Showing papers in "Bernoulli in 2005"

Profile likelihood inferences on semiparametric varying-coefficient partially linear models

Abstract: Varying-coefficient partially linear models are frequently used in statistical modelling, but their estimation and inference have not been systematically studied. This paper proposes a profile least-squares technique for estimating the parametric component and studies the asymptotic normality of the profile least-squares estimator. The main focus is the examination of whether the generalized likelihood technique developed by Fan et al. is applicable to the testing problem for the parametric component of semiparametric models. We introduce the profile likelihood ratio test and demonstrate that it follows an asymptotically χ2 distribution under the null hypothesis. This not only unveils a new Wilks type of phenomenon, but also provides a simple and useful method for semiparametric inferences. In addition, the Wald statistic for semiparametric models is introduced and demonstrated to possess a sampling property similar to the profile likelihood ratio statistic. A new and simple bandwidth selection technique is proposed for semiparametric inferences on partially linear models and numerical examples are presented to illustrate the proposed methods.

On adaptive Markov chain Monte Carlo algorithms

Abstract: We look at adaptive Markov chain Monte Carlo algorithms that generate stochastic processes based on sequences of transition kernels, where each transition kernel is allowed to depend on the history of the process. We show under certain conditions that the stochastic process generated is ergodic, with appropriate stationary distribution. We use this result to analyse an adaptive version of the random walk Metropolis algorithm where the scale parameter σ is sequentially adapted using a Robbins-Monro type algorithm in order to find the optimal scale parameter σopt. We close with a simulation example.

On covariance estimation of non-synchronously observed diffusion processes

Abstract: We consider the problem of estimating the covariance of two diffusion processes when they are observed only at discrete times in a non-synchronous manner. The modern, popular approach in the literature, the realized covariance estimator, which is based on (regularly spaced) synchronous data, is problematic because the choice of regular interval size and data interpolation scheme may lead to unreliable estimation. We propose a new estimator which is free of any `synchronization' processing of the original data, hence free of bias or other problems caused by it.

Diffusion-type models with given marginal distribution and autocorrelation function

Abstract: Flexible stationary diffusion-type models are developed that can fit both the marginal distribution and the correlation structure found in many time series from, for example, finance and turbulence. Diffusion models with linear drift and a known and prespecified marginal distribution are studied, and the diffusion coefficients corresponding to a large number of common probability distributions are found explicitly. An approximation to the diffusion coefficient based on saddlepoint techniques is developed for use in cases where there is no explicit expression for the diffusion coefficient. It is demonstrated theoretically as well as in a study of turbulence data that sums of diffusions with linear drift can fit complex correlation structures. Any infinitely divisible distribution satisfying a weak regularity condition can be obtained as a marginal distribution.

Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes

Abstract: We consider nonparametric estimation of the Levy measure of a hidden Levy process driving a stationary Omstein-Uhlenbeck process which is observed at discrete time points. This Levy measure can be expressed in terms of the canonical function of the stationary distribution of the Omstein-Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this canonical function based on a preliminary estimator of the characteristic function of the stationary distribution. We provide a suppport-reduction algorithm for the numerical computation of the estimator, and show that the estimator is asymptotically consistent under various sampling schemes. We also define a simple consistent estimator of the intensity parameter of the process. Along the way, a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is shown that the Oenstein-Uhlenbeck process is β-mixing. Some general results on uniform convergence of random characteristic functions are included. © 2005 ISI/BS.

On the convergence of the spectral empirical process of Wigner matrices

Abstract: It is well known that the spectral distribution Fn of a Wigner matrix converges to Wigner's semicircle law. We consider the empirical process indexed by a set of functions analytic on an open domain of the complex plane including the support of the semicircle law. Under fourth-moment conditions, we prove that this empirical process converges to a Gaussian process. Explicit formulae for the mean function and the covariance function of the limit process are provided.

Asymptotics for L2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances

Abstract: Weighted L2 functionals of the empirical quantile process appear as a component of many test statistics, in particular in tests of fit to location-scale families of distributions based on weighted Wasserstein distances. An essentially complete set of distributional limit theorems for the squared empirical quantile process integrated with respect to general weights is presented. The results rely on limit theorems for quadratic forms in exponential random variables, and the proofs use only simple asymptotic theory for probability distributions in Rn. The limit theorems are then applied to determine the asymptotic distribution of the test statistics on which weighted Wasserstein tests are based. In particular, this paper contains an elementary derivation of the limit distribution of the Shapiro-Wilk test statistic under normality.

