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ISSN (Print) 1435-926X - ISSN (Online) 0026-1335
• Hermite ranks and $U$ -statistics
• Abstract: Abstract We focus on the asymptotic behavior of $U$ -statistics of the type \begin{aligned} \sum _{1\le i\ne j\le n} h(X_i,X_j)\\ \end{aligned} in the long-range dependence setting, where $(X_i)_{i\ge 1}$ is a stationary mean-zero Gaussian process. Since $(X_i)_{i\ge 1}$ is Gaussian, $h$ can be decomposed in Hermite polynomials. The goal of this paper is to compare the different notions of Hermite rank and to provide conditions for the remainder term in the decomposition to be asymptotically negligeable.
PubDate: 2013-12-10

• Nonparametric density estimation in compound Poisson processes using convolution power estimators
• Abstract: Abstract Consider a compound Poisson process which is discretely observed with sampling interval $\Delta$ until exactly $n$ nonzero increments are obtained. The jump density and the intensity of the Poisson process are unknown. In this paper, we build and study parametric estimators of appropriate functions of the intensity, and an adaptive nonparametric estimator of the jump size density. The latter estimation method relies on nonparametric estimators of $m$ th convolution powers density. The $L^2$ -risk of the adaptive estimator achieves the optimal rate in the minimax sense over Sobolev balls. Numerical simulation results on various jump densities enlight the good performances of the proposed estimator.
PubDate: 2013-12-06

• Note on the existence and modulus of continuity of the ${\textit{SLE}}_8$ curve
• Abstract: Abstract We review one method for estimating the modulus of continuity of a Schramm–Loewner evolution (SLE) curve in terms of the inverse Loewner map. Then we prove estimates about the distribution of the inverse Loewner map, which underpin the difficulty in bounding the modulus of continuity of SLE for $\kappa =8$ . The main idea in the proof of these estimates is applying the Girsanov theorem to reduce the problem to estimates about one-dimensional Brownian motion.
PubDate: 2013-12-05

• Preface to the GPSD 2012 special issue of Metrika
• PubDate: 2013-12-03

• The covariation for Banach space valued processes and applications
• Abstract: Abstract This article focuses on a recent concept of covariation for processes taking values in a separable Banach space $B$ and a corresponding quadratic variation. The latter is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace $\chi$ of the dual of the projective tensor product of $B$ with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of $\bar{\nu }_0$ -semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark–Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.
PubDate: 2013-12-01

• Modified maximum spacings method for generalized extreme value distribution and applications in real data analysis
• Abstract: Abstract This paper analyzes weekly closing price data of the S&P 500 stock index and electrical insulation element lifetimes data based on generalized extreme value distribution. A new estimation method, modified maximum spacings (MSP) method, is proposed and obtained by using interior penalty function algorithm. The standard error of the proposed method is calculated through Bootstrap method. The asymptotic properties of the modified MSP estimators are discussed. Some simulations are performed, which show that the proposed method is not only available for the whole shape parameter space, but is also of high efficiency. The benchmark risk index, value at risk (VaR), is evaluated according to the proposed method, and the confidence interval of VaR is also calculated through Bootstrap method. Finally, the results are compared with those derived by empirical calculation and some existing methods.
PubDate: 2013-11-29

• Goodness-of-fit tests for parametric nonhomogeneous Markov processes
• Abstract: Abstract Tests for parametric nonhomogeneous and homogeneous Markov processes are given. Asymptotic distribution of test statistics is investigated. Tests for various well-known models are discussed as examples.
PubDate: 2013-11-27

• Discussion of dynamic programming and linear programming approaches to stochastic control and optimal stopping in continuous time
• Abstract: Abstract This paper seeks to highlight two approaches to the solution of stochastic control and optimal stopping problems in continuous time. Each approach transforms the stochastic problem into a deterministic problem. Dynamic programming is a well-established technique that obtains a partial/ordinary differential equation, variational or quasi-variational inequality depending on the type of problem; the solution provides the value of the problem as a function of the initial position (the value function). The other method recasts the problems as linear programs over a space of feasible measures. Both approaches use Dynkin’s formula in essential but different ways. The aim of this paper is to present the main ideas underlying these approaches with only passing attention paid to the important and necessary technical details.
PubDate: 2013-11-27

