Unitarity of Generalized Fourier–Gauss Transforms

Abstract

A generalized Fourier–Gauss transform is an operator acting in a Boson Fock space and is formulated as a continuous linear operator acting on the space of test white noise functions. It does not admit, in general, a unitary extension with respect to the norm of the Boson Fock space induced from the Gaussian measure with variance 1 but is extended to a unitary isomorphism if the Gaussian measure is replaced with the ones with different covariance operators. As an application, unitarity of a generalized dilation is discussed.     

On localization of source by hidden Gaussian processes with small noise

We consider the problem of identification of the position of some source by observations of K detectors receiving signals from this source. The time of arriving of the signal to the k-th detector depends of the distance between this detector and the source. The signals are observed in the presence of small Gaussian noise. The properties of the MLE and Bayesian estimators are studied in the asymptotic of small noise.

     

A Non-Linear Filter

Abstract

Using a measure change related to Bayes' rule recursive estimates are obtained for an approximate conditional density where a state process has deterministic dynamics and it is observed in additive Gaussian noise.     

On the zeros of the spectrogram of white noise

In a recent paper, Flandrin [16] proposed filtering based on the zeros of a spectrogram with Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time–frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram of continuous white Gaussian noise, we explicitly characterize their statistical distribution, and we investigate the computational and statistical underpinnings of the practical implementation of signal detection based on the statistics of the zeros of the spectrogram. The crux of our analysis is that the zeros of the spectrogram of white Gaussian noise correspond to the zeros of a Gaussian analytic function, a topic of recent independent mathematical interest [24].     

Testing the Cointegrating Rank with Uncorrelated but Dependent Errors

Abstract

We study the asymptotic behavior of the reduced rank estimator of the cointegrating space and adjustment space for vector error correction time series models with nonindependent innovations. It is shown that the distribution of the adjustment space can be quite different for models with iid innovations and models with nonindependent innovations. It is also shown that the likelihood ratio test remains valid when the assumption of iid Gaussian errors is relaxed. Monte Carlo experiments illustrate the finite sample performance of the likelihood ratio test using various kinds of weak error processes.     

Subspace clustering for the finite mixture of generalized hyperbolic distributions

The finite mixture of generalized hyperbolic distributions is a flexible model for clustering, but its large number of parameters for estimation, especially in high dimensions, can make it computationally expensive to work with. In light of this issue, we provide an extension of the subspace clustering technique developed for finite Gaussian mixtures to that of generalized hyperbolic distribution. The methodology will be demonstrated with numerical experiments.

     

Asymptotic Normality of the Minimum Non-Hilbertian Distance Estimators for a Diffusion Process with Small Noise

A one-dimensional diffusion type process with small noise is observed up to the time T. It depends on an unknown real parameter. Some minimum distance estimators of this parameter are considered. These estimators are defined using the L ^p-metric or the uniform metric. The limiting distribution of the normalizing minimum distance estimators (as the noise vanishing) is known to be the distribution of a random variable. The distribution of this random variable is studied as the time T goes to the infinity. We will prove under some conditions that it has a limiting Gaussian law. This revised version was published online in June 2006 with corrections to the Cover Date.     

A simulation study to compare robust clustering methods based on mixtures

The following mixture model-based clustering methods are compared in a simulation study with one-dimensional data, fixed number of clusters and a focus on outliers and uniform “noise”: an ML-estimator (MLE) for Gaussian mixtures, an MLE for a mixture of Gaussians and a uniform distribution (interpreted as “noise component” to catch outliers), an MLE for a mixture of Gaussian distributions where a uniform distribution over the range of the data is fixed (Fraley and Raftery in Comput J 41:578–588, 1998), a pseudo-MLE for a Gaussian mixture with improper fixed constant over the real line to catch “noise” (RIMLE; Hennig in Ann Stat 32(4): 1313–1340, 2004), and MLEs for mixtures of t-distributions with and without estimation of the degrees of freedom (McLachlan and Peel in Stat Comput 10(4):339–348, 2000). The RIMLE (using a method to choose the fixed constant first proposed in Coretto, The noise component in model-based clustering. Ph.D thesis, Department of Statistical Science, University College London, 2008) is the best method in some, and acceptable in all, simulation setups, and can therefore be recommended.     

Asymptotic properties of some goodness-of-fit tests based on the L 1-norm

Some goodness-of-fit tests based on the L ₁-norm are considered. The asymptotic distribution of each statistic under the null hypothesis is the distribution of the L ₁-norm of the standard Wiener process on [0,1]. The distribution function, the density function and a table of some percentage points of the distribution are given. A result for the asymptotic tail probability of the L ₁-norm of a Gaussian process is also obtained. The result is useful for giving the approximate Bahadur efficiency of the test statistics whose asymptotic distributions are represented as the L ₁-norms of Gaussian processes.     

Asymptotic properties of the MLE for the autoregressive process coefficients under stationary Gaussian noise

In this paper we study the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order p with regular stationary Gaussian noise. We prove the large sample asymptotic properties of the MLE under very mild conditions. We do simulations for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).     

