Showing papers on "Central limit theorem published in 2007"
TL;DR: In this paper, the authors proposed a method to address the problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem.
Abstract: This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem (Bassett and Koenker (1982)). The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural distribution and quantile functions using data on Vietnam veteran status and earnings.
487 citations
TL;DR: The proposed distribution turns out to be a very convenient tool for modelling cascaded Nakagami-m fading channels and analyzing the performance of digital communications systems operating over such channels.
Abstract: A generic and novel distribution, referred to as Nakagami, constructed as the product of N statistically independent, but not necessarily identically distributed, Nakagami-m random variables (RVs), is introduced and analyzed. The proposed distribution turns out to be a very convenient tool for modelling cascaded Nakagami-m fading channels and analyzing the performance of digital communications systems operating over such channels. The moments-generating, probability density, cumulative distribution, and moments functions of the N *Nakagami distribution are developed in closed form using the Meijer's G -function. Using these formulas, generic closed-form expressions for the outage probability, amount of fading, and average error probabilities for several binary and multilevel modulation signals of digital communication systems operating over the N *Nakagami fading and the additive white Gaussian noise channel are presented. Complementary numerical and computer simulation performance evaluation results verify the correctness of the proposed formulation. The suitability of the N *Nakagami fading distribution to approximate the lognormal distribution is also being investigated. Using Kolmogorov--Smirnov tests, the rate of convergence of the central limit theorem as pertaining to the multiplication of Nakagami-m RVs is quantified.
329 citations
Posted Content•
TL;DR: In this paper, a new notion of G-normal distributions is introduced for sublinear expectation theory, which is very similar to the situation of linear expectation in the classical probability theory.
Abstract: We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Ito's type (Ito's integral, Ito's formula, Ito's equation) through the corresponding G-Brownian motion. We will also present analytical calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty.
Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y.
Two important theorems have been proved: The law of large number and the central limit theorem.
200 citations
TL;DR: In this paper, a new family of Fisher information and entropy power inequalities for sums of independent random variables are presented, which relate the information in the sum of n independent variables to the information contained in sums over subsets of the random variables, for an arbitrary collection of subsets.
Abstract: New families of Fisher information and entropy power inequalities for sums of independent random variables are presented. These inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables, for an arbitrary collection of subsets. As a consequence, a simple proof of the monotonicity of information in central limit theorems is obtained, both in the setting of independent and identically distributed (i.i.d.) summands as well as in the more general setting of independent summands with variance-standardized sums.
177 citations
Book•
06 Sep 2007
TL;DR: Random Systems with Covariance inequalities Moment and Maximal Inequalities Central Limit Theorem Almost sure Convergence Invariance Principles Law of the Iterated Logarithm Statistical Applications Integral Functionals as discussed by the authors.
Abstract: Random Systems with Covariance Inequalities Moment and Maximal Inequalities Central Limit Theorem Almost Sure Convergence Invariance Principles Law of the Iterated Logarithm Statistical Applications Integral Functionals.
167 citations
Posted Content•
TL;DR: In this article, a unified technique for deriving gaussian central limit theorems via relatively soft arguments is presented. But the proofs of such results are usually rather difficult, involving hard computations specific to the model in question.
Abstract: Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of `second order Poincar\'e inequalities': just as ordinary Poincar\'e inequalities give variance bounds, second order Poincar\'e inequalities give central limit theorems. The proof of the main result employs Stein's method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.
145 citations
Posted Content•
TL;DR: In this paper, the authors proposed a particle filter scheme for a class of partially-observed multivariate diffusions, which does not require approximations of the transition and/or the observation density using timediscretisations.
Abstract: In this paper we introduce a novel particle filter scheme for a class of partially-observed multivariate diffusions. %continuous-time dynamic models where the %signal is given by a multivariate diffusion process. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Unlike currently available methods, our particle filters do not require approximations of the transition and/or the observation density using time-discretisations. Instead, they build on recent methodology for the exact simulation of the diffusion process and the unbiased estimation of the transition density as described in \cite{besk:papa:robe:fear:2006}. %In particular, w We introduce the Generalised Poisson Estimator, which generalises the Poisson Estimator of \cite{besk:papa:robe:fear:2006}. %Thus, our filters avoid the systematic biases caused by %time-discretisations and they have significant computational %advantages over alternative continuous-time filters. These %advantages are supported theoretically by a A central limit theorem is given for our particle filter scheme.
120 citations
TL;DR: In this article, an extension of some limit theorems for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramer's condition was proved.
Abstract: Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramer's condition. The large deviation x-region under consideration is broader than in the classical Cramer's theorem, and the estimate of the remainder is uniform with respect to x. The corresponding asymptotic expansion with arbitrarily many summands is also obtained.
