The parameters of stable law parameters can be estimated using a regression based approach involving the empirical characteristic function. One approach is to use a fixed number of points for all ...parameters of the distribution to estimate the characteristic function. In this work the results are derived where all points in an interval is used to estimate the empirical characteristic function, thus least squares estimators of a linear function of the parameters, using an infinite number of observations. It was found that the procedure performs very good in small samples.
The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable ...distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.
We consider an approximation of a cumulative distribution function F(x) by the cumulative distributionfunction G(x) of the Irwin law. In this case, a function F(x) can be cumulative distribution ...functions of sums (products) ofindependent (dependent) random variables. Remainder term of the approximation is estimated by the cumulant method.The cumulant method is used by introducing special cumulants, satisfying the V. Statulevičius type condition. The mainresult is a nonuniform bound for the difference |F(x)-G(x)| in terms of special cumulants of the symmetric cumulativedistribution function F(x).
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.
The simulation of a discrete sample path of a Lévy process reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with exponential moments, the inverse transform ...method proposed in Glasserman and Liu 2010 is reliable and efficient. The values of the cumulative distribution function (cdf) are computed by inverting the characteristic function and tabulated on a uniform grid. The inverse of the cumulative distribution function is obtained by linear interpolation. In this article, we apply a Hilbert transform method for the characteristic function inversion. The Hilbert transform representation for the cdf can be discretized using a simple rule highly accurately. Most importantly, the error estimates admit explicit and computable expressions, which allow us to compute the cdf to any desired accuracy. We present an explicit bound for the estimation bias in terms of the range of the grid where probabilities are tabulated, the step size of the grid, and the approximation error for the probabilities. The bound can be computed from the characteristic function directly and allows one to determine the size and fineness of the grid and numerical parameters for evaluating the Hilbert transforms for any given bias tolerance level in one-dimensional problems. For multidimensional problems, we present a procedure for selecting the grid and the numerical parameters that is shown to converge theoretically and works well practically. The inverse transform method is attractive not only for Lévy processes that are otherwise not easy to simulate, but also for processes with special structures that could be simulated in different ways. The method is very fast and accurate when combined with quasi-Monte Carlo schemes and variance reduction techniques. The main results we derived are not limited to Lévy processes and can be applied to simulating from tabulated cumulative distribution functions in general and characteristic functions in our analytic class in particular.
This article studies a new procedure to test for the equality of k regression curves in a fully non-parametric context. The test is based on the comparison of empirical estimators of the ...characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi-squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n–1/2. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.
In this paper, we propose an image splicing detecting method using the characteristic function moments for the inter-scale co-occurrence matrix in the wavelet domain. We construct the co-occurrence ...matrices by using a pair of wavelet difference values across inter-scale wavelet subbands. In this process, we do not adopt the thresholding operation to prevent information loss. We extract the high-order characteristic function moments of the two-dimensional joint density function generated by the inter-scale co-concurrent matrices in order to detect image splicing forgery. Our method can be applied regardless of the color or gray image dataset using only luminance component of an image. By performing experimental simulations, we demonstrate that the proposed method achieves good performance in splicing detection. Our results show that the detection accuracy was greater than 95 % on average with well-known four splicing detection image datasets.
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