Infinite divisibility and generalized subexponentiality

Abstract: We introduce a new class of distributions by generalizing the subexponential class to investigate the asymptotic relation between the tails of an infinitely divisible distribution and its Levy measure. We call a one-sided distribution μ O-subexponential if it has positive tail satisfying limsup x →∞ μ*μ(x,∞)/μ(x,∞)<∞ . Necessary and sufficient conditions for an infinitely divisible distribution to be O-subexponential are given in a similar way to the subexponential case in work by Embrechts et al. It is of critical importance that the O-subexponential is not closed under convolution roots. This property leads to the difference between our result and that corresponding to the subexponential class. Moreover, under the assumption that an infinitely divisible distribution has exponential tail, it is shown that an infinitely divisible distribution is convolution equivalent if and only if the ratio of its tail and its Levy measure goes to a positive constant as x goes to infinity. Additionally, the upper and lower limits of the ratio of the tails of a semi-stable distribution and its Levy measure are given.

Optimal quantization methods for nonlinear filtering with discrete-time observations

Abstract: We develop an optimal quantization approach for numerically solving nonlinear filtering problems associated with discrete-time or continuous-time state processes and discrete-time observations. Two quantization methods are discussed: a marginal quantization and a Markovian quantization of the signal process. The approximate filters are explicitly solved by a finite-dimensional forward procedure. A posteriori error bounds are stated, and we show that the approximate error terms are minimal at some specific grids that may be computed off-line by a stochastic gradient method based on Monte Carlo simulations. Some numerical experiments are carried out: the convergence of the approximate filter as the accuracy of the quantization increases and its stability when the latent process is mixing are emphasized.

Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model

Abstract: We consider a semiparametric convolution model where the noise has known Fourier transform which decays asymptotically as an exponential with unknown scale parameter; the deconvolution density is less smooth than the noise in the sense that the tails of the Fourier transform decay more slowly, ensuring the identifiability of the model. We construct a consistent estimation procedure for the noise level and prove that its rate is optimal in the minimax sense. Two convergence rates are distinguished according to different smoothness properties for the unknown density. If the tail of its Fourier transform does not decay faster than exponentially, the asymptotic optimal rate and exact constant are evaluated, while if it does not decay faster than polynomially, this rate is evaluated up to a constant. Moreover, we construct a consistent estimator of the unknown density, by using a plug-in method in the classical kernel estimation procedure. We establish that the rates of estimation of the deconvolution density are slower than in the case of an entirely known noise distribution. In fact, nonparametric rates of convergence are equal to the rate of estimation of the noise level, and we prove that these rates are minimax. In a few specific cases the plug-in method converges at even slower rates.

Estimation of the extreme-value index and generalized quantile plots

Abstract: In extreme-value analysis, a central topic is the adaptive estimation of the extreme-value index γ Hitherto, most of the attention in this area has been devoted to the case γ>0, that is, when F ¯ is a regularly varying function with index -1/γ In addition to the well-known Hill estimator, many other estimators are currently available Among the most important are the kernel-type estimators and the weighted least-squares slope estimators based on the Pareto quantile plot or the Zipf plot, as reviewed by Csorgo and Viharos Using an exponential regression model (ERM) for spacings between successive extreme order statistics, both Beirlant et al and Feuerverger and Hall introduced bias-reduced estimators

Identification of multifractional Brownian motion

Abstract: We develop a method for estimating the Hurst function of a multifractional Brownian motion, which is an extension of the fractional Brownian motion in the sense that the path regularity can now vary with time. This method is based on a local estimation of the second-order moment of a unique discretized filtered path. The effectiveness of our procedure is investigated in a short simulation study.

Passage times for a spectrally negative Lévy process with applications to risk theory

Abstract: The distributions of the last passage time at a given level and the joint distributions of the last passage time, the first passage time and their difference for a general spectrally negative process are derived in the form of Laplace transforms. The results are applied to risk theory.

Approximation of sums of conditionally independent variables by the translated Poisson distribution

Abstract: It is shown that the distribution of the sum of a Poisson random variable and an independent approximately normally distributed integer-valued random variable can be well approximated in total variation by a translated Poisson distribution, and further that a mixed translated Poisson distribution is close to a mixed translated Poisson distribution with the same random shift but fixed variance. Using these two results, a general approach is then presented for the approximation of sums of integer-valued random variables, having some conditional independence structure, by a translated Poisson distribution. We illustrate the method by means of two examples. The proofs are mainly based on Stein's method for distributional approximation.

