## Recursos de colección

1. #### Local times of multifractional Brownian sheets

Meerschaert, Mark; Wu, Dongsheng; Xiao, Yimin
Denote by H(t)=(H1(t), …, HN(t)) a function in t∈ℝ+N with values in (0, 1)N. Let {BH(t)(t)}={BH(t)(t), t∈ℝ+N} be an (N, d)-multifractional Brownian sheet (mfBs) with Hurst functional H(t). Under some regularity conditions on the function H(t), we prove the existence, joint continuity and the Hölder regularity of the local times of {BH(t)(t)}. We also determine the Hausdorff dimensions of the level sets of {BH(t)(t)}. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets.

2. #### Tail probabilities for infinite series of regularly varying random vectors

Hult, Henrik; Samorodnitsky, Gennady
A random vector X with representation X=∑j≥0AjZj is considered. Here, (Zj) is a sequence of independent and identically distributed random vectors and (Aj) is a sequence of random matrices, ‘predictable’ with respect to the sequence (Zj). The distribution of Z1 is assumed to be multivariate regular varying. Moment conditions on the matrices (Aj) are determined under which the distribution of X is regularly varying and, in fact, ‘inherits’ its regular variation from that of the (Zj)’s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums and stochastic integrals.

3. #### Asymptotic expansions at any time for scalar fractional SDEs with Hurst index H>1/2

Darses, Sébastien; Nourdin, Ivan
We study the asymptotic expansions with respect to h of ¶ E[Δhf(Xt)],  E[Δhf(Xt)|ℱtX] and E[Δhf(Xt)|Xt], ¶ where Δhf(Xt)=f(Xt+h)−f(Xt), when f:ℝ→ℝ is a smooth real function, t≥0 is a fixed time, X is the solution of a one-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst index H>1/2 and ℱX is its natural filtration.

4. #### Central limit theorems for double Poisson integrals

Peccati, Giovanni; Taqqu, Murad S.
Motivated by second order asymptotic results, we characterize the convergence in law of double integrals, with respect to Poisson random measures, toward a standard Gaussian distribution. Our conditions are expressed in terms of contractions of the kernels. To prove our main results, we use the theory of stable convergence of generalized stochastic integrals developed by Peccati and Taqqu. One of the advantages of our approach is that the conditions are expressed directly in terms of the kernel appearing in the multiple integral and do not make any explicit use of asymptotic dependence properties such as mixing. We illustrate our techniques...

5. #### Probability measures, Lévy measures and analyticity in time

Barndorff-Nielsen, Ole E.; Hubalek, Friedrich
We investigate the relation of the semigroup probability density of an infinite activity Lévy process to the corresponding Lévy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the Lévy measure and the third method uses the analytic continuation of the Lévy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.

6. #### Kshirsagar–Tan independence property of beta matrices and related characterizations

Bobecka, Konstancja; Wesołowski, Jacek
A new independence property of univariate beta distributions, related to the results of Kshirsagar and Tan for beta matrices, is presented. Conversely, a characterization of univariate beta laws through this independence property is proved. A related characterization of a family of 2×2 random matrices including beta matrices is also obtained. The main technical challenge was a problem involving the solution of a related functional equation.

7. #### Smoothed weighted empirical likelihood ratio confidence intervals for quantiles

Ren, Jian-Jian
Thus far, likelihood-based interval estimates for quantiles have not been studied in the literature on interval censored case 2 data and partly interval censored data, and, in this context, the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval censored data. The fourth order expansion of the weighted empirical log-likelihood ratio is derived and the theoretical coverage accuracy equation for...

8. #### Adaptive wavelet-based estimator of the memory parameter for stationary Gaussian processes

Bardet, Jean-Marc; Bibi, Hatem; Jouini, Abdellatif
This work is intended as a contribution to the theory of a wavelet-based adaptive estimator of the memory parameter in the classical semi-parametric framework for Gaussian stationary processes. In particular, we introduce and develop the choice of a data-driven optimal bandwidth. Moreover, we establish a central limit theorem for the estimator of the memory parameter with the minimax rate of convergence (up to a logarithm factor). The quality of the estimators is demonstrated via simulations.

9. #### Evaluation and selection of models for out-of-sample prediction when the sample size is small relative to the complexity of the data-generating process

Leeb, Hannes
In regression with random design, we study the problem of selecting a model that performs well for out-of-sample prediction. We do not assume that any of the candidate models under consideration are correct. Our analysis is based on explicit finite-sample results. Our main findings differ from those of other analyses that are based on traditional large-sample limit approximations because we consider a situation where the sample size is small relative to the complexity of the data-generating process, in the sense that the number of parameters in a ‘good’ model is of the same order as sample size. Also, we allow...

10. #### Testing for changes in polynomial regression

Aue, Alexander; Horváth, Lajos; Hušková, Marie; Kokoszka, Piotr
We consider a nonlinear polynomial regression model in which we wish to test the null hypothesis of structural stability in the regression parameters against the alternative of a break at an unknown time. We derive the extreme value distribution of a maximum-type test statistic which is asymptotically equivalent to the maximally selected likelihood ratio. The resulting test is easy to apply and has good size and power, even in small samples.

