Abstract:
GARCH models are useful tools in the investigation of phenomena, where volatility changes are prominent features, like most financial data. The parameter estimation via quasi maximum likelihood (QMLE) and its properties are by now well understood. However, there is a gap between practical applications and the theory, as in reality there are usually not enough observations for the limit results to be valid approximations. We try to fill this gap by this paper, where the properties of a recent bootstrap methodology in the context of GARCH modeling are revealed. The results are promising as it turns out that this remarkably simple method has essentially the same limit distribution, as the original estimatorwith the advantage of easy confidence interval construction, as it is demonstrated in the paper. The finite-sample properties of the suggested estimators are investigated through a simulation study, which ensures that the results are practically applicable for sample sizes as low as a thousand. On the other hand, the results are not 100% accurate until sample size reaches 100 thousands - but it is shown that this property is not a feature of our bootstrap procedure only, as it is shared by the original QMLE, too.

Abstract:
The paper deals with unconditional wavelet bases in weighted spaces. Inhomogeneous wavelets of Daubechies type are considered. Necessary and sufficient conditions for weights are found for which the wavelet system is an unconditional basis in weighted spaces in dependence on .

Abstract:
The integral wavelet transform is defined in weighted Sobolevspaces, in which some properties of the transform as well as itsasymptotical behaviour for small dilation parameter are studied.

Abstract:
With increasing internationalization of financial transactions, the foreign exchange market has been profoundly transformed and became more competitive and volatile. This places the accurate and reliable measurement of market risks in a crucial position for both investment decision and hedging strategy designs. This paper deals with the measurement of risks from a Value at Risk (VaR) perspective. A Wavelet-ARMA-GARCH refinement method to VaR estimate is used and compared with classical ARMA-GARCH approach. Performances of both approaches have been tested and compared using Kupiec backtesting procedures. Experiment results suggest that the performance of Wavelet-ARMA-GARCH refinement method to VaR estimate improves the reliability of VaR estimates at all confidence levels which offers considerable flexibility and potential performance improvement for Foreign exchange dealers. Furthermore, the appropriate selection and combination of parameters can lead to comprehensive performance improvement in reliability.

Abstract:
This paper investigates the asymptotic theory of the quasi-maximum exponential likelihood estimators (QMELE) for ARMA--GARCH models. Under only a fractional moment condition, the strong consistency and the asymptotic normality of the global self-weighted QMELE are obtained. Based on this self-weighted QMELE, the local QMELE is showed to be asymptotically normal for the ARMA model with GARCH (finite variance) and IGARCH errors. A formal comparison of two estimators is given for some cases. A simulation study is carried out to assess the performance of these estimators, and a real example on the world crude oil price is given.

Abstract:
The purpose of this paper is to assess the statistical characterization of weighted networks in terms of the generalization of the relevant parameters, namely, average path length, degree distribution, and clustering coefficient. Although the degree distribution and the average path length admit straightforward generalizations, for the clustering coefficient several different definitions have been proposed in the literature. We examined the different definitions and identified the similarities and differences between them. In order to elucidate the significance of different definitions of the weighted clustering coefficient, we studied their dependence on the weights of the connections. For this purpose, we introduce the relative perturbation norm of the weights as an index to assess the weight distribution. This study revealed new interesting statistical regularities in terms of the relative perturbation norm useful for the statistical characterization of weighted graphs.

Abstract:
The purpose of this paper is to assess the statistical characterization of weighted networks in terms of the generalization of the relevant parameters, namely average path length, degree distribution and clustering coefficient. Although the degree distribution and the average path length admit straightforward generalizations, for the clustering coefficient several different definitions have been proposed in the literature. We examined the different definitions and identified the similarities and differences between them. In order to elucidate the significance of different definitions of the weighted clustering coefficient, we studied their dependence on the weights of the connections. For this purpose, we introduce the relative perturbation norm of the weights as an index to assess the weight distribution. This study revealed new interesting statistical regularities in terms of the relative perturbation norm useful for the statistical characterization of weighted graphs.

Abstract:
We investigate the estimation of a weighted density taking the form $g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated distribution function and $w$ is a known (non-negative) weight. Such a class encompasses many examples, including those arising in order statistics or when $g$ is related to the maximum or the minimum of $N$ (random or fixed) independent and identically distributed (\iid) random variables. We here construct a new adaptive non-parametric estimator for $g$ based on a plug-in approach and the wavelets methodology. For a wide class of models, we prove that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with $p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared error) over Besov balls. The theoretical findings are illustrated through several simulations.

Abstract:
A novel weighted wavelet is constructed by using the subdivision scheme andrecursive average interpolated method.Its characteristics is respectively proved to be compactsupported,smooth,vanishing moment,and orthogonal.Cornpared with the common waveletby the Fourier transform,the decay of weighted wavelet coefficient is faster than that of thecommon wavelet.Furthermore,the method is simple and easily applied.Its computationalcomplexity is greatly reduced.The simulation demonstrate that the weighted wavelet issufficiently smooth even though the weighted function has a large jump,and that it canapproximate functions exactly and the approximated convergence is faster.

Abstract:
We introduce a scale of weighted Carleson norms, which depend on an integrability parameter p, where p=2 corresponds to the classical Carleson measure condition. Relations between the weighed BMO norm of a vector-valued function f:R->X, and the Carleson norm of the sequence of its wavelet coefficients, are established. These extend the results of Harboure-Salinas-Viviani, also in the scalar-valued case when p is not 2.