The functional delta method for deriving asymptotic distributions is presented. Assuming our interest lies in T θ 0 $$ T\left({\boldsymbol{\theta}}_0\right) $$ where θ 0 $$ {\boldsymbol{\theta}}_0 $$ is an unknown, infinite-dimensional parameter and T $$ T $$ is a known functional. Like the delta method the functional delta method allows to immediately obtain an approximation of the distribution of the plug-in estimator T θ ̂ n $$ T\left({\hat{\boldsymbol{\theta}}}_n\right) $$ through the asymptotic distribution of r n T θ ̂ n − T θ 0 $$ {r}_n\left(T\left({\hat{\boldsymbol{\theta}}}_n\right)-T\left({\boldsymbol{\theta}}_0\right)\right) $$ subject to (a) the asymptotic distribution of r n θ ̂ n − θ 0 $$ {r}_n\left({\hat{\boldsymbol{\theta}}}_n-{\boldsymbol{\theta}}_0\right) $$ being known and (b) the existence of an appropriate functional derivative of T $$ T $$ at θ 0 $$ {\boldsymbol{\theta}}_0 $$ . This article is categorized under:

We present new concentration inequalities for classical and smoothed empirical processes of independent and dependent real-valued random variables. In the case of i.i.d. random variables, we show that Massart's celebrated refinement of the Dvoretzky-Kiefer-Wolfowitz inequality for the canonical empirical process extends to the empirical process indexed by classes of functions of uniformly bounded variation. The famous Talagrand inequality for empirical processes does not seem to give a similar result. Furthermore, we show that versions of both Massart's exponential bound and the generalisation we prove can also be obtained for the smoothed empirical process. In the case where the underlying random variables follow a linear process, we use a recently shown concentration inequality for the canonical empirical process to obtain, to the best of our knowledge, the first concentration inequalities for the smoothed empirical process of dependent random variables.

Observation-driven models for time series have a long history in statistics and econometrics, but are typically studied under the assumption of correct dynamic specification. We develop an in-fill asymptotic framework to study the limiting behavior of the estimated (i.e., filtered) time-varying parameter paths obtained with such models in (severely) mis-specified settings. We show that despite such mis-specification, the filtered paths, particularly those from the class of score-driven models of Creal et al. (2011, 2013) and Harvey (2013), still converge in probability to the Kullback-Leibler optimal time-varying parameter paths, even in severely mis-specified settings. We obtain distributional convergence results for the filtering errors and formulate the observation-driven filter that minimizes the asymptotic filter error variance. Such an optimal filter again has score-driven features. The results substantially generalize earlier findings, which we demonstrate by applying the new theory to time-varying tail shape models, dynamic copulas, and time-varying regression models. We further highlight the practical relevance of the asymptotic results by using them to construct pointwise intervals that quantify the uncertainty of filtered parameter paths based on observation-driven filters and apply these to the volatility path of intraday Pfizer log-returns.

Various approaches have been proposed for the two-sample and the k-sample problem. This includes, for instance, characteristic function based approaches, empirical likelihood methods, kernel based approaches, so-called smooth tests which are based on orthogonal polynomials and approaches based on Cram & eacute;r-von Mises type statistics. Regardless of the approach a common assumption is that the two-samples are independent and that each sample consists of independent random variables. Here we consider the following settings: a) We have dependency between the samples which consist of independent random variables; b) The samples are independent but each sample consists of (possibly) dependent random variables; c) The samples are dependent and each sample consists of (possibly) dependent random variables. More concretely, we consider the two-sample smooth test recently proposed by [47], show consistency of this test and derive its asymptotic distributions in these three settings. It turns out that in all settings considered the asymptotic distribution of the test statistic is free of the marginal distribution under the null hypothesis. Moreover, various bootstrap schemes for this test are introduced. These bootstrap schemes cover the three settings mentioned above. Bootstrap consistency is also shown for these three cases. The results for this smooth test cover several cases that, to the best of our knowledge, are not encompassed by any of the other approaches to the two-sample problem.