11. #### Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model

Hu, Jianhua; Yan, Guohua
Let $\mathbf{Y}=\mathbf{X}\bolds{\Theta}\mathbf{Z}'+\bolds{\mathcal {E}}$ be the growth curve model with $\bolds{\mathcal{E}}$ distributed with mean 0 and covariance In⊗Σ, where Θ, Σ are unknown matrices of parameters and X, Z are known matrices. For the estimable parametric transformation of the form γ=CΘD' with given C and D, the two-stage generalized least-squares estimator γ̂(Y) defined in (7) converges in probability to γ as the sample size n tends to infinity and, further, $\sqrt{n}[\hat{\bolds{\gamma}}(\mathbf{Y})-{\bolds{\gamma}}]$ converges in distribution to the multivariate normal distribution $\mathcal{N}(\mathbf{0},(\mathbf{C}\mathbf{R}^{-1}\mathbf{C}')\otimes(\mathbf{D}(\mathbf{Z}'\bolds{\Sigma }^{-1}\mathbf{Z})^{-1}\mathbf{D}'))$ under the condition that limn→∞ X'X/n=R for some positive definite matrix R. Moreover, the unbiased and invariant quadratic estimator Σ̂(Y) defined...

12. #### The central limit theorem under random truncation

Stute, Winfried; Wang, Jane-Ling
Under left truncation, data (Xi, Yi) are observed only when Yi≤Xi. Usually, the distribution function F of the Xi is the target of interest. In this paper, we study linear functionals ∫ϕ dFn of the nonparametric maximum likelihood estimator (MLE) of F, the Lynden-Bell estimator Fn. A useful representation of ∫ϕ dFn is derived which yields asymptotic normality under optimal moment conditions on the score function ϕ. No continuity assumption on F is required. As a by-product, we obtain the distributional convergence of the Lynden-Bell empirical process on the whole real line.

13. #### Marginal likelihood for parallel series

McCullagh, Peter
Suppose that k series, all having the same autocorrelation function, are observed in parallel at n points in time or space. From a single series of moderate length, the autocorrelation parameter β can be estimated with limited accuracy, so we aim to increase the information by formulating a suitable model for the joint distribution of all series. Three Gaussian models of increasing complexity are considered, two of which assume that the series are independent. This paper studies the rate at which the information for β accumulates as k increases, possibly even beyond n. The profile log likelihood for the model...

14. #### Consistency of the α-trimming of a probability. Applications to central regions

Cascos, Ignacio; López-Díaz, Miguel
The sequence of α-trimmings of empirical probabilities is shown to converge, in the Painlevé–Kuratowski sense, on the class of probability measures endowed with the weak topology, to the α-trimming of the population probability. Such a result is applied to the study of the asymptotic behaviour of central regions based on the trimming of a probability.

15. #### Density estimation with heteroscedastic error

Delaigle, Aurore; Meister, Alexander
It is common, in deconvolution problems, to assume that the measurement errors are identically distributed. In many real-life applications, however, this condition is not satisfied and the deconvolution estimators developed for homoscedastic errors become inconsistent. In this paper, we introduce a kernel estimator of a density in the case of heteroscedastic contamination. We establish consistency of the estimator and show that it achieves optimal rates of convergence under quite general conditions. We study the limits of application of the procedure in some extreme situations, where we show that, in some cases, our estimator is consistent, even when the scaling parameter...

16. #### Augmented GARCH sequences: Dependence structure and asymptotics

Hörmann, Siegfried
The augmented GARCH model is a unification of numerous extensions of the popular and widely used ARCH process. It was introduced by Duan and besides ordinary (linear) GARCH processes, it contains exponential GARCH, power GARCH, threshold GARCH, asymmetric GARCH, etc. In this paper, we study the probabilistic structure of augmented GARCH(1, 1) sequences and the asymptotic distribution of various functionals of the process occurring in problems of statistical inference. Instead of using the Markov structure of the model and implied mixing properties, we utilize independence properties of perturbed GARCH sequences to directly reduce their asymptotic behavior to the case of...

17. #### GARCH modelling in continuous time for irregularly spaced time series data

Maller, Ross A.; Müller, Gernot; Szimayer, Alex
The discrete-time GARCH methodology which has had such a profound influence on the modelling of heteroscedasticity in time series is intuitively well motivated in capturing many ‘stylized facts’ concerning financial series, and is now almost routinely used in a wide range of situations, often including some where the data are not observed at equally spaced intervals of time. However, such data is more appropriately analyzed with a continuous-time model which preserves the essential features of the successful GARCH paradigm. One possible such extension is the diffusion limit of Nelson, but this is problematic in that the discrete-time GARCH model and...

18. #### Stochastic calculus for convoluted Lévy processes

Bender, Christian; Marquardt, Tina
We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation Lévy process with a Volterra-type kernel. This class of processes contains, for example, fractional Lévy processes as studied by Marquardt [Bernoulli 12 (2006) 1090–1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an Itô formula which separates the different contributions from the memory due to the convolution and from the jumps.

19. #### Estimation of the Brownian dimension of a continuous Itô process

Jacod, Jean; Lejay, Antoine; Talay, Denis
In this paper, we consider a d-dimensional continuous Itô process which is observed at n regularly spaced times on a given time interval [0, T]. This process is driven by a multidimensional Wiener process and our aim is to provide asymptotic statistical procedures which give the minimal dimension of the driving Wiener process, which is between 0 (a pure drift) and d. We exhibit several different procedures, all similar to asymptotic testing hypotheses.

20. #### A unifying framework for k-statistics, polykays and their multivariate generalizations

Di Nardo, Elvira; Guarino, Giuseppe; Senato, Domenico
Through the classical umbral calculus, we provide a unifying syntax for single and multivariate k-statistics, polykays and multivariate polykays. From a combinatorial point of view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for k-statistics that results in a very fast algorithm to generate such estimators.

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