodic and rules the diffusion coefficient (volatility) of (Y_t). We study the ergodicity and mixing properties of the observations (Y_i͉). For this purpose, we first present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (V_t). When the stochastic differential equation of (V_t) depends on unknown parameters, we derive moment-type estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n^1/2. Examples of models coming from finance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.

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Exponential-polynomial families and the term structure of interest rates

Damir Filipovic

Bernoulli 6 (6), 1081-1107, (December 2000)
KEYWORDS: consistent Itôprocess, diffusion process, exponential-polynomial family, forward rate curve, interest rate model, inverse problem

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Exponential-polynomial families like the Nelson-Siegel or Svensson family are widely used to estimate the current forward rate curve. We investigate whether these methods go well with intertemporal modelling. We characterize the consistent Itô processes which have the property to provide an arbitrage-free interest rate model when representing the parameters of some bounded exponential-polynomial type function. This includes diffusion processes in particular. We show that there is a strong limitation on their choice. Bounded exponential-polynomial families are best not used for modelling the term structure of interest rates.

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The central limit theorem under censoring

Michael G. Akritas

Bernoulli 6 (6), 1109-1120, (December 2000)
KEYWORDS: distributions with atoms, i.i.d. representation, Kaplan-Meier integrals, martingales for counting processes

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The central limit theorem for integrals of the Kaplan-Meier estimator is obtained. The basic tools are the martingale methods developed by Gill and the identities and inequalities of Efron and Johnstone. The assumptions needed are both weaker and more transparent than those in the recent literature, and the resulting variance expression is simpler, especially for distributions with atoms.

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Minimal sufficient statistics in location-scale parameter models

Lutz Mattner

Bernoulli 6 (6), 1121-1134, (December 2000)
KEYWORDS: characterization, complete sufficient statistics, Equivariance, exponential family, independence, infinitely divisible distribution, mean periodic functions, normal distribution, order statistics, transformation model

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Let f be a probability density on the real line, let n be any positive integer, and assume the condition (R) that logf is locally integrable with respect to Lebesgue measure. Then either logf is almost everywhere equal to a polynomial of degree less than n, or the order statistic of n independent and identically distributed observations from the location-scale parameter model generated by f is minimal sufficient. It follows, subject to (R) and n≥3, that a complete sufficient statistic exists in the normal case only. Also, for f with (R) infinitely divisible but not normal, the order statistic is always minimal sufficient for the corresponding location-scale parameter model. The proof of the main result uses a theorem on the harmonic analysis of translation- and dilation-invariant function spaces, attributable to Leland (1968) and Schwartz (1947).

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Back Matter

Annual index Bernoulli Journal, vol. 6,.(2000)

Bernoulli 6 (6), (December 2000)

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Annual index Bernoulli Journal, vol. 6,.(2000)

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Acknowledgement of refereeing work

Bernoulli 6 (6), (December 2000)

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Acknowledgement of refereeing work

Bernoulli Journal, vol. 6,.(2000)

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