120 citations
TL;DR: A general formula for the distribution of time-averaged observables for systems modeled according to the subdiffusive continuous time random walk is found and a weakly nonergodic statistical mechanical framework is constructed, which is based on Lévy's generalized central limit theorem.
Abstract: We find a general formula for the distribution of time-averaged observables for systems modeled according to the subdiffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly nonergodic statistical mechanical framework is constructed, which is based on Levy's generalized central limit theorem. As an example we calculate the distribution of X, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that X exhibits large fluctuations compared with the ensemble average .
119 citations
TL;DR: In this paper, the authors introduce Wiener integrals with respect to the Hermite process and prove a non-central limit theorem in which the Wiener integral appears as limit.
Abstract: We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.
117 citations
TL;DR: In this article, a general method to study dependent data in a binary tree was proposed, where an individual in one generation gives rise to two different offspring, one of type 0 and another of type 1, in the next generation.
Abstract: We propose a general method to study dependent data in a binary tree, where an individual in one generation gives rise to two different offspring, one of type 0 and one of type 1, in the next generation. For any specific characteristic of these individuals, we assume that the characteristic is stochastic and depends on its ancestors’ only through the mother’s characteristic. The dependency structure may be described by a transition probability P(x, dy dz) which gives the probability that the pair of daughters’ characteristics is around (y, z), given that the mother’s characteristic is x. Note that y, the characteristic of the daughter of type 0, and z, that of the daughter of type 1, may be conditionally dependent given x, and their respective conditional distributions may differ. We then speak of bifurcating Markov chains. We derive laws of large numbers and central limit theorems for such stochastic processes. We then apply these results to detect cellular aging in Escherichia Coli, using the data of Stewart et al. and a bifurcating autoregressive model.
Book•
01 Jan 2007
TL;DR: Brownian Motion: The LIL and some Fine-Scale Properties as mentioned in this paper, Skorokhod Embedding and Donsker's Invariance Principle, and a Historical Note on Brownian Motion.
Abstract: Random Maps, Distribution, and Mathematical Expectation.- Independence, Conditional Expectation.- Martingales and Stopping Times.- Classical Zero-One Laws, Laws of Large Numbers and Deviations.- Weak Convergence of Probability Measures.- Fourier Series, Fourier Transform, and Characteristic Functions.- Classical Central Limit Theorems.- Laplace Transforms and Tauberian Theorem.- Random Series of Independent Summands.- Kolmogorov's Extension Theorem and Brownian Motion.- Brownian Motion: The LIL and Some Fine-Scale Properties.- Skorokhod Embedding and Donsker's Invariance Principle.- A Historical Note on Brownian Motion.
Posted Content•
TL;DR: In this article, it was shown that the limit distribution of the central limit theorem is a G-normal distribution, which is the same as that of the normal distribution in the classic probability theory.
Abstract: The law of large numbers (LLN) and central limit theorem (CLT) are long and widely been known as two fundamental results in probability theory.
Recently problems of model uncertainties in statistics, measures of risk and superhedging in finance motivated us to introduce, in [4] and [5] (see also [2], [3] and references herein), a new notion of sublinear expectation, called \textquotedblleft% $G$-expectation\textquotedblright, and the related \textquotedblleft$G$-normal distribution\textquotedblright from which we were able to define G-Brownian motion as well as the corresponding stochastic calculus. The notion of G-normal distribution plays the same important rule in the theory of sublinear expectation as that of normal distribution in the classic probability theory. It is then natural and interesting to ask if we have the corresponding LLN and CLT under a sublinear expectation and, in particular, if the corresponding limit distribution of the CLT is a G-normal distribution. This paper gives an affirmative answer. The proof of our CLT is short since we borrow a deep interior estimate of fully nonlinear PDE in [6] which extended a profound result of [1] (see also [7]) to parabolic PDEs. The assumptions of our LLN and CLT can be still improved. But the discovered phenomenon plays the same important rule in the theory of nonlinear expectation as that of the classical LLN and CLT in classic probability theory.
TL;DR: In this article, a central limit theorem for bounded test functions on independent random marked vectors with a common density is given, where the measure is defined as a measure determined by the (suitably rescaled) set of points near the vertices of vertices.
Abstract: Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $
u_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a central limit theorem for $
u_n(f)$, and deduce weak convergence of $
u_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and $k$-nearest neighbours graph.
Posted Content•
TL;DR: In this paper, the authors prove central and non-central limit theorems for renormalized weighted power variations of order q>=2 of the fractional Brownian motion with Hurst parameter H in (0, 1), where q is an integer.
