Showing papers in "Bernoulli in 1999"

Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists

Abstract: Author(s): Aldous, DJ | Abstract: Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x, y)/N, were AT is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x, y) = 1 and K(x, y) = xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear. © 1999 ISI/BS.

An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions

Abstract: The Radon-Nikodym derivative between a centred fractional Brownian motion Z and the same process with constant drift is derived by finding an integral transformation which changes Z to a process with independent increments. A representation of Z through a standard Brownian motion on a finite interval is given. The maximum-likelihood estimator of the drift and some other applications are presented.

Estimating equations based on eigenfunctions for a discretely observed diffusion process

Abstract: A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discrete-time observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study

Prequential probability: principles and properties

Abstract: Forecaster has to predict, sequentially, a string of uncertain quantities ( X 1,X 2,...) , whose values are determined and revealed, one by one, by Nature. Various criteria may be proposed to assess Forecaster's empirical performance. The weak prequential principle requires that such a criterion should depend on Forecaster's behaviour or strategy only through the actual forecasts issued. A wide variety of appealing criteria are shown to respect this principle. We further show that many such criteria also obey the strong prequential principle, which requires that, when both Nature and Forecaster make their choices in accordance with a common joint distribution P for ( X 1,X 2,...) , certain stochastic properties, underlying and justifying the criterion and inferences based on it, hold regardless of the detailed specification of P . In order to understand further this compliant behaviour, we introduce the prequential framework, a game-theoretic basis for probability theory in which it is impossible to violate the prequential principles, and we describe its connections with classical probability theory. In this framework, in order to show that some criterion for assessing Forecaster's empirical performance is valid, we have to exhibit a winning strategy for a third player, Statistician, in a certain perfect-information game. We demonstrate that many performance criteria can be formulated and are valid in the framework and, therefore, satisfy both prequential principles.

Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes

Abstract: The area-interaction process and the continuum random-cluster model are characterized in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpler to analyse and simulate. Using this correspondence we devise a two-component Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a Swendsen-Wang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.

The exponential statistical manifold: mean parameters, orthogonality and space transformations

Abstract: Let ( X,cal X,μ) be a measure space, and let cal M (X,cal X,μ) denote the set of the μ -almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on cal M (X,cal X,μ) can be realized by an affine atlas whose charts are defined locally by the mappings cal M (X,cal X,μ)⊃cal U p∋q↦log(q/p)+K(p,q)∈B p , where cal U p is a suitable open set containing p , K (p,q) is the Kullback--Leibler relative information and B p is the vector space of centred and exponentially ( p⋅μ) -integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, i.e. measurable transformation of the sample space. A generalization of the mixed parametrization method for exponential models is also presented.

Parameter estimation for discretely observed stochastic volatility models

Abstract: This paper deals with parameter estimation for stochastic volatility models. We consider a two-dimensional diffusion process (Y-t, V-t). Only (Y-t) is observed at n discrete times with a regular sampling interval. The unobserved coordinate (V-t) rules the diffusion coefficient (volatility) of (Y-t) and is an ergodic diffusion depending on unknown parameters. We build estimators of the parameters present in the stationary distribution of (V-t), based on appropriate functions of the observations. Consistency is proved under the asymptotic framework that the sampling interval tends to 0, while the number of observations and the length of the observation time tend to infinity. Asymptotic normality is obtained under an additional condition on the rate of convergence of the sampling interval. Examples of models from finance are treated, and numerical simulation results are given.

The concept of duality and applications to Markov processes arising in neutral population genetics models

Abstract: One possible and widely used definition of the duality of Markov processes employs functions H relating one process to another in a certain way. For given processes X and Y the space U of all such functions H, called the duality space of X and Y, is studied in this paper. The algebraic structure of U is closely related to the eigenvalues and eigenvectors of the transition matrices of X and Y. Often as for example in physics (interacting particle systems) and in biology (population genetics models) dual processes arise naturally by looking forwards and backwards in time. In particular, time-reversible Markov processes are self-dual. In this paper, results on the duality space are presented for classes of haploid and two-sex population models. For example dim U = N+ 3 for the classical haploid Wright-Fisher model with fixed population size N.

