a b s t r a c tThe classical functional delta method (FDM) provides a convenient tool for deriving the asymptotic distribution of statistical functionals from the weak convergence of the respective empirical processes. However, for many interesting functionals depending on the tails of the underlying distribution this FDM cannot be applied since the method typically relies on Hadamard differentiability w.r.t. the uniform sup-norm. In this article, we present a version of the FDM which is suitable also for nonuniform sup-norms, with the outcome that the range of application of the FDM enlarges essentially. On one hand, our FDM, which we shall call the modified FDM, works for functionals that are ''differentiable'' in a weaker sense than Hadamard differentiability. On the other hand, it requires weak convergence of the empirical process w.r.t. a nonuniform sup-norm. The latter is not problematic since there exist strong respective results on weighted empirical processes obtained by Shorack and Wellner (1986) [25], Shao and Yu (1996) [23], Wu (2008) [32], and others. We illustrate the gain of the modified FDM by deriving the asymptotic distribution of plug-in estimates of popular risk measures that cannot be treated with the classical FDM.

Many problems in financial engineering involve the estimation of unknown conditional expectations across a time interval. Often Least Squares Monte Carlo techniques are used for the estimation. One method that can be combined with Least Squares Monte Carlo is the "Regress-Later" method. Unlike conventional methods where the value function is regressed on a set of basis functions valued at the beginning of the interval, the "Regress-Later" method regresses the value function on a set of basis functions valued at the end of the interval. The conditional expectation across the interval is then computed exactly for each basis function. We provide sufficient conditions under which we derive the convergence rate of Regress-Later estimators. Importantly, our results hold on non-compact sets. We show that the Regress-Later method is capable of converging significantly faster than conventional methods and provide an explicit example. Achieving faster convergence speed provides a strong motivation for using Regress-Later methods in estimating conditional expectations across time.

It is commonly acknowledged that V-functionals with an unbounded kernel are not Hadamard differentiable and that therefore the asymptotic distribution of U-and V-statistics with an unbounded kernel cannot be derived by the Functional Delta Method (FDM). However, in this article we show that V-functionals are quasi-Hadamard differentiable and that therefore a modified version of the FDM (introduced recently in (J. Multivariate Anal. 101 (2010) 2452-2463)) can be applied to this problem. The modified FDM requires weak convergence of a weighted version of the underlying empirical process. The latter is not problematic since there exist several results on weighted empirical processes in the literature; see, for example, (J. 313-333). The modified FDM approach has the advantage that it is very flexible w.r.t. both the underlying data and the estimator of the unknown distribution function. Both will be demonstrated by various examples. In particular, we will show that our FDM approach covers mainly all the results known in literature for the asymptotic distribution of U-and V-statistics based on dependent data -and our assumptions are by tendency even weaker. Moreover, using our FDM approach we extend these results to dependence concepts that are not covered by the existing literature.

Sequential order statistics have been introduced to model sequential k-out-of-n systems which, as an extension of k-out-of-n systems, allow the failure of some components of the system to influence the remaining ones. Based on an independent sample of vectors of sequential order statistics, the maximum likelihood estimators of the model parameters of a sequential k-out-of-n system are derived under order restrictions. Special attention is paid to the simultaneous maximum likelihood estimation of the model parameters and the distribution parameters for a flexible location-scale family. Furthermore, order restricted hypothesis tests are considered for making the decision whether the usual k-out-of-n model or the general sequential k-out-of-n model is appropriate for a given data.

The functional delta-method provides a convenient tool for deriving bootstrap consistency of a sequence of plug-in estimators w.r.t. a given functional from bootstrap consistency of the underlying sequence of estimators. It has recently been shown in [7] that the range of applications of the functional delta-method for establishing bootstrap consistency in probability of the sequence of plug-in estimators can be considerably enlarged by replacing the usual condition of Hadamard differentiability of the given functional by the weaker condition of quasi-Hadamard differentiability. Here we introduce the notion of uniform quasi-Hadamard differentiability and show that this notion extends the set of functionals for which almost sure bootstrap consistency of the corresponding sequence of plug-in estimators can be obtained by the functional delta-method. We illustrate the benefit of our results by means of the Average Value at Risk functional as well as the composition of the Average Value at Risk functional and the compound convolution functional. For the latter we use a chain rule to be proved here. In our examples we consider the weighted exchangeable bootstrap for independent observations and the blockwise bootstrap for β-mixing observations.