86 research outputs found
Continuous mapping approach to the asymptotics of - and -statistics
We derive a new representation for - and -statistics. Using this
representation, the asymptotic distribution of - and -statistics can be
derived by a direct application of the Continuous Mapping theorem. That novel
approach not only encompasses most of the results on the asymptotic
distribution known in literature, but also allows for the first time a unifying
treatment of non-degenerate and degenerate - and -statistics. Moreover,
it yields a new and powerful tool to derive the asymptotic distribution of very
general - and -statistics based on long-memory sequences. This will be
exemplified by several astonishing examples. In particular, we shall present
examples where weak convergence of - or -statistics occurs at the rate
and , respectively, when is the rate of weak convergence
of the empirical process. We also introduce the notion of asymptotic (non-)
degeneracy which often appears in the presence of long-memory sequences.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ508 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Functional delta-method for the bootstrap of quasi-Hadamard differentiable functionals
The functional delta-method provides a convenient tool for deriving the
asymptotic distribution of a plug-in estimator of a statistical functional from
the asymptotic distribution of the respective empirical process. Moreover, it
provides a tool to derive bootstrap consistency for plug-in estimators from
bootstrap consistency of empirical processes. It has recently been shown that
the range of applications of the functional delta-method for the asymptotic
distribution can be considerably enlarged by employing the notion of
quasi-Hadamard differentiability. Here we show in a general setting that this
enlargement carries over to the bootstrap. That is, for quasi-Hadamard
differentiable functionals bootstrap consistency of the plug-in estimator
follows from bootstrap consistency of the respective empirical process. This
enlargement often requires convergence in distribution of the bootstrapped
empirical process w.r.t.\ a nonuniform sup-norm. The latter is not problematic
as will be illustrated by means of examples
A Residual Bootstrap for Conditional Value-at-Risk
This paper proposes a fixed-design residual bootstrap method for the two-step
estimator of Francq and Zako\"ian (2015) associated with the conditional
Value-at-Risk. The bootstrap's consistency is proven for a general class of
volatility models and intervals are constructed for the conditional
Value-at-Risk. A simulation study reveals that the equal-tailed percentile
bootstrap interval tends to fall short of its nominal value. In contrast, the
reversed-tails bootstrap interval yields accurate coverage. We also compare the
theoretically analyzed fixed-design bootstrap with the recursive-design
bootstrap. It turns out that the fixed-design bootstrap performs equally well
in terms of average coverage, yet leads on average to shorter intervals in
smaller samples. An empirical application illustrates the interval estimation
Identifiability issues of age-period and age-period-cohort models of the Lee-Carter type
The predominant way of modelling mortality rates is the Lee-Carter model and
its many extensions. The Lee-Carter model and its many extensions use a latent
process to forecast. These models are estimated using a two-step procedure that
causes an inconsistent view on the latent variable. This paper considers
identifiability issues of these models from a perspective that acknowledges the
latent variable as a stochastic process from the beginning. We call this
perspective the plug-in age-period or plug-in age-period-cohort model. Defining
a parameter vector that includes the underlying parameters of this process
rather than its realisations, we investigate whether the expected values and
covariances of the plug-in Lee-Carter models are identifiable. It will be seen,
for example, that even if in both steps of the estimation procedure we have
identifiability in a certain sense it does not necessarily carry over to the
plug-in models
Bootstrapping Average Value at Risk of Single and Collective Risks
Almost sure bootstrap consistency of the blockwise bootstrap for the Average Value at Risk
of single risks is established for strictly stationary β-mixing observations. Moreover, almost sure
bootstrap consistency of a multiplier bootstrap for the Average Value at Risk of collective risks is
established for independent observations. The main results rely on a new functional delta-method
for the almost sure bootstrap of uniformly quasi-Hadamard differentiable statistical functionals, to be
presented here. The latter seems to be interesting in its own right
Asymptotics for statistical functionals of long-memory sequences
AbstractWe present two general results that can be used to obtain asymptotic properties for statistical functionals based on linear long-memory sequences. As examples for the first one we consider L- and V-statistics, in particular tail-dependent L-statistics as well as V-statistics with unbounded kernels. As an example for the second result we consider degenerate V-statistics. To prove these results we also establish a weak convergence result for empirical processes of linear long-memory sequences, which improves earlier ones
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