Observed information in semi-parametric models

Abstract: We discuss the estimation of the asymptotic covariance matrix of semi-parametric maximum likelihood estimators by the observed profile information We show that a discretized version of the second derivative of the profile likelihood function yields consistent estimators of minus the efficient information matrix

Asymptotic normality of posterior distributions in high-dimensional linear models

Abstract: We study consistency and asymptotic normality of posterior distributions of the regression coefficient in a linear model when the dimension of the parameter grows with increasing sample size. Under certain growth restrictions on the dimension (depending on the design matrix), we show that the posterior distributions concentrate in neighbourhoods of the true parameter and can be approximated by an appropriate normal distribution.

Nonlinear Wavelet Estimation of Time-Varying Autoregressive Processes

Abstract: We consider nonparametric estimation of the parameter functions a(i)(.), i = 1, ..., p, of a time-varying autoregressive process. Choosing an orthonormal wavelet basis representation of the Functions a(i), the empirical wavelet coefficients are derived from the time series data as the solution of a least-squares minimization problem. In order to allow the a(i) to be functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coefficients and obtain locally smoothed estimates of the a(i). We show that the resulting estimators attain the usual minimax L-2 rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The finite-sample behaviour of our procedure is demonstrated by application to two typical simulated examples.

Approximate nonlinear filtering by projection on exponential manifolds of densities

Abstract: This paper introduces in detail a new systematic method to construct approximate finite-dimensional solutions for the nonlinear filtering problem. Once a finite-dimensional family is selected, the nonlinear filtering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection filter for the chosen family. The general definition of the projection filter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is defined. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented. The classical concept of assumed density filter (ADF) is compared with the projection filter. It is shown that the concept of ADF is inconsistent in the sense that the resulting filters depend on the choice of a stochastic calculus, i.e. the Ito or the Stratonovich calculus. It is shown that in the context of exponential families, the projection filter coincides with the Stratonovich-based ADF. An example is provided, which shows that this does not hold in general, for non-exponential families of densities.

Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative

Abstract: We study the problem of testing a simple hypothesis for a nonparametric ''signal + white-noise'' model. It is assumed under the null hypothesis that the ''signal'' is completely specified, e.g., that no signal is present. This hypothesis is tested against a composite alternative of the following form: the underlying function (the signal) is separated away from the null in the L2 norm and, in addition, it possesses some smoothness properties. We focus on the case of an inhomogeneous alternative when the smoothness properties of the signal are measured in an Lp norm with p<2. We consider tests whose errors have probabilities which do not exceed prescribed values and we measure the quality of testing by the minimal distance between the null and the alternative set for which such testing is still possible. We evaluate the optimal rate of decay for this distance to zero as the noise level tends to zero. Then a rate-optimal test is proposed which essentially uses a pointwise-adaptive estimation procedure.

A new mixing notion and functional central limit theorems for a sieve bootstrap in time series

Abstract: We study a bootstrap method for stationary real-valued time series, which is based on the sieve of autoregressive processes. Given a sample X 1 ,...,X n from a linear process { X t} t ∈ℤ , we approximate the underlying process by an autoregressive model with order p =p(n) , where p (n)→∞,p(n)=o(n) as the sample size n →∞ . Based on such a model, a bootstrap process { X t *} t ∈ℤ is constructed from which one can draw samples of any size. We show that, with high probability, such a sieve bootstrap process { X t *} t ∈ℤ satisfies a new type of mixing condition. This implies that many results for stationary mixing sequences carry over to the sieve bootstrap process. As an example we derive a functional central limit theorem under a bracketing condition.

On pointwise adaptive nonparametric deconvolution

Abstract: We consider estimating an unknown function f from indirect white noise observations with particular emphasis on the problem of nonparametric deconvolution. Nonparametric estimators that can adapt to unknown smoothness of f are developed. The adaptive estimators are specified under two sets of assumptions on the kernel of the convolution transform. In particular, kernels having the Fourier transform with polynomially and exponentially decaying tails are considered. It is shown that the proposed estimates possess, in a sense, the best possible abilities for pointwise adaptation.

