The estimation for Lévy processes in high frequency data

In this article, a generalized Lévy model is proposed and its parameters are estimated in high-frequency data settings. An infinitesimal generator of Lévy processes is used to study the asymptotic properties of the drift and volatility estimators. They are consistent asymptotically and are independent of other parameters making them better than those in Chen et al. (2010 Chen, S. X., Delaigle, A., Hall, P. (2010). Nonparametric estimation for a class of Lévy processes. Journal of Econometrics 157:257–271.[Crossref], [Web of Science ®] , [Google Scholar]). The estimators proposed here also have fast convergence rates and are simple to implement.     

Discussion of ‘An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models’ by Battaglia and Protopapas

This discussion focuses on threshold nonstationary?Cnonlinear time series modelling; it raises various issues to do with identifiability and model complexity. It also gives some background history concerning smooth threshold/transition autoregressive models and hidden Markov switching models.     

The uniform local asymptotics for a Lévy process and its overshoot and undershoot

ABSTRACT

In this article, we obtain the uniform local asymptotics for a Lévy process with a heavy-tailed Lévy measure and for the overshoot and undershoot of the Lévy process. As applications, we get the uniform asymptotics of the finite-time ruin probability and the local ruin probability for the Lévy risk model with a heavy-tailed Lévy measure. By the above results, we find that in the compound Poisson model perturbed by a Brownian motion, the effect of the Brownian component on the asymptotics of the finite-time ruin probability and the local ruin probability washes out.     

On the likelihood function of small time variance Gamma Lévy processes

We investigate the likelihood function of small generalized Laplace laws and variance gamma Lévy processes in the short time framework. We prove the local asymptotic normality property in statistical inference for the variance gamma Lévy process under high-frequency sampling with its associated optimal convergence rate and Fisher information matrix. The location parameter is required to be given in advance for this purpose, while the remaining three parameters are jointly well behaved with an invertible Fisher information matrix. The results are discussed with relation to equivalent formulations of the variance gamma Lévy process, that is, as a time-changed Brownian motion and as a difference of two independent gamma processes.     

Discussion of “An analysis of global warming in the Alpine region based of nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

We discuss the scientific contribution of Battaglia and Protopapas?? paper concerning the debate on global warming supported by an extensive analysis of temperature time series in the Alpine region. In the work, Authors use several exploratory and modelling tools for assessing and discriminating the presence of different patterns in the data. We add some general and specific considerations mainly devoted to the modelling stage of their analysis.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

Approximation to two independent Gaussian processes from a unique Lévy process and applications

Abstract

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively.     

Discussion of “An analysis of global warming in the Alpine region based on nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

The paper by Battaglia and Protopapas (Stat Method Appl 2012) is stimulating. It gives an elegant mathematical generalization of autoregressive models (the nine types). It explains state-of-the-art model fitting techniques (genetic algorithm combined with fitness function and least squares). It is written in a fluent and authoritative manner. Important for having a wider impact: it is accessible to non-statisticians. Finally, it has interesting results on the temperature evolution over the instrumental period (roughly the past 200?years). These merits make this paper an important contribution to applied statistics as well as climatology. As a climate researcher, coming from Physics and having had an affiliation with a statistical institute only as postdoc, I re-analyse here three data series with the aim of providing motivation for model selection and interpreting the results from the climatological perspective.     

Fréchet differentiability in statistical inference for time series

It is shown how the method of Fréchet differentiability can simplify the asymptotic derivations in an important range of robust inferential problems for stationary and related time series models. The uniform root-n consistency of the empirical distribution function for the Cramer von Mises norm under a weak mixing condition is indicated. Various regularity conditions naturally implemented and leading to the differentiability are discussed. A simulation study supplementing the theoretical discussion is included.     

A comment on “An analysis of global warming in the Alpine Region based on nonlinear nonstationary time series models” by F. Battaglia and M. K. Protopapas

We discuss the paper by Battaglia and Protopapas concerning the analysis of global warming phenomenon in the Alpine Region. The Authors consider a nonlinear model which takes into account regimes in time and levels. In this contribution some of the statistical results presented in the paper are commented and a different approach to the problem is proposed. It is based on a temporal aggregation analysis and it can help to highlight some features in the data.     

Lévy Processes,Phase-Type Distributions,and Martingales

Lévy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance gamma processes, CGMY Lévy processes (tempered stable processes), NIG (normal inverse Gaussian) Lévy processes, and hyperbolic Lévy processes. We consider here a dense class of Lévy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Lévy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Lévy process with reflection in 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also, the relation to fluid models with a Brownian component is discussed.     

On Error Rates in Normal Approximations and Simulation Schemes for Lévy Processes

Let X=(X(t) : t≥0) be a Lévy process. In simulation, one often wants to know at what size it is possible to truncate the small jumps while retaining enough accuracy. A useful tool here is the Edgeworth expansion. We provide a third order expansion together with a uniform error bound, assuming third Lévy moment is 0. We next discuss approximating X in the finite variation case. Truncating the small jumps, we show that, adding their expected value, and further, including their variability by approximating by a Brownian motion, gives successively better results in general. Finally, some numerical illustrations involving a normal inverse Gaussian Lévy process are given.     

Parameter estimation for Ornstein–Uhlenbeck processes of the second kind driven by α-stable Lévy motions

In this article, we study the problem of parameter estimation for Ornstein–Uhlenbeck processes of the second kind driven by α-stable Lévy motions, based on continuous and discrete observations, respectively. Using the trajectory fitting method combined with the weighted least-squares technique, we discuss the consistency and the asymptotic distributions of the estimators for general weights in both the ergodic and the non ergodic cases.     

Karhunen–Loève expansions of Lévy processes

We derive the basis functions and joint distribution of the stochastic coefficients of the Karhunen–Loève expansion of a square-integrable Lévy process. Further, we demonstrate a method for simulating the coefficients via a shot-noise representation.     

A note on consistent estimation of Kullback–Leibler discrepancy in Poisson regression

A recent theorem by Hannig and Lee on consistency of their estimator of Kullback–Leibler discrepancy is re-proved under assumptions suitably modified to correct a fault in the original proof.     

Wavelets and estimation of long memory in nonstationary models: Does anything beat the exact local whittle estimator?

In this article, we analyze the performance of five estimation methods for the long memory parameter d. The goal of our article is to construct a wavelet estimate for the fractional differencing parameter in nonstationary long memory processes that dominate the well-known estimate of Shimotsu and Phillips (2005) Shimotsu, K., Phillips, P. (2005). Exact local whittle estimation of fractional integration. Annals of statistics 20:87–127. [Google Scholar]. The simulation results show that the wavelet estimation method of Lee (2005) Lee, J. (2005). Estimating memory parameter in the US inflation rate. Economics Letters 87:207–210. [Google Scholar] with several tapering techniques performs better under most cases in nonstationary long memory. The comparison is based on the empirical root mean squared error of each estimate.     

Valuation of contingent claims with stochastic interest rate and mortality driven by Lévy processes

Abstract

In this paper, we consider a model with stochastic interest rate and stochastic mortality, which is driven by a Lévy process. Under the assumption that the stochastic mortality and interest rate are dependent, we discuss the valuation of life insurance contracts. Employing the method of change of measure together with the Bayes’ rule, we present the pricing formulas in closed form for the survival and death benefit models. Finally, numerical experiments illustrate the effects of some parameters.