Abstract: In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q>=2 of the fractional Brownian motion with Hurst parameter H in (0,1), where q is an integer. The central limit holds for 1/(2q) 1-1/(2q), we show the convergence in L^2 to a stochastic integral with respect to the Hermite process of order q.
TL;DR: The probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables, is investigated and numerical evidence that in this case the probability density converges to a q -Gaussian leads to a power-law generalization of the CLT.
Abstract: We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A central limit theorem (CLT) is valid only if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a q -Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e., relevant for large classes of critical dynamical systems.
TL;DR: In this paper, asymptotic properties of M-estimates of regression parameters in linear models in which errors are dependent are derived and weak and strong Bahadur representations are derived.
Abstract: We study asymptotic properties of M-estimates of regression parameters in linear models in which errors are dependent. Weak and strong Bahadur representations of the M-estimates are derived and a central limit theorem is established. The results are applied to linear models with errors being short-range dependent linear processes, heavy-tailed linear processes and some widely used nonlinear time series.
TL;DR: In this article, the authors review some applications of central limit theorems and extreme values statistics in the context of disordered systems and discuss several problems, in particular concerning random matrix theory and the generalization of the Tracy-Widom distribution when the disorder has "fat tails".
Abstract: We review some applications of central limit theorems and extreme values statistics in the context of disordered systems. We discuss several problems, in particular concerning random matrix theory and the generalization of the Tracy–Widom distribution when the disorder has 'fat tails'. We underline the relevance of power-law tails for directed polymers and mean-field spin glasses and we point out various open problems and conjectures on these matters. We find that, in many instances, the assumption of Gaussian disorder cannot be taken for granted.
TL;DR: In this article, the authors study Markov chain Monte Carlo algorithms for exploring the intractable posterior density that results when the probit regression likelihood is combined with a flat prior on β.
Abstract: Consider a probit regression problem in which Y 1 ,..., Y n are independent Bernoulli random variables such that Pr(Y i = 1) = Φ(x i T β) where x i is a p-dimensional vector of known covariates that are associated with Y i , β is a p-dimensional vector of unknown regression coefficients and Φ(·) denotes the standard normal distribution function. We study Markov chain Monte Carlo algorithms for exploring the intractable posterior density that results when the probit regression likelihood is combined with a flat prior on β. We prove that Albert and Chib's data augmentation algorithm and Liu and Wu's PX-DA algorithm both converge at a geometric rate, which ensures the existence of central limit theorems for ergodic averages under a second-moment condition. Although these two algorithms are essentially equivalent in terms of computational complexity, results of Hobert and Marchev imply that the PX-DA algorithm is theoretically more efficient in the sense that the asymptotic variance in the central limit theorem under the PX-DA algorithm is no larger than that under Albert and Chib's algorithm. We also construct minorization conditions that allow us to exploit regenerative simulation techniques for the consistent estimation of asymptotic variances. As an illustration, we apply our results to van Dyk and Meng's lupus data. This example demonstrates that huge gains in efficiency are possible by using the PX-DA algorithm instead of Albert and Chib's algorithm.
TL;DR: In this paper, it was shown that the almost sure invariance principle (approximation by Brownian mo- tion) holds for geometric Lorenz attractors as well as the central limit theorem, the law of the iterated logarithm, and functional versions of these results.
Abstract: We prove statistical limit laws for Holder observations of the Lorenz at- tractor, and more generally for geometric Lorenz attractors. In particular, we prove the almost sure invariance principle (approximation by Brownian mo- tion). Standard consequences of this result include the central limit theorem, the law of the iterated logarithm, and the functional versions of these results.
TL;DR: In this paper, a local U-statistic process is introduced and central limit theorems in various norms are obtained for it, which involves the development of several inequalities for U-processes that may be useful in other contexts.
Abstract: A notion of local U-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for U-processes that may be useful in other contexts. This local U-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517-525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the Lp norms are obtained for these estimators.
TL;DR: A new stochastic fading channel model called cascaded Weibull fading is introduced and the associated capacity is derived in closed form and the statistical basis of the lognormal distribution is investigated.
Abstract: A new stochastic fading channel model called cascaded Weibull fading is introduced and the associated capacity is derived in closed form. This model is generated by the product of independent, but not necessarily identically distributed, Weibull random variables (RVs). By quantifying the convergence rate of the central limit theorem as pertaining to the multiplication of Weibull distributed RVs, the statistical basis of the lognormal distribution is investigated. By performing Kolmogorov–Smirnov tests, the null hypothesis for this product to be approximated by the lognormal distribution is studied. Another null hypothesis is also examined for this product to be approximated by a Weibull distribution with properly adjusted statistical parameters.
Journal Article•
TL;DR: In this paper, the authors consider the asymptotic behavior of a sequence (theta(n)), theta (n) = tau(n) o tau (n - 1)... o tAU(1), where (tau n))(n >= 1) are non-singular transformations on a probability space.