Random-design regression under long-range dependent errors

Abstract: We consider the random-design nonparametric regression model with long-range dependent errors that may also depend on the independent and identically distributed explanatory variables. Disclosing a smoothing dichotomy, we show that the finite-dimensional distributions of the Nadaraya-Watson kernel estimator of the regression fimunction converge either to those of a degenerate process with completely dependent marginals or to those of a Gaussian white-noise process. The first case occurs when the bandwidths are large enough in a specified sense to allow long-range dependence to prevail. The second case is for bandwidths that are small in the given sense, when both the required norming sequence and the limiting process are the same as if the errors were independent. This conclusion is also derived for all bandwidths if the errors are short-range dependent. The borderline situation results in a limiting convolution of the two cases. The main results contrast with previous findings for deterministic-design regression.

Likelihood ratio tests in contamination models

Abstract: We study the asymptotic distribution of the likelihood ratio statistic to test whether the contamination of a known density f0 by another density of the same parametric family reduces to f0. The classical asymptotic theory for the likelihood ratio statistic fails, and we propose a general reparametrization which ensures regularity properties. Under the null hypothesis, the likelihood ratio statistic converges to the supremum of a squared truncated Gaussian process. The result is extended to the case of the contamination of a mixture of p known densities by q other densities of the same family.

Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution

Abstract: The Poisson-binomial distribution is approximated by a Poisson law with respect to a new multi-metric (difference metric) unifying a broad class of probability metrics between discrete distributions. The accompanying non-metric situation is also considered leading to moderate- and large-deviation results. Using the Charlier B expansion and Fourier arguments, sharp bounds and asymptotic relations are given.

Some asymptotic properties of the local time of the uniform empirical process

Abstract: We study the almost sure asymptotic properties of the local time of the uniform empirical process. In particular, we obtain two versions of the law of the iterated logarithm for the integral of the square of the local time. It is interesting to note that the corresponding problems for the Wiener process remain open. Properties of Lp-norms of the local time are studied. We also characterize the joint asymptotics of the local time at a fixed level and the maximum local time.

Lp estimation of the diffusion coefficient

Abstract: We study the functional estimation of the space-dependent diffusion coefficient in a one-dimensional framework. The sample path is observed at discrete times. We study global L p -loss errors ( 1≤p<+∞) over Besov spaces B sp ∞ . We show that, under suitable conditions, the minimax rate of convergence is the usual n - s/(1+2s) . Linking our model to nonparametric regression, we provide an estimating procedure based on a linear wavelet method which is optimal in the minimax sense.

Asymptotic Inference for a Linear Stochastic Differential Equation with Time Delay

Abstract: For the stochastic differential equation d X(t)=a X(t)+bX(t-1),dt+dW(t),t≥0, the local asymptotic properties of the likelihood function are studied. They depend strongly on the true value of the parameter θ =(a,b) * . Eleven different cases are possible if θ runs through ℝ 2 . Let θ T be the maximum likelihood estimator of θ based on ( X(t),t≤T) . Applications to the asymptotic behaviour of θ T as T →∞ are given.

The p-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure

Abstract: In recent years the problem of finding martingale measures for a stochastic process has found applications in the field of mathematical finance, e.g. the famous Black-Scholes formula for evaluating a European call option can be seen as the expectational value of a random variable with respect to the (in this case unique) martingale measure for the discounted stock price process. In general there is no unique martingale measure for a stochastic process. So one is confronted with the problem of choosing a proper martingale measure. Very popular possibilities are the so-called minimal martingale measure, which has been introduced by F611mer and Schweizer (1991), or the variance-optimal measure (Schweizer 1995; Delbaen and Schachermayer 1996; Delbaen et al. 1997). The latter is characterized by minimizing the L2 norm of the Radon-Nikodym derivative of the new measure with respect to the original measure among all signed martingale measures for the process. The former exhibits this feature locally (for a more exact description see F611mer and Schweizer (1991)). Another possibility is the minimal-entropy martingale measure. It has been shown by Frittelli (1996) that for a bounded process a unique martingale measure, which minimizes relative entropy between the original measure and the martingale measure, always exists. In addition, if the relative entropy is finite, the two measures are equivalent. For an economic interpretation of the variance-optimal and minimal-entropy measures see Delbaen et al. (1997) and see Frittelli (1996) and Platen and Rebolledo (1995) respectively. The aim of this paper is to find a connection between these two concepts in discrete time with a finite horizon. It turns out that the missing link is given by martingale measures, which we call p optimal and which are characterized by minimizing the I' norm instead of