• Asymptotic behaviour of near-maxima of Gaussian sequences
• Abstract: Abstract Let $(X_1,X_2,\ldots ,X_n)$ be a Gaussian random vector with a common correlation coefficient $\rho _n,\,0\le \rho _n<1$ , and let $M_n= \max (X_1,\ldots , X_n),\,n\ge 1$ . For any given $a>0$ , define $T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)$ and $S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1$ . In this paper, we obtain the limit distributions of $(K_n(a))$ and $(S_n(a))$ , under the assumption that $\rho _n\rightarrow \rho$ as $n\rightarrow \infty ,$ for some $\rho \in [0,1)$ .
PubDate: 2013-11-16

• Second order longitudinal dynamic models with covariates: estimation and forecasting
• Abstract: Abstract In this paper, we propose an extension to the first-order branching process with immigration in the presence of fixed covariates and unobservable random effects. The extension permits the possibility that individuals from the second generation of the process may contribute to the total number of offsprings at time $t$ by producing offsprings of their own. We will study the basic properties of the second order process and discuss a generalized quasilikelihood (GQL) estimation of the mean and variance parameters and the generalized method of moments estimation of the correlation parameters. We will discuss the asymptotic distribution of the GQL estimator by first deriving the influence curve of the estimator. For the fixed effects model we shall derive a forecasting function and the variance of the forecast error. The performance of the proposed estimators and forecasts will be examined through a simulation study.
PubDate: 2013-11-07

• Stochastic comparisons for the number of working components of a system in random environment
• Abstract: Abstract In terms of stochastic orders, the purpose of this paper is to show how the random environment can affect the number of working components of a system with heterogeneous components sharing a common random environment. Applications to a class of semiparametric mixture models, stress-strength model and warm standby system are presented.
PubDate: 2013-11-01

• Large sample properties for a class of copulas in bivariate survival analysis
• Abstract: Abstract This work is concerned with asymptotic properties of the bivariate survival function estimator using the functional relationship between marginal survival functions and a class of copulas for the dependence structure. Specifically, we study consistency and weak convergence of the bivariate survival function estimator obtained considering a two-step procedure of estimation. The obtained results are found from a key decomposition of the bivariate survival function in quantities that can be studied separately. In particular, we use relating results to almost sure and weak convergence of estimators, almost sure convergence of uniformly equicontinuous functions, and the delta method for functionals.
PubDate: 2013-11-01

• Polynomial spline estimation for generalized varying coefficient partially linear models with a diverging number of components
• Abstract: Abstract Generalized varying coefficient partially linear models are a flexible class of semiparametric models that deal with data with different types of responses. In this paper, we focus on polynomial spline estimator as a computationally easier alternative to the more commonly used local polynomial regression approach, since one can directly take advantage of many existing implementations for generalized linear models. Furthermore, motivated by the high dimensionality characteristics that accompany many modern data sets nowadays, we investigate its asymptotic properties when both the number of nonparametric and the number of parametric components grows with, but is still smaller than, the sample size. Simulations and a real data example are used to illustrate our proposal.
PubDate: 2013-11-01

• Parametric estimation of hidden stochastic model by contrast minimization and deconvolution
• Abstract: Abstract We study a new parametric approach for particular hidden stochastic models. This method is based on contrast minimization and deconvolution and can be applied, for example, for ecological and financial state space models. After proving consistency and asymptotic normality of the estimation leading to asymptotic confidence intervals, we provide a thorough numerical study, which compares most of the classical methods that are used in practice (Quasi-Maximum Likelihood estimator, Simulated Expectation Maximization Likelihood estimator and Bayesian estimators) to estimate the Stochastic Volatility model. We prove that our estimator clearly outperforms the Maximum Likelihood Estimator in term of computing time, but also most of the other methods. We also show that this contrast method is the most robust with respect to non Gaussianity of the error and also does not need any tuning parameter.
PubDate: 2013-11-01

