Showing papers in "Bernoulli in 2011"

Absolute regularity and ergodicity of Poisson count processes

Abstract: We consider a class of observation-driven Poisson count processes where the current value of the accompanying intensity process depends on previous values of both processes. We show under a contractive condition that the bivariate process has a unique stationary distribution and that the stationary version of the count process is absolutely regular. Moreover, since the intensities can be written as measurable functionals of the count variables we conclude that the bivariate process is ergodic. As an important application of these results, we show how a test method previously used in the case of independent Poisson data can be used in the case of Poisson count processes.

Estimating conditional quantiles with the help of the pinball loss

Abstract: Using the so-called pinball loss for estimating conditional quantiles is a well-known tool in both statistics and machine learning. So far, however, only little work has been done to quantify the efficiency of this tool for non-parametric (modified) empirical risk minimization approaches. The goal of this work is to fill this gap by establishing inequalities that describe how close approximate pinball risk minimizers are to the corresponding conditional quantile. These inequalities, which hold under mild assumptions on the data-generating distribution, are then used to establish so-called variance bounds which recently turned out to play an important role in the statistical analysis of (modified) empirical risk minimization approaches. To illustrate the use of the established inequalities, we then use them to establish an oracle inequality for support vector machines that use the pinball loss. Here, it turns out that we obtain learning rates which are optimal in a min-max sense under some standard assumptions on the regularity of the conditional quantile function.

Nonparametric tests for pathwise properties of semimartingales

Abstract: We propose two nonparametric tests for investigating the pathwise properties of a signal modeled as the sum of a Levy process and a Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the quadratic variation, we design a test for the presence of a continuous martingale component in the process and a test for establishing whether the jumps have finite or infinite variation, based on observations on a discrete-time grid. We evaluate the performance of our tests using simulations of various stochastic models and use the tests to investigate the fine structure of the DM/USD exchange rate fluctuations and SPX futures prices. In both cases, our tests reveal the presence of a non-zero Brownian component and a finite variation jump component.

Integral representations and properties of operator fractional Brownian motions

Abstract: Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time-reversibility, the defining properties (i)–(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here by means of their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and taking the Fourier transforms of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time-reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of an OFBM has a rigid structure, here called the dichotomy principle. The notion of operator Brownian motions is also explored.

A goodness-of-fit test for bivariate extreme-value copulas

Abstract: It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. In this paper, a procedure is proposed for testing whether this function belongs to a given parametric family. The test is based on a Cramer-von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers [Ann. Statist. 37 (2009) 2990-3022]. As the limiting distribution of the test statistic depends on unknown parameters, it must be estimated via a parametric bootstrap procedure, the validity of which is established. Monte Carlo simulations are used to assess the power of the test and an extension to dependence structures that are left-tail decreasing in both variables is considered. © 2011 ISI/BS.

Multipower Variation for Brownian Semistationary Processes

Abstract: In this paper we study the asymptotic behaviour of power and multipower variations of processes $Y$: $$Y_t = ∫_{−∞}^tg(t − s)σ_sW(\mathrm{d}s) + Z_t,$$ where $g : (0, ∞) → ℝ$ is deterministic, $σ > 0$ is a random process, $W$ is the stochastic Wiener measure and $Z$ is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency $σ$. The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of $Y$ as a basis for studying properties of the intermittency process $σ$. Notably the processes $Y$ are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given.

Probability distributions with summary graph structure

Abstract: A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of edges that couple node pairs. One important class contains regression graphs. Regression graphs are a type of so-called chain graph and describe stepwise processes, in which at each step single or joint responses are generated given the relevant explanatory variables in their past. For joint densities that result after possible marginalising or conditioning, we introduce summary graphs. These graphs reflect the independence structure implied by the generating process for the reduced set of variables and they preserve the implied independences after additional marginalising and conditioning. They can identify generating dependences that remain unchanged and alert to possibly severe distortions due to direct and indirect confounding. Operators for matrix representations of graphs are used to derive these properties of summary graphs and to translate them into special types of paths in graphs.

