We consider estimation for a class of Levy processes, modelled as a sum of a drift, a symmetric stable process and a compound Poisson process. We propose a nonparametric approach to estimating unknown parameters of ou...
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We consider estimation for a class of Levy processes, modelled as a sum of a drift, a symmetric stable process and a compound Poisson process. We propose a nonparametric approach to estimating unknown parameters of our model, including the drift, the scale and index parameters in the stable law, the mean of the Poisson process and the underlying jump size distribution. We show that regression and nonparametric deconvolution methods, based on the empirical characteristic function, can be used for inference. Interesting connections are shown to exist between properties of our estimators and of those found in conventional deconvolution. (C) 2010 Elsevier B.V. All rights reserved.
This paper deals with uniform rates of convergence for the empirical distribution function and the empirical characteristic function for a class of stationary mixing processes. We obtain that ‖Fn -F‖=o(n-1/2(1nn)1/2...
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This paper deals with uniform rates of convergence for the empirical distribution function and the empirical characteristic function for a class of stationary mixing processes. We obtain that ‖Fn -F‖=o(n-1/2(1nn)1/2mn) a. s. for the -mixing process, where {mn} is a positiv sequence satisfying nrlimmr=∞, lim Σ (i)/m2r=lim ( ) 1/2=0, 1≤nr≤T, and T (nr)/nr≤A≤∞, and lim‖Fn -F‖ / (cn ln2n)=0 a. s. for the geometrically a-mixing process, where {cn} is an arbitrary positive sequence satisfying limcn=∞. For the empirical characteristic function a uniform rate of Convergence is also given.
Tests are proposed for the assumption that the conditional distribution of a multivariate GARCH process is elliptic. These tests are of Kolmogorov-Smirnov and Cramer-von Mises-type and make use of the common geometry ...
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Tests are proposed for the assumption that the conditional distribution of a multivariate GARCH process is elliptic. These tests are of Kolmogorov-Smirnov and Cramer-von Mises-type and make use of the common geometry underlying the characteristicfunction of any spherically symmetric distribution. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. It is shown that both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of the innovations. Therefore a conditional Monte Carlo procedure is used to actually carry out the tests. The validity of this resampling scheme is formally justified. Results on the behavior of the new tests in finite-samples are included along with comparisons with other tests. (C) 2016 Elsevier B.V. All rights reserved.
Stable distributions are characterized by four parameters which can be estimated via a number of methods, and although approximate maximum likelihood estimation techniques have been proposed, they are computationally ...
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Stable distributions are characterized by four parameters which can be estimated via a number of methods, and although approximate maximum likelihood estimation techniques have been proposed, they are computationally intensive and difficult to implement. This article describes a fast, wavelet-based, regression-type method for estimating the parameters of a stable distribution. Fourier domain representations, combined with a wavelet multiresolution approach, are shown to be effective and highly efficient tools for inference in stable law families. Our procedures are illustrated and compared with other estimation methods using simulated data, and an application to a real data example is explored. One novel aspect of this work is that here wavelets are being used to solve a parametric problem, rather than a nonparametric one, which is the more typical context in wavelet applications.
Consistent procedures are constructed for testing the goodness-of-fit of the error distribution in nonparametric regression models. The test starts with a kernel-type regression fit and proceeds with the construction ...
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Consistent procedures are constructed for testing the goodness-of-fit of the error distribution in nonparametric regression models. The test starts with a kernel-type regression fit and proceeds with the construction of a test statistic in the form of an L (2) distance between a parametric and a nonparametric estimates of the residual characteristicfunction. The asymptotic null distribution and the behavior of the test statistic under alternatives are investigated. A simulation study compares bootstrap versions of the proposed test to corresponding procedures utilizing the empirical distribution function.
Parameter estimation in linear errors-in-variables models typically requires that the measurement error distribution be known or estimable from replicate data. A generalized method of moments approach can be used to e...
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Parameter estimation in linear errors-in-variables models typically requires that the measurement error distribution be known or estimable from replicate data. A generalized method of moments approach can be used to estimate model parameters in the absence of knowledge of the error distributions, but it requires the existence of a large number of model moments. In this paper, parameter estimation based on the phase function, a normalized version of the characteristicfunction, is considered. This approach requires the model covariates to have asymmetric distributions, while the error distributions are symmetric. Parameters are estimated by minimizing a distance function between the empirical phase functions of the noisy covariates and the outcome variable. No knowledge of the measurement error distribution is needed to calculate this estimator. Both asymptotic and finite-sample properties of the estimator are studied. The connection between the phase function approach and method of moments is also discussed. The estimation of standard errors is considered and a modified bootstrap algorithm for fast computation is proposed. The newly proposed estimator is competitive with the generalized method of moments, despite making fewer model assumptions about the moment structure of the measurement error. Finally, the proposed method is applied to a real dataset containing measurements of air pollution levels.
We introduce the characteristic symmetry function, based on the characteristicfunction of the underlying distribution, whose behaviour is indicative of symmetry or its absence. A statistic is proposed for testing sym...
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We introduce the characteristic symmetry function, based on the characteristicfunction of the underlying distribution, whose behaviour is indicative of symmetry or its absence. A statistic is proposed for testing symmetry about an unspecified centre, derived from the empiricalcharacteristic symmetry function. The statistic is readily computible, it utilizes information in the empirical characteristic function over an interval, and does not require the estimation of the centre of symmetry. Under general symmetry the asymptotic null distribution of the statistic is folded normal. The empirical power for selected alternatives is studied by a small-scale simulation and a numerical illustration is given.
A test for the equality of error distributions in two nonparametric regression models is proposed. The test statistic is based on comparing the empirical characteristic functions of the residuals calculated from indep...
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A test for the equality of error distributions in two nonparametric regression models is proposed. The test statistic is based on comparing the empirical characteristic functions of the residuals calculated from independent samples of the models. The asymptotic null distribution of the test statistic cannot be used to estimate its null distribution because it is unknown, since it depends on the unknown common error distribution. To approximate the null distribution, a weighted bootstrap estimator is studied, providing a consistent estimator. The finite sample performance of this approximation as well as the power of the resulting test are evaluated by means of a simulation study. The procedure can be extended to testing for the equality of d>2 error distributions.
We consider goodness-of-fit methods for multivariate symmetric and asymmetric stable Paretian random vectors in arbitrary dimension. The methods are based on the empirical characteristic function and are implemented b...
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We consider goodness-of-fit methods for multivariate symmetric and asymmetric stable Paretian random vectors in arbitrary dimension. The methods are based on the empirical characteristic function and are implemented both in the i.i.d. context as well as for innovations in GARCH models. Asymptotic properties of the proposed procedures are discussed, while the finite-sample properties are illustrated by means of an extensive Monte Carlo study. The procedures are also applied to real data from the financial markets.
In this paper we construct estimators of certain nonlinear functionals of the mth derivative of a probability density function, based on the empirical characteristic function. Using empirical processes techniques we g...
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In this paper we construct estimators of certain nonlinear functionals of the mth derivative of a probability density function, based on the empirical characteristic function. Using empirical processes techniques we give a CLT for these estimators. (C) 1999 Elsevier Science B.V. All rights reserved.
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