On data depth and distribution-free discriminant analysis using separating surfaces

Abstract: A very well-known traditional approach in discriminant analysis is to use some linear (or nonlinear) combination of measurement variables which can enhance class separability. For instance, a linear (or a quadratic) classifier finds the linear projection (or the quadratic function) of the measurement variables that will maximize the separation between the classes. These techniques are very useful in obtaining good lower-dimensional views of class separability. Fisher's discriminant analysis, which is primarily motivated by the multivariate normal distribution, uses the first- and second-order moments of the training sample to build such classifiers. These estimates, however, are highly sensitive to outliers, and they are not reliable for heavy-tailed distributions. This paper investigates two distribution-free methods for linear classification, which are based on the notions of statistical depth functions. One of these classifiers is closely related to Tukey's half-space depth, while the other is based on the concept of regression depth. Both these methods can be generalized for constructing nonlinear surfaces to discriminate among competing classes. These depth-based methods assume some finite-dimensional parametric form of the discriminating surface and use the distributional geometry of the data cloud to build the classifier. We use a few simulated and real data sets to examine the performance of these discriminant analysis tools and study their asymptotic properties under appropriate regularity conditions.

Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates

Abstract: This paper is concerned with two-person zero-sum games for continuous-time Markov chains, with possibly unbounded payoff and transition rate functions, under the discounted payoff criterion. We give conditions under which the existence of the value of the game and a pair of optimal stationary strategies is ensured by using the optimality (or Shapley) equation. We prove the convergence of the value iteration scheme to the game’s value and to a pair of optimal stationary strategies. Moreover, when the transition rates are bounded we further show that the convergence of value iteration is exponential. Our results are illustrated with a controlled queueing system with unbounded transition and reward rates.

Distance-reducing Markov bases for sampling from a discrete sample space

Abstract: We study Markov bases for sampling from a discrete sample space equipped with a convenient metric. Starting from any two states in the sample space, we ask whether we can always move closer by an element of a Markov basis. We call a Markov basis distance-reducing if this is the case. The particular metric we consider in this paper is the L1-norm on the sample space. Some characterizations of L1-norm-reducing Markov bases are derived.

Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws

Abstract: We are interested in the one-dimensional scalar conservation law ∂ t u(t,x)=νD αu(t,x)-∂ xA(u(t,x)) with fractional viscosity operator Dαv(x) = F-1(|ξ|αF(v)(ξ))(x) is the cumulative distribution function of a signed measure on R. We associate a nonlinear martingale problem with the Fokker-Planck equation obtained by spatial differentiation of the conservation law. After checking uniqueness for both the conservation law and the martingale problem, we prove existence thanks to a propagation-of-chaos result for systems of interacting particles with fixed intensity of jumps related to ν. The empirical cumulative distribution functions of the particles converge to the solution of the conservation law. As a consequence, it is possible to approximate this solution numerically by simulating the stochastic differential equation which gives the evolution of particles. Finally, when the intensity of jumps vanishes (ν→0) as the number of particles tends to +∞, we obtain that the empirical cumulative distribution functions converge to the unique entropy solution of the inviscid (ν=0) conservation law.

Sharp estimates in signed Poisson approximation of Poisson mixtures

Abstract: We give sharp estimates in total variation and certain kinds of stop-loss metrics in signed Poisson approximation of Poisson mixtures. We provide closed-form solutions to the problem of best choice of the Poisson parameter in simple Poisson approximation with respect to the total variation distance. The important special case of the negative binomial distribution is also discussed. To obtain our results, we apply a differential calculus based on different Taylor formulae for the Poisson process which allows us to give simple unified proofs.

Central limit theorem and convergence to stable laws in Mallows distance

Abstract: We give a new proof of the classical Central Limit Theorem, in the Mallows (L r -Wasserstein) distance. Our proof is elementary in the sense that it does not require complex analysis, but rather makes use of a simple subadditive inequality related to this metric. The key is to analyse the case where equality holds. We provide some results concerning rates of convergence. We also consider convergence to stable distributions, and obtain a bound on the rate of such convergence.

Nonparametric bootstrap prediction

Abstract: Ensemble learning has recently been intensively studied in the field of machine learning. `Bagging' is a method of ensemble learning and uses bootstrap data to construct various predictors. The required prediction is then obtained by averaging the predictors. Harris proposed using this technique with the parametric bootstrap predictive distribution to construct predictive distributions, and showed that the parametric bootstrap predictive distribution gives asymptotically better prediction than a plug-in distribution with the maximum likelihood estimator. In this paper, we investigate nonparametric bootstrap predictive distributions. The nonparametric bootstrap predictive distribution is precisely that obtained by applying bagging to the statistical prediction problem. We show that the nonparametric bootstrap predictive distribution gives predictions asymptotically as good as the parametric bootstrap predictive distribution.