Conditioning on an extreme component: Model consistency with regular variation on cones

Abstract: Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector necessitating that each component satisfies a marginal domain of attraction condition. [12] and [11] developed an approximation to the joint distribution of the random vector by conditioning that one of the components be extreme. Prior papers left unresolved the consistency of dierent models obtained by conditioning on dierent components being extreme and we provide understanding of this issue. We also clarify the relationship between these conditional distributions, multivariate extreme value theory, and standard regular variation on cones of the form [0,1] ◊ (0,1].

Asymptotics of supremum distribution of a Gaussian process over a Weibullian time

Abstract: Let $\{X(t) :t∈[0, ∞)\}$ be a centered Gaussian process with stationary increments and variance function $σ_X^2(t)$. We study the exact asymptotics of $ℙ(\sup _{t∈[0, T]}X(t)>u)$ as $u→∞$, where $T$ is an independent of $\{X(t)\}$ non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion.

Chain graph models of multivariate regression type for categorical data

Abstract: We discuss a class of chain graph models for categorical variables defined by what we call a multivariate regression chain graph Markov property. First, the set of local independencies of these models is shown to be Markov equivalent to those of a chain graph model recently defined in the literature. Next we provide a parametrization based on a sequence of generalized linear models with a multivariate logistic link function that captures all independence constraints in any chain graph model of this kind.

Some intriguing properties of Tukey’s half-space depth

Abstract: For multivariate data, Tukey’s half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties. In this article, we derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey’s half-space depth and the related median for probability distributions on any Banach space (which may be finite- or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces.

Quasi Ornstein–Uhlenbeck processes

Abstract: The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of the Wold–Karhunen decomposition theorem, such solutions are, in principle, representable as a moving average (plus a drift-like term) but the kernel in the moving average is generally not available in explicit form. A class of cases is determined where an explicit expression of the kernel can be given, and this is used to obtain information on the asymptotic behavior of the associated autocorrelation functions, both for small and large lags. Applications to Gaussian- and Levy-driven fractional Ornstein–Uhlenbeck processes are presented. A Fubini theorem for Levy bases is established as an element in the derivations.

Support vector machines with a reject option

Abstract: This paper studies $l_1$ regularization with high-dimensional features for support vector machines with a built-in reject option (meaning that the decision of classifying an observation can be withheld at a cost lower than that of misclassification). The procedure can be conveniently implemented as a linear program and computed using standard software. We prove that the minimizer of the penalized population risk favors sparse solutions and show that the behavior of the empirical risk minimizer mimics that of the population risk minimizer. We also introduce a notion of classification complexity and prove that our minimizers adapt to the unknown complexity. Using a novel oracle inequality for the excess risk, we identify situations where fast rates of convergence occur.

Measuring the roughness of random paths by increment ratios

Abstract: A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a Levy process with $\alpha-$stable tangent process is established and can be used to estimate the fractional parameter $\alpha \in (0,2)$ following a central limit theorem.

On nonlinear Markov chain Monte Carlo

Abstract: Let $\mathscr{P}(E)$ be the space of probability measures on a measurable space $(E,\mathcal{E})$. In this paper we introduce a class of nonlinear Markov chain Monte Carlo (MCMC) methods for simulating from a probability measure $\pi\in\mathscr{P}(E)$. Nonlinear Markov kernels (see [Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Springer]) $K:\mathscr{P}(E)\times E\rightarrow\mathscr{P}(E)$ can be constructed to, in some sense, improve over MCMC methods. However, such nonlinear kernels cannot be simulated exactly, so approximations of the nonlinear kernels are constructed using auxiliary or potentially self-interacting chains. Several nonlinear kernels are presented and it is demonstrated that, under some conditions, the associated approximations exhibit a strong law of large numbers; our proof technique is via the Poisson equation and Foster–Lyapunov conditions. We investigate the performance of our approximations with some simulations.

The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables

Abstract: We propose a new algorithm to compute numerically the distribution function of the sum of $d$ dependent, non-negative random variables with given joint distribution.