Bayesian smoothing in the estimation of the pair potential function of Gibbs point processes

Abstract: A flexible Bayesian method is suggested for the pair potential estimation with a high-dimensional parameter space The method is based on a Bayesian smoothing technique, commonly applied in statistical image analysis For the calculation of the posterior mode estimator a new Monte Carlo algorithm is developed The method is illustrated through examples with both real and simulated data, and its extension into truly nonparametric pair potential estimation is discussed

The asymptotic minimax constant for sup-norm loss in nonparametric density estimation

Abstract: We develop the exact constant of the risk asymptotics in the uniform norm for density estimation. This constant has already been found for nonparametric regression and for signal estimation in Gaussian white noise. Holder classes for arbitrary smoothness index β>0 on the unit interval are considered. The constant involves the value of an optimal recovery problem as in the white noise case, but in addition it depends on the maximum of densities in the function class.

Nonparametric estimation of quadratic regression functionals

Abstract: Quadratic regression functionals are important for bandwidth selection of nonparametric regression techniques and for nonparametric goodness-of-fit test. Based on local polynomial regression, we propose estimators for weighted integrals of squared derivatives of regression functions. The rates of convergence in mean square error are calculated under various degrees of smoothness and appropriate values of the smoothing parameter. Asymptotic distributions of the proposed quadratic estimators are considered with the Gaussian noise assumption. It is shown that when the estimators are pseudo-quadratic (linear components dominate quadratic components), asymptotic normality with rate n-1/2 can be achieved.

On multivariate median regression

Abstract: An extension of the concept of least absolute deviation regression for problems with multivariate response is considered. The approach is based on a transformation and retransformation technique that chooses a data-driven coordinate system for transforming the response vectors and then retransforms the estimate of the matrix of regression parameters, which is obtained by performing coordinatewise least absolute deviations regression on the transformed response vectors. It is shown that the estimates are equivariant under non-singular linear transformations of the response vectors. An algorithm called TREMMER (Transformation Retransformation Estimates in Multivariate MEdian Regression) has been suggested, which adaptively chooses the optimal data-driven coordinate system and then computes the regression estimates. We have also indicated how resampling techniques like the bootstrap can be used to conveniently estimate the standard errors of TREMMER estimates. It is shown that the proposed estimate is more efficient than the non-equivariant coordinatewise least absolute deviations estimate, and it outperforms ordinary least-squares estimates in the case of heavy-tailed non-normal multivariate error distributions. Asymptotic normality and some other optimality properties of the estimate are also discussed. Some interesting examples are presented to motivate the need for affine equivariant estimation in multivariate median regression and to demonstrate the performance of the proposed methodology.

Consistency and strong inconsistency of group-invariant predictive inferences

Abstract: Consider a statistical model which is invariant under a group of transformations that acts transitively on the parameter space. In this situation, the problem of constructing invariant predictive distributions is considered. It is shown, under certain assumptions, that Fisherian pivoting and the use of right Haar measure as an improper prior distribution both yield the same invariant predictive distribution. Furthermore, it is shown that any other invariant predictive distribution is strongly inconsistent in the sense of Stone.

Laplace transforms related to excursions of a one-dimensional diffusion

Abstract: Various known expressions in terms of hyperbolic functions for the Laplace transforms of random times related to one-dimensional Brownian motion are derived in a unified way by excursion theory and extended to one-dimensional diffusions.

A note on limit theorems for multivariate martingales

Abstract: Multivariate versions of the law of large numbers and the central limit theorem for martingales are given in a generality that is often necessary when studying statistical inference for stochastic process models To illustrate the usefulness of the results, we consider estimation for a multidimensional Gaussian diffusion, where results on consistency and asymptotic normality of the maximum likelihood estimator are obtained in cases that were not covered by previously published limit theorems The results are also applied to martingales of a different nature, which are typical of the problems occurring in connection with statistical inference for stochastic delay equations