On classifying processes

Abstract: for all processes not in G. Of course, if gn is a classification in a pointwise sense then it is a classification in probability, but a classification in probability is not necessarily a classification in a pointwise sense. For the class M k of k-step mixing Markov chains of fixed order k, there is a pointwise classification of the type we have just described. (For mixing Markov chains, see Proposition 1.2.10 in Shields 1996.) It was carried out in detail for independent processes by Bailey (1976). (Actually, he proved the result only for independent processes and indicated how to generalize his result for the class of Mk.) For the class Mmix U 0 Mk of mixing Markov chains of any order, Bailey showed that no such classification exists. See Omstein and Weiss (1990) for further results on this kind of question. Our concern in this paper is with the class of finitarily Markovian processes which is defined as follows. Let {Xn}1 be a stationary and ergodic binary time series. A one-sided stationary time series

Bootstrap prediction and Bayesian prediction under misspecified models

Abstract: We consider a statistical prediction problem under misspecified models. In a sense, Bayesian prediction is an optimal prediction method when an assumed model is true. Bootstrap prediction is obtained by applying Breiman's 'bagging' method to a plug-in prediction. Bootstrap prediction can be considered to be an approximation to the Bayesian prediction under the assumption that the model is true. However, in applications, there are frequently deviations from the assumed model. In this paper, both prediction methods are compared by using the Kullback-Leibler loss under the assumption that the model does not contain the true distribution. We show that bootstrap prediction is asymptotically more effective than Bayesian prediction under misspecified models.

Adaptive estimation of linear functionals under different performance measures

Abstract: Lower bounds are given for probabilistic error subject to a mean squared error constraint. Consequences for the expected length of variable length confidence intervals centred on adaptive estimators are given. It is shown that in many contexts centring confidence intervals on adaptive estimators must lead either to poor coverage probability or unnecessarily long intervals.

Minimax expected measure confidence sets for restricted location parameters

Abstract: We study confidence sets for a parameter θ∈Θ that have minimax expected measure among random sets with at least 1-α coverage probability. We characterize the minimax sets using duality, which helps to find confidence sets with small expected measure and to bound improvements in expected measure compared with standard confidence sets. We construct explicit minimax expected length confidence sets for a variety of one-dimensional statistical models, including the bounded normal mean with known and with unknown variance. For the bounded normal mean with unit variance, the minimax expected measure 95% confidence interval has a simple form for Θ= [-τ, τ] with τ≤3.25. For Θ= [-3, 3], the maximum expected length of the minimax interval is about 14% less than that of the minimax fixed-length affine confidence interval and about 16% less than that of the truncated conventional interval [X -1.96, X + 1.96] ∩[-3,3].

On Hipp's compound Poisson approximations via concentration functions

Abstract: This paper is devoted to a refinement of Hipp’s method in the compound Poisson approximation to the distribution of the sum of independent but not necessarily identically distributed random variables. Approximations by related Kornya–Presman signed measures are also considered. By using alternative proofs, we show that several constants in the upper bounds for the Kolmogorov and the stop-loss distances can be reduced. Concentration functions play an important role in Hipp’s method. Therefore, we provide an improvement of the constant in Le Cam’s bound for concentration functions of compound Poisson distributions. But we also follow Hipp’s idea to estimate such concentration functions with the help of Kesten’s concentration function bound for sums of independent random variables. In fact, under the assumption that the summands are identically distributed, we present a smaller constant in Kesten’s bound, the proof of which is based on a slight sharpening of Le Cam’s version of the Kolmogorov–Rogozin inequality.

On L2-convergence of controlled branching processes with random control function

Abstract: Consider, on the same probability space, two independent sets of non-negative integervalued random variables, fX nj : n 1⁄4 0, 1, . . .; j 1⁄4 1, 2, . . .g and f n(k) : n 1⁄4 0, 1, . . .; k 1⁄4 0, 1, . . . ,g where the X nj are independent and identically distributed, and the f n(k)gk>0 are independent stochastic processes with identical one-dimensional probability distributions. The common probability law of the X nj is pk :1⁄4 P(X 01 1⁄4 k), k 1⁄4 0, 1, . . . , called the offspring distribution. The n(k) are called control variables. The controlled branching process with random control function fZ ngn>0 is then defined as

Convergence results for conditional expectations

Abstract: Let E,F be two Polish spaces and [Xn,Yn],[X,Y] random variables with values in E×F (not necessarily defined on the same probability space). We show some conditions which are sufficient in order to assure that, for each bounded continuous function f on E×F, the conditional expectation of f(Xn,Yn) given Yn converges in distribution to the conditional expectation of f(X,Y) given Y.

Empirical likelihood based inference for the derivative of the nonparametric regression function

Abstract: We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative We show that the limiting distribution of the empirical likelihood ratio is a scaled χ 1 2 distribution rather than the usual (unscaled) χ 1 2 distribution We use this limiting distribution to construct pointwise confidence intervals for the derivative Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals A small simulation study also suggests that they are more accurate

On the unlimited growth of a class of homogeneous multitype Markov chains

Abstract: We consider a homogeneous multitype Markov chain whose states have non-negative integer coordinates, and give criteria for deciding whether or not the chain grows indefinitely with positive probability. The results are applied to study the extinction problem in a general class of controlled multitype branching processes.