From Schoenberg to Pick–Nevanlinna: Toward a complete picture of the variogram class

Abstract: We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg–Levy type and demonstrate some important properties of rotationally invariant variograms.

Explicit identities for Lévy processes associated to symmetric stable processes

Abstract: In this paper we introduce a new class of Levy processes which we call hypergeometric- stable Levy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Levy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.

Coupling for Ornstein–Uhlenbeck processes with jumps

Abstract: Consider the linear stochastic differential equation (SDE) on $ℝ^n$: $$\mathrm{d}X_t = AX_t \mathrm{d}t + B \mathrm{d}L_t,$$ where $A$ is a real $n × n$ matrix, $B$ is a real $n × d$ real matrix and $L_t$ is a Levy process with Levy measure $ν$ on $ℝ^d$. Assume that $ν(\mathrm{d}z) ≥ ρ_0(z)\mathrm{d}z$ for some ${ρ_0} \geq 0$. If $A \leq 0$, Rank$(B) = n$ and $∫_{\{|z−z_0| \leq e\}}ρ_0(z)^{−1} \mathrm{d}z 0$, then the associated Markov transition probability $P_t(x, \mathrm{d}y)$ satisfies $$‖P_t(x, ⋅) − P_t(y, ⋅)‖_{\mathrm{var}} \le \frac{C(1 + |x − y|)}{√t}, x, y ∈ ℝ^d, t > 0,$$ for some constant $C > 0$, which is sharp for large $t$ and implies that the process has successful couplings. The Harnack inequality, ultracontractivity and the strong Feller property are also investigated for the (conditional) transition semigroup.

Characterization theorems for the Gneiting class of space-time covariances

Abstract: We characterize the Gneiting class of space–time covariance functions and give more relaxed conditions on the functions involved. We then show necessary conditions for the construction of compactly supported functions of the Gneiting type. These conditions are very general since they do not depend on the Euclidean norm.

An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

Abstract: For an absolutely continuous (integer-valued) r.v. X of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order k holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237–260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. X, expressions that seem to be known only in particular cases (for the Normal, see [Houdre and Kagan, J. Theoret. Probab. 8 (1995) 23–30]; see also [Houdre and Perez-Abreu, Ann. Probab. 23 (1995) 400–419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

On a fractional linear birth-death process

Abstract: In this paper, we introduce and examine a fractional linear birth–death process $N_ν(t), t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^ν(t), t>0, k≥0$. We present a subordination relationship connecting $N_ν(t), t>0$, with the classical birth–death process $N(t), t>0$, by means of the time process $T_{2ν}(t), t>0$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^ν(t)$ and the state probabilities $p_k^ν(t), t>0, k≥1$, in the three relevant cases $λ>μ, λ<μ, λ=μ$ (where $λ$ and $μ$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth–death process with the fractional pure birth process. Finally, the mean values $\mathbb{E}N_{ u}(t)$ and $\operatorname{\mathbb{V}ar}N_{ u}(t)$ are derived and analyzed.

Mixing properties of ARCH and time-varying ARCH processes

Abstract: There exist very few results on mixing for non-stationary processes. However, mixing is often required in statistical inference for non-stationary processes such as time-varying ARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic process are derived in terms of the conditional densities of the process. These bounds are used to obtain the $α$, 2-mixing and $β$-mixing rates of the non-stationary time-varying ARCH($p$) process and ARCH($∞$) process. It is shown that the mixing rate of the time-varying ARCH($p$) process is geometric, whereas the bound on the mixing rate of the ARCH($∞$) process depends on the rate of decay of the ARCH($∞$) parameters. We note that the methodology given in this paper is applicable to other processes.