The spectrum of the random environment and localization of noise

We consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized. The graphs of the noise covariance structure for d = 4, 3, 2.1 from above.     

Central limit theorem for constrained Poisson systems

We prove that for a class of constrained Poisson white noise fields, the scaling (continuum) limit exists and equals Gaussian white noise, indexed by mean zero test functions. Under natural conditions on the Lévy measure, the (Poisson) moments converge to their Gaussian counterparts.     

Score tests for inverse Gaussian mixtures

The mixed inverse Gaussian given by Whitmore (biScand. J. Statist., 13 , 1986, 211–220) provides a convenient way for testing the goodness‐of‐fit of a pure inverse Gaussian distribution. The test is a one‐sided score test with the null hypothesis being the pure inverse Gaussian (i.e. the mixing parameter is zero) and the alternative a mixture. We devise a simple score test and study its finite sample properties. Monte Carlo results show that it compares favourably with the smooth test of Ducharme ( Test , 10 , 2001, 271‐290). In practical applications, when the pure inverse Gaussian distribution is rejected, one is interested in making inference about the general values of the mixing parameter. However, as it is well known that the inverse Gaussian mixture is a defective distribution; hence, the standard likelihood inference cannot be applied. We propose several alternatives and provide score tests for the mixing parameter. Finite sample properties of these tests are examined by Monte Carlo simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Fractional Brownian motion approximation based on fractional integration of a white noise

We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We consider correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similarity properties of the approximation to fractional Brownian motion, namely, `τ^H laws' for the structure function and the range. We conclude that the models proposed serve as a convenient tool for modelling of natural processes and testing and improvement of methods aimed at analysis and interpretation of experimental data.     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

A test for independence of two multivariate samples

The paper presents a new nonparametric test for independence of two vectors. The idea is based on zonotope approach by G. Koshevoy, H. Oja and others, see [4, 5]. Under the independence hypothesis the test statistic converges in distribution to the supremum of a certain Gaussian field, and its asymptotic distribution is found using the theory of extrema of random Gaussian fields developed by V. Piterbarg and Yu. Tyurin, see [6, 8]. In contrast to traditional correlation coefficients the formula is not symmetric.      

Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform

This paper considers two flexible classes of omnibus goodness-of-fit tests for the inverse Gaussian distribution. The test statistics are weighted integrals over the squared modulus of some measure of deviation of the empirical distribution of given data from the family of inverse Gaussian laws, expressed by means of the empirical Laplace transform. Both classes of statistics are connected to the first nonzero component of Neyman's smooth test for the inverse Gaussian distribution. The tests, when implemented via the parametric bootstrap, maintain a nominal level of significance very closely. A large-scale simulation study shows that the new tests compare favorably with classical goodness-of-fit tests for the inverse Gaussian distribution, based on the empirical distribution function.     

Variable selection via quantile regression with the process of Ornstein-Uhlenbeck type

Based on the data-cutoff method,we study quantile regression in linear models,where the noise process is of Ornstein-Uhlenbeck type with possible jumps.In single-level quantile regression,we allow the noise process to be heteroscedastic,while in composite quantile regression,we require that the noise process be homoscedastic so that the slopes are invariant across quantiles.Similar to the independent noise case,the proposed quantile estimators are root-n consistent and asymptotic normal.Furthermore,the adaptive least absolute shrinkage and selection operator(LASSO)is applied for the purpose of variable selection.As a result,the quantile estimators are consistent in variable selection,and the nonzero coefficient estimators enjoy the same asymptotic distribution as their counterparts under the true model.Extensive numerical simulations are conducted to evaluate the performance of the proposed approaches and foreign exchange rate data are analyzed for the illustration purpose.     

On a Characterization Theorem for Locally Compact Abelian Groups Containing an Element of Order 2

According to the well-known Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. While coefficients of linear forms are topological automorphisms of a group.

     

Simulation and evaluation of the distribution of interest rate risk

We study methods to simulate term structures in order to measure interest rate risk more accurately. We use principal component analysis of term structure innovations to identify risk factors and we model their univariate distribution using GARCH-models with Student’s t-distributions in order to handle heteroscedasticity and fat tails. We find that the Student’s t-copula is most suitable to model co-dependence of these univariate risk factors. We aim to develop a model that provides low ex-ante risk measures, while having accurate representations of the ex-post realized risk. By utilizing a more accurate term structure estimation method, our proposed model is less sensitive to measurement noise compared to traditional models. We perform an out-of-sample test for the U.S. market between 2002 and 2017 by valuing a portfolio consisting of interest rate derivatives. We find that ex-ante Value at Risk measurements can be substantially reduced for all confidence levels above 95%, compared to the traditional models. We find that that the realized portfolio tail losses accurately conform to the ex-ante measurement for daily returns, while traditional methods overestimate, or in some cases even underestimate the risk ex-post. Due to noise inherent in the term structure measurements, we find that all models overestimate the risk for 10-day and quarterly returns, but that our proposed model provides the by far lowest Value at Risk measures.