Catherine Larédo

Stochastic volatility models as hidden Markov models and statistical applications

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 Delta. 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-i Delta). 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.

Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments

The main goal of the asymptotic equivalence theory of Le Cam (1986) is to approximate general statistical models by simple ones. We develop here a global asymptotic equivalence result for nonparametric drift estimation of a discretely observed diffusion process and its Euler scheme. The asymptotic equivalences are established by constructing explicit equivalence mappings. The impact of such asymptotic equivalence results is that it justifies the use in many applications of the Euler scheme instead of the diffusion process. We especially investigate the case of diffusions with non constant diffusion coefficient. To obtain asymptotic equivalence, experiments obtained by random change of times are introduced.

Estimation for stochastic differential equations with mixed effects

We consider the long-term behaviour of a one-dimensional mixed effects diffusion process with a multivariate random effect φ in the drift coefficient. We first study the estimation of the random variable φ based on the observation of one sample path on the time interval as T tends to infinity. The process is not Markov and we characterize its invariant distributions. We build moments and maximum likelihood-type estimators of the random variable φ which are consistent and asymptotically mixed normal with rate . Moreover, we obtain non-asymptotic bounds for the moments of these estimators. Examples with a bivariate random effect are detailed. Afterwards, the estimation of parameters in the distribution of the random effect from N i.i.d. processes is investigated. Estimators are built and studied as both N and tend to infinity. We prove that the convergence rate of estimators differs when deterministic components are present in the random effects. For true random effects, the rate of convergence is whereas for deterministic components, the rate is . Illustrative examples are given.

Coalescent-Based DNA Barcoding: Multilocus Analysis and Robustness

DNA barcoding is the assignment of individuals to species using standardized mitochondrial sequences. Nuclear data are sometimes added to the mitochondrial data to increase power. A barcoding method for analysing mitochondrial and nuclear data is developed. It is a Bayesian method based on the coalescent model. Then this method is assessed using simulated and real data. It is found that adding nuclear data can reduce the number of ambiguous assignments. Finally, the robustness of coalescent-based barcoding to departures from model assumptions is studied using simulations. This method is found to be robust to past population size variations, to within-species population structures, and to designs that poorly sample populations within species. Supplementary Material is available online at www.liebertonline.com/cmb.

Estimation of Plant Demographic Parameters from Stage-Structured Censuses

This article presents some statistical methods for estimating the parameters of a population dynamics model for annual plants. The model takes account of reproduction, immigration, seed survival in a seed bank, and plant growth. The data consist of the number of plants in several developmental stages that were measured in a number of populations for a few consecutive years; they are incomplete since seeds could not be counted. It is assumed that there are no measurement errors or that measurement errors are binomial and not frequent. Some statistical methods are developed within the framework of estimating equations or Bayesian inference. These methods are applied to oilseed rape data.

OPTIMAL SAMPLING DESIGNS FOR STUDIES OF GENE FLOW: A COMMENT ON ASSUNÇÃO AND JACOBI

The number of studies concerning the estimation of gene flow has drastically increased in the last few years. This is partly due to new possibilities of estimating both migration rates between populations and the shape of dispersal curves. In particular, genetic engineering allows the construction of genetically modified marked plants that are easy to trace. The use of herbicide resistance (Scheffler et al. 1993; Lavigne et al. 1998), antibiotic resistance (McPartlan and Dale 1994; Paul et al. 1995), and insecticidal genes (Llewellyn and Fitt 1996) has allowed these researchers to score up to millions of individuals to estimate the gene dispersal curves of a variety of crop plants. All these studies consist of a source of plants producing marked pollen surrounded by recipient plants on which progeny is sampled and scored for the presence of the marker gene. Before the work of Assunqao and Jacobi (1996), the sampling design had not been investigated theoretically. This led these authors to suggest an algorithm to optimize it, both in terms of sampling stations and of numbers of plants to sample at each station. However, as we show here, their intiutive conclusions are based on two very restrictive assumptions.