Fractional Levy-driven Ornstein-Uhlenbeck processes and stochastic differential equations

Abstract: Using Riemann-Stieltjes methods for integrators of bounded p-variation we define a pathwise in- tegral driven by a fractional Levy process (FLP). To explicitly solve general fractional stochastic differential equations (SDEs) we introduce an Ornstein-Uhlenbeck model by a stochastic inte- gral representation, where the driving stochastic process is an FLP. To achieve the convergence of improper integrals, the long-time behavior of FLPs is derived. This is sufficient to define the fractional Levy-Ornstein-Uhlenbeck process (FLOUP) pathwise as an improper Riemann- Stieltjes integral. We show further that the FLOUP is the unique stationary solution of the corresponding Langevin equation. Furthermore, we calculate the autocovariance function and prove that its increments exhibit long-range dependence. Exploiting the Langevin equation, we consider SDEs driven by FLPs of bounded p-variation for p < 2 and construct solutions us- ing the corresponding FLOUP. Finally, we consider examples of such SDEs, including various state space transforms of the FLOUP and also fractional Levy-driven Cox-Ingersoll-Ross (CIR) models.

A martingale approach to continuous-time marginal structural models

Abstract: Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. The key point is that this can be understood in terms of Girsanov’s change of measure. This offers a mathematical interpretation of marginal structural models that has not been available before. We consider both a model of an observational study and a model of a hypothetical randomized trial. These models correspond to different martingale measures – the observational measure and the randomized trial measure – on some underlying space. We describe situations where the randomized trial measure is absolutely continuous with respect to the observational measure. The resulting continuous-time likelihood ratio process with respect to these two probability measures corresponds to the weights in discrete-time marginal structural models. In order to do inference for the hypothetical randomized trial, we can simulate samples using observational data weighted by this likelihood ratio.

Invariance principles for linear processes with application to isotonic regression

Abstract: In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range dependence and the limiting distribution is a fractional Brownian motion. The proofs are based on new approximations by a linear process with martingale difference innovations. The results are then applied to study an estimator of the isotonic regression when the error process is a (possibly long-range dependent) time series.

Self-similar scaling limits of non-increasing Markov chains

Abstract: We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from $n$ and appropriately rescaled, converges in distribution, as $n → ∞$, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in $Λ$-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in our paper, Scaling limits of Markov branching trees, with applications to Galton–Watson and random unordered trees (2010).

On the functional central limit theorem via martingale approximation

Abstract: In this paper, we develop necessary and sufficient conditions for the validity of a martingale approximation for the partial sums of a stationary process in terms of the maximum of consecutive errors. Such an approximation is useful for transferring the conditional functional central limit theorem from the martingale to the original process. The condition found is simple and well adapted to a variety of examples, leading to a better understanding of the structure of several stochastic processes and their asymptotic behaviors. The approximation brings together many disparate examples in probability theory. It is valid for classes of variables defined by familiar projection conditions such as the Maxwell–Woodroofe condition, various classes of mixing processes, including the large class of strongly mixing processes, and for additive functionals of Markov chains with normal or symmetric Markov operators.

Renorming divergent perpetuities

Abstract: We consider a sequence of random variables ($R_n$) defined by the recurrence $R_n = Q_n + M_nR_{n−1}, n ≥ 1$, where $R_0$ is arbitrary and ($Q_n, M_n), n ≥ 1$, are i.i.d. copies of a two-dimensional random vector ($Q, M$), and ($Q_n, M_n$) is independent of $R_{n−1}$. It is well known that if $E \ln|M| < 0$ and $E \ln^+|Q| < ∞$, then the sequence ($R_n$) converges in distribution to a random variable $R$ given by $R\stackrel d=\sum_{k=1}^\infty Q_k\prod_{j=1}^{k-1}M_j$, and usually referred to as perpetuity. In this paper we consider a situation in which the sequence ($R_n$) itself does not converge. We assume that $E \ln|M|$ exists but that it is non-negative and we ask if in this situation the sequence ($R_n$), after suitable normalization, converges in distribution to a non-degenerate limit.

Approximations of fractional Brownian motion

Abstract: Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the one-parameter fractional Brownian motion is constructed using a two-parameter Poisson process. The proof involves the tightness and identification of finite-dimensional distributions.