• Asymptotic behavior of the estimated weights and of the estimated performance measures of the minimum VaR and the minimum CVaR optimal portfolios for dependent data
• Abstract: Abstract In this paper we derive the asymptotic distributions of the estimated weights and of estimated performance measures of the minimum value-at-risk portfolio and of the minimum conditional value-at-risk portfolio assuming that the asset returns follow a strictly stationary process. It is proved that the estimated weights as well as the estimated performance measures are asymptotically multivariate normally distributed. We also present an asymptotic test for the weights and a joint test for the characteristics of both portfolios. Moreover, the asymptotic densities of the estimated performance measures are compared with the corresponding exact densities. It is shown that the asymptotic approximation performs well even for the moderate sample size.
PubDate: 2013-11-01

• On the maxima of heterogeneous gamma variables with different shape and scale parameters
• Abstract: Abstract In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let $X_{1}$ and $X_{2}$ be two independent gamma random variables with $X_{i}$ having shape parameter $r_{i}>0$ and scale parameter $\lambda _{i}$ , $i=1,2$ , and let $X^{*}_{1}$ and $X^{*}_{2}$ be another set of independent gamma random variables with $X^{*}_{i}$ having shape parameter $r_{i}^{*}>0$ and scale parameter $\lambda _{i}^{*}$ , $i=1,2$ . Denote by $X_{2:2}$ and $X^{*}_{2:2}$ the corresponding maxima, respectively. It is proved that, among others, if $(r_{1},r_{2})$ majorize $(r_{1}^{*},r_{2}^{*})$ and $(\lambda _{1},\lambda _{2})$ weakly majorize $(\lambda _{1}^{*},\lambda _{2}^{*})$ , then $X_{2:2}$ is stochastically larger that $X^{*}_{2:2}$ in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.
PubDate: 2013-10-26

• Testing equality of shape parameters in several inverse Gaussian populations
• Abstract: Abstract Due to the strikingly resemblance to the normal theory and inference methods, the inverse Gaussian (IG) distribution is commonly applied to model positive and right-skewed data. As the shape parameter in the IG distribution is greatly related to other important quantities such as the mean, skewness, kurtosis and the coefficient of variation, it plays an important role in distribution theory. This paper focuses on testing the equality of shape parameters in several inverse Gaussian distributions. Three tests are suggested: the exact generalized inference-based test, the asymptotic test and a test that is based on parametric bootstrap approximation. Simulation studies are undertaken to examine the performances of the these methods, and three real data examples are analyzed for illustration.
PubDate: 2013-10-19

• Testing for the bivariate Poisson distribution
• Abstract: Abstract This paper studies goodness-of-fit tests for the bivariate Poisson distribution. Specifically, we propose and study several Cramér–von Mises type tests based on the empirical probability generating function. They are consistent against fixed alternatives for adequate choices of the weight function involved in their definition. They are also able to detect local alternatives converging to the null at a certain rate. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approximation and compares their powers for finite sample sizes. Extensions for testing goodness-of-fit for the multivariate Poisson distribution are also discussed.
PubDate: 2013-10-17

• Some results on constructing general minimum lower order confounding $2^{n-m}$ designs for $n\le 2^{n-m-2}$
• Abstract: Abstract Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC $2^{n-m}$ designs with $N/4+1\le n\le N-1$ were constructed by Li et al. (Stat Sinica 21:1571–1589, 2011), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, 2010) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, 2010), where $N=2^{n-m}$ is run number and $n$ is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC $2^{n-m}$ designs respectively with the three parameter cases of $n\le N-1$ : (i) $m\le 4$ , (ii) $m\ge 5$ and $n=(2^m-1)u+r$ for $u>0$ and $r=0,1,2$ , and (iii) $m\ge 5$ and $n=(2^m-1)u+r$ for $u\ge 0$ and $r=2^m-3,2^m-2$ .
PubDate: 2013-10-16

• Optimal and robust designs for trigonometric regression models
• Abstract: Abstract This article presents discussions on the optimal and robust designs for trigonometric regression models under different optimality criteria. First, we investigate the classical Q-optimal designs for estimating the response function in a full trigonometric regression model with a given order. The equivalencies of Q-, A-, and G-optimal designs for trigonometric regression in general are also articulated. Second, we study minimax designs and their implementation in the case of trigonometric approximation under Q-, A-, and D-optimality. Then, We indicate the existence of the symmetric designs that are D-optimal minimax designs for general trigonometric regression models, and prove the existence of the symmetric designs that are Q- or A-optimal minimax designs for two particular trigonometric regression models under certain conditions.
PubDate: 2013-10-10

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