共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n discrete time observations with step Δ. The asymptotic framework is: n tends to infinity, Δ=Δn tends to zero while nΔn tends to infinity. First, we use a Fourier approach (“frequency domain”): this allows us to construct an adaptive nonparametric estimator and to provide a bound for the global L2-risk. Second, we use a direct approach (“time domain”) which allows us to construct an estimator on a given compact interval. We provide a bound for L2-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework. 相似文献
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In this article, we review the concept of a Lévy copula to describe the dependence structure of a bivariate compound Poisson process. In this first statistical approach we consider a parametric model for the Lévy copula and estimate the parameters of the full dependent model based on a maximum likelihood approach. This approach ensures that the estimated model remains in the class of multivariate compound Poisson processes. A simulation study investigates the small sample behaviour of the MLEs, where we also suggest a new simulation algorithm. Finally, we apply our method to Danish fire insurance data. 相似文献
4.
Reg Kulperger 《Journal of multivariate analysis》1979,9(1):101-115
A simple branching diffusion process is given as an elementary model of spatial evolution. A parametric estimation theory is presented for this model. As side results, a spatial central limit theorem and spatial strong law of large numbers are also obtained. 相似文献
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Tomasz J. Kozubowski Anna K. Panorska Krzysztof Podgrski 《Journal of multivariate analysis》2008,99(7):1418-1437
The joint distribution of X and N, where N has a geometric distribution and X is the sum of N IID exponential variables (independent of N), is infinitely divisible. This leads to a bivariate Lévy process {(X(t),N(t)),t≥0}, whose coordinates are correlated negative binomial and gamma processes. We derive basic properties of this process, including its covariance structure, representations, and stochastic self-similarity. We examine the joint distribution of (X(t),N(t)) at a fixed time t, along with the marginal and conditional distributions, joint integral transforms, moments, infinite divisibility, and stability with respect to random summation. We also discuss maximum likelihood estimation and simulation for this model. 相似文献
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In the estimation of parametric models for stationary spatial or spatio-temporal data on a d-dimensional lattice, for d?2, the achievement of asymptotic efficiency under Gaussianity, and asymptotic normality more generally, with standard convergence rate, faces two obstacles. One is the “edge effect”, which worsens with increasing d. The other is the possible difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, due mainly to the Jacobian term. This is especially a problem in “multilateral” models, which are naturally expressed in terms of lagged values in both directions for one or more of the d dimensions. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the computational problem, but when subjected to a standard device for avoiding the edge effect has disastrous asymptotic performance, along with finite sample numerical drawbacks, the objective function lacking a minimum-distance interpretation and losing any global convexity properties. We overcome these problems by first optimizing a standard, guaranteed non-negative, discrete-frequency, Whittle function, without edge-effect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almost-unbiased periodogram, the desired asymptotic properties being achieved after finitely many steps. The asymptotic regime allows increase in both directions of all d dimensions, with the central limit theorem established after re-ordering as a triangular array. However our work offers something new for “unilateral” models also. When the data are non-Gaussian, asymptotic variances of all parameter estimates may be affected, and we propose consistent, non-negative definite estimates of the asymptotic variance matrix. 相似文献
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We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations X0,XΔ,…,XnΔ, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞ and Δ→0) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided. 相似文献
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The asymptotic distribution of the quasi-maximum likelihood (QML) estimator is established for generalized autoregressive conditional heteroskedastic (GARCH) processes, when the true parameter may have zero coefficients. This asymptotic distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions. For an important subclass of models, no moment condition is imposed on the GARCH process. The main practical implication of these results concerns the estimation of overidentified GARCH models. 相似文献
11.
Christian Hering 《Journal of multivariate analysis》2010,101(6):1428-1433
A probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by group-specific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas. 相似文献
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Mathieu Rosenbaum Peter Tankov 《Stochastic Processes and their Applications》2011,121(7):1607-1632
We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions. 相似文献
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Kamal C. Chanda 《Annals of the Institute of Statistical Mathematics》1983,35(1):439-446
Summary LetX
t
, ...,X
n
be random variables forming a realization from a linear process
where {Z
t
} is a sequence of independent and identically distributed random variables with E|Z
t
|<∞ for some ε>0, andg
r
→0 asr→∞ at some specified rate. LetX
1 have a probability density functionf. It is then established that for every realx, the standard kernel type estimator
based onX
t
(1≦t≦n) is, under some general regularity conditions, asymptotically normal and converges a.s. tof(x) asn→∞.
Research was supported in part by the Air Force Office of Scientific Research Grant No. AFOSR-81-0058. 相似文献
14.
For a strictly stationary sequence of random vectors in Rd we study convergence of partial sum processes to a Lévy stable process in the Skorohod space with J1-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing. 相似文献
15.
Parametric models for tail copulas are being used for modeling tail dependence and maximum likelihood estimation is employed to estimate unknown parameters. However, two important questions seem unanswered in the literature: (1) What is the asymptotic distribution of the MLE and (2) how does one test the parametric model? In this paper, we answer these two questions in the case of a single parameter for ease of illustration. A simulation study is provided to investigate the finite sample performance of the proposed estimator and test. 相似文献
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We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained. 相似文献
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Let (Ut,Vt) be a bivariate Lévy process, where Vt is a subordinator and Ut is a Lévy process formed by randomly weighting each jump of Vt by an independent random variable Xt having cdf F. We investigate the asymptotic distribution of the self-normalized Lévy process Ut/Vt at 0 and at ∞. We show that all subsequential limits of this ratio at 0 (∞) are continuous for any nondegenerate F with finite expectation if and only if Vt belongs to the centered Feller class at 0 (∞). We also characterize when Ut/Vt has a non-degenerate limit distribution at 0 and ∞. 相似文献
18.
Michael G. Akritas Richard A. Johnson 《Annals of the Institute of Statistical Mathematics》1982,34(1):579-589
Summary We considerpth order autoregressive time series where the shocks need not be normal. By employing the concept of contiguity, we obtain
the sysmptotic power for tests of hypothesis concerning the autoregressive parameters. Our approach allows consideration of
the double exponential and other thicker-tailed distributions for the shocks. We derive a new result in the contiguity framework
that leads directly to an expression for the Pitman efficiencies of tests as well as estimators.
The numerical values of the efficiencies suggest a lack of robustness for the normal theory least squares estimators when
the shock distribution is thick tailed or an outlier prone mixed normal. An important alternative test statistic is proposed
that competes with the normal theory tests.
This research was supported by the Office of Naval Research under Grant No. N00014-78-C-0722 and by the Army Research Office. 相似文献
19.
We consider a multidimensional diffusion X with drift coefficient b(Xt,α) and diffusion coefficient εa(Xt,β) where α and β are two unknown parameters, while ε is known. For a high frequency sample of observations of the diffusion at the time points k/n, k=1,…,n, we propose a class of contrast functions and thus obtain estimators of (α,β). The estimators are shown to be consistent and asymptotically normal when n→∞ and ε→0 in such a way that ε−1n−ρ remains bounded for some ρ>0. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function. 相似文献
20.
Nakahiro Yoshida 《Probability Theory and Related Fields》1993,95(4):429-450
Summary Using the Malliavin calculus we derived asymptotic expansion of the distributions of the Bayes estimators for small diffusions. The second order efficiency of the Bayes estimator is proved. 相似文献