Theoretical Characteristic Functions Research Articles

Continuous-Time Random Walk for a Particle in a Periodic Potential

Continuous-time random walks offer powerful coarse-grained descriptions of transport processes. We here microscopically derive such a model for a Brownian particle diffusing in a deep periodic potential. We determine both the waiting-time and the jump-length distributions in terms of the parameters of the system, from which we analytically deduce the non-Gaussian characteristic function. We apply this continuous-time random walk model to characterize the underdamped diffusion of single cesium atoms in a one-dimensional optical lattice. We observe excellent agreement between experimental and theoretical characteristic functions, without any free parameter.

On Properties of the MixedTS Distribution and Its Multivariate Extension

A review of the univariate MixedTS is given and some new results on the asymptotic tail behaviour are derived. The multivariate version of the Mixed Tempered Stable, which is a generalisation of the Normal Variance Mean Mixtures, is discussed. Characteristics of this distribution, its capacity in fitting tails and in capturing dependence structure between components are investigated. We discuss a random number generating procedure and introduce an estimation methodology based on the minimisation of a distance between empirical and theoretical characteristic functions. Asymptotic tail behaviour of the univariate Mixed Tempered Stable is exploited in the estimation procedure in order to obtain a better fitting on tails. Advantages of the multivariate Mixed Tempered Stable distribution are discussed and illustrated via numerical analysis.

Inference for the Generalized Normal Laplace Distribution

The generalized normal Laplace distribution has been used in financial modeling because of its skewness and excess kurtosis. To estimate its parameters, we use a method based on minimizing the quadratic distance between the real and imaginary parts of the empirical and theoretical characteristic functions. The quadratic distance estimator (QDE) obtained is shown to be robust, consistent, and with an asymptotically normal distribution. The goodness-of-fit test statistics presented follow an asymptotic chi-square distribution. The performance of the QDE is illustrated by simulation results and an application to financial data.

Bayesian Inference in Multivariate Stable Distributions

In this paper we take up Bayesian inference in general, multivariate stable distributions. We use approximate Bayesian computation (ABC) along with carefully crafted proposal distributions for the implementation of MCMC. The problem of selecting summary statistics in ABC is resolved through the use of the characteristic function. Two important problems in multivariate stable distributions are: (i) the selection of an optimal con guration of the grid points where the empirical and theoretical characteristic functions are compared, and (ii) the estimation of the spectral measure through which the distributions are de ned in Rd. The problems are resolved successfully and certain new approximations to the spectral measure are proposed and implemented. E fficient proposal/importance distributions are constructed, and tested thoroughly, to ensure good performance in connection with ABC. The new techniques are applied to exchange rate and stock return data and are supplemented by Monte Carlo simulations. In addition, critical values are provided for a closeness statistic between the empirical and theoretical characteristic functions, resolving successfully another major problem in ABC-related inference.

Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variates

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.

Estimation for multivariate stable distributions with generalized empirical likelihood

This paper considers the generalized empirical likelihood (GEL) method for estimating the parameters of the multivariate stable distribution. The GEL method is considered to be an extension of the generalized method of moments (GMM). The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. We treat the spectral measure, which summarizes scale and asymmetry, by discretization. In order to estimate all the model parameters simultaneously, we apply the estimating function constructed by equating empirical and theoretical characteristic functions. The efficacy of the proposed GEL method is demonstrated in Monte Carlo studies. An illustrative example involving daily returns of market indexes is also included.

Empirical Likelihood Estimation for a Class of Stable Processes

We investigate the empirical likelihood estimation for a parameter of linear processes whose innovations have i.i.d. symmetric α-stable distributions. To construct the estimating function for the empirical likelihood method, we make use of the empirical and theoretical characteristic functions. The asymptotic normality of the maximum empirical likelihood estimator is derived. The behavior of the asymptotic variance with respect to infinitesimal perturbations of the dependence effect is studied and we find that it is dependence robust when α > 1. Numerical studies are also given.

A Novel Fiber-Optic Slide-Sensing Scheme

A novel fiber-optic slide sensor is presented in this paper. The slide sensor is designed for robust hands or intelligent mechanical clamping apparatuses. In the slide sensor designing, a slide ball has been used as a slide transfer device of the sensor. In the center of the inside-sliding ball, a plane reflective mirror has been fixed with the ball to determine the two-dimensional rotation angles, and the angles were measured by a five-fiber-optic probe. The theoretical characteristic functions of the slide sensor are deduced and the experimental results are confirmed theoretical results.

The Estimation of the Parameters of the Stable Laws

Estimation of the parameters of the stable laws is considered for samples of size at least 50. The modulus of the difference between the empirical and theoretical characteristic functions is weighted and integrated over the real line; this function of the sample values and the parameters is then minimized with respect to the parameters via a gradient search routine nested within a sequential search. Extensive simulation experiments validate the effectiveness of the estimation procedure over the entire parameter space. Application of the procedure to stock market price behaviour is made.

The estimation of the parameters of the stable laws

SUMMARY Estimation of the parameters of the stable laws is considered for samples of size at least 50. The modulus of the difference between the empirical and theoretical characteristic functions is weighted and integrated over the real line; this function of the sample values and the parameters is then minimized with respect to the parameters via a gradient search routine nested within a sequential search. Extensive simulation experiments validate the effectiveness of the estimation procedure over the entire parameter space. Application of the procedure to stock market price behaviour is made.