Abstract: We consider the asymptotic behaviour of a sequence (theta(n)), theta(n) = tau(n) o tau(n - 1) . . . o tau(1), where (tau(n))(n >= 1) are non-singular transformations on a probability space. After briefly discussing some definitions and problems in this general framework, we consider the case of piecewise expanding transformations of the interval. Exactness and statistical properties (a central limit theorem for BV functions after a moving centering) can be shown for some families of such transformations. The method relies on an extension of the spectral theory of transfer operators to the case of a sequence of transfer operators.
TL;DR: In this article, the asymptotic behavior of realized power variations, and more generally of sums of a given function evaluated at the increments of a Levy process between the successive times i Δn for i = 0, 1,...,n.
Abstract: We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Levy process between the successive times i Δn for i = 0,1,...,n . One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/or centering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function f . As for the associated central limit theorem, one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.
TL;DR: In this article, a notion of local $U$-statistic process is introduced and central limit theorems in various norms are obtained for it, which involves the development of several inequalities for the process that may be useful in other contexts.
Abstract: A notion of local $U$-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for $U$-processes that may be useful in other contexts. This local $U$-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517--525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the $L_p$ norms are obtained for these estimators.
TL;DR: In this paper, the consequences of fitting long autoregressions under regularity conditions that allow for these two situations and where an infinite autoregressive representation of the process need not exist are investigated.
Abstract: Autoregressive models are commonly employed to analyze empirical time series. In practice, however, any autoregressive model will only be an approximation to reality and in order to achieve a reasonable approximation and allow for full generality the order of the autoregression, h say, must be allowed to go to infinity with T, the sample size. Although results are available on the estimation of autoregressive models when h increases indefinitely with T such results are usually predicated on assumptions that exclude (1) non-invertible processes and (2) fractionally integrated processes. In this paper we will investigate the consequences of fitting long autoregressions under regularity conditions that allow for these two situations and where an infinite autoregressive representation of the process need not exist. Uniform convergence rates for the sample autocovariances are derived and corresponding convergence rates for the estimates of AR(h) approximations are established. A central limit theorem for the coefficient estimates is also obtained. An extension of a result on the predictive optimality of AIC to fractional and non-invertible processes is obtained.
TL;DR: In this paper, the central limit theorem and its weak invariance principle for sums of nonadapted stationary sequences, under different normalizations, have been studied for linear processes with dependent innovations and regular functions of linear processes.
Abstract: In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.
TL;DR: In this paper, a nonparametric maximum likelihood estimator for Gaussian locally stationary processes is constructed by minimizing a frequency domain likelihood over a class of functions and the asymptotic behavior of the resulting estimator is studied.
Abstract: This paper deals with nonparametric maximum likelihood estimation for Gaussian locally stationary processes. Our nonparametric MLE is constructed by minimizing a frequency domain likelihood over a class of functions. The asymptotic behavior of the resulting estimator is studied. The results depend on the richness of the class of functions. Both sieve estimation and global estimation are considered. Our results apply, in particular, to estimation under shape constraints. As an example, autoregressive model fitting with a monotonic variance function is discussed in detail, including algorithmic considerations. A key technical tool is the time-varying empirical spectral process indexed by functions. For this process, a Bernstein-type exponential inequality and a central limit theorem are derived. These results for empirical spectral processes are of independent interest.
TL;DR: In this paper, the authors examined the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi-Bayesian estimation carried out using Metropolis random walks and showed that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension $d, and, in particular, is of stochastic order $d^2$ in the leading cases after the burn-in period.
Abstract: In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi-Bayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the Laplace-Bernstein-Von Mises central limit theorem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using the conditions required for the central limit theorem to hold, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases where the underlying log-likelihood or extremum criterion function is possibly non-concave, discontinuous, and with increasing parameter dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner.
Under minimal assumptions required for the central limit theorem to hold under the increasing parameter dimension, we show that the Metropolis algorithm is theoretically efficient even for the canonical Gaussian walk which is studied in detail. Specifically, we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension $d$, and, in particular, is of stochastic order $d^2$ in the leading cases after the burn-in period. We then give applications to exponential families, curved exponential families, and Z-estimation of increasing dimension.
TL;DR: Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.
Abstract: We provide a quantitative analysis of the phenomenon of crowding of near-extreme events by computing exactly the density of states (DOS) near the maximum of a set of independent and identically distributed random variables. We show that the mean DOS converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential, faster than pure exponential, or as a pure exponential function. We argue that some of these results would remain valid even for certain correlated cases and verify it for power-law correlated stationary Gaussian sequences. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.