The delta method for deriving asymptotic distributions is presented. Assume interest lies in f(theta(0)) where theta(0) is an unknown parameter and f is a known function. The delta method allows to immediately obtain an approximation of the distribution of the plug-in estimator f((theta) over cap (n)) through the asymptotic distribution of a(n)(f ((theta) over cap (n)) - f( theta(0)))whenever the asymptotic distribution of a(n)((theta) over cap (-)(n) theta(0)) is known and f is differentiable at theta(0). This article is categorized under: Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

A fixed-design residual bootstrap method is proposed for the two-step estimator of Francq and Zakoïan(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.

First an overview of a class of models for recurrent events is given. The class of models considered is known as virtual or effective age models. One of the strengths of this class of models is their ability to account for intervention effects after an event occurrence. Some of the models within this class allow to account for the effects of covariates and the impact of the number of already observed events. After having provided an overview of this class of models, non- and semiparametric inference methods for these models are reviewed. Several open problems in non- and semiparametric inference methods for these models are also described.

Recently, Radulovic and Wegkamp introduced a new technique to show convergence in distribution of the empirical process indexed by functions of bounded variation. This method of proof allows to directly extend convergence results known for the canonical empirical process to convergence in distribution of the empirical process indexed by functions of bounded variation. The purpose of this article is twofold. First, we extend the mentioned technique to index functions of locally bounded variation. Second, and more importantly, we demonstrate that this technique provides a new approach to show convergence in distribution of the smoothed empirical process based on kernel density estimators. Using this approach we can prove to the best of our knowledge the first results on convergence in distribution of the smoothed empirical process of dependent data. Our results cover both weak and strong dependence as well as index sets of functions of locally bounded variation. Moreover our results cover an MISE optimal choice of the bandwidth for the kernel density estimator which to some extent is the plug-in property in the Bickel-Ritov sense. In the case of i.i.d. data our results extend a seminal result of Gine and Nickl.

We develop a Feasible Generalized Least Squares estimator of the date of a structural break in level and/or trend. The estimator is based on a consistent estimate of a T-dimensional inverse autocovariance matrix. A cubic polynomial transformation of break date estimates can be approximated by a nonstandard yet nuisance parameter free distribution asymptotically. The new limiting distribution captures the asymmetry and bimodality in finite samples and is applicable for inference with a single, known, set of critical values. We consider the confidence intervals/sets for break dates based on both Wald-type tests and by inverting multiple likelihood ratio (LR) tests. A simulation study shows that the proposed estimator increases the empirical concentration probability in a small neighborhood of the true break date and potentially reduces the mean squared errors. The LR-based confidence intervals/sets have good coverage while maintaining informative length even with highly persistent errors and small break sizes.

To quantify uncertainty around point estimates of conditional objects such as conditional means or variances, parameter uncertainty has to be taken into account. Attempts to incorporate parameter uncertainty are typically based on the unrealistic assumption of observing two independent processes, where one is used for parameter estimation, and the other for conditioning upon. Such unrealistic foundation raises the question whether these intervals are theoretically justified in a realistic setting. This paper presents an asymptotic justification for this type of intervals that does not require such an unrealistic assumption, but relies on a sample-split approach instead. By showing that our sample-split intervals coincide asymptotically with the standard intervals, we provide a novel, and realistic, justification for confidence intervals of conditional objects. The analysis is carried out for a rich class of time series models.

Virtual age models are very useful to analyse recurrent events.Among the strengths of these models is their ability to account for treatment (or intervention) effects after an event occurrence.Despite their flexibility for modeling recurrent events, the number of applications is limited.This seems to be a result of the fact that in the semiparametric setting all the existing results assume the virtual age function that describes the treatment (or intervention) effects to be known.This shortcoming can be overcome by considering semiparametric virtual age models with parametrically specified virtual age functions.Yet, fitting such a model is a difficult task.Indeed, it has recently been shown that for these models the standard profile likelihood method fails to lead to consistent estimators.Here we show that consistent estimators can be constructed by smoothing the profile log-likelihood function appropriately.We show that our general result can be applied to most of the relevant virtual age models of the literature.Our approach shows that empirical process techniques may be a worthwhile alternative to martingale methods for studying asymptotic properties of these inference methods.A simulation study is provided to illustrate our consistency results together with an application to real data.

Ethane is the most abundant non-methane hydrocarbon in the Earth's atmosphere and an important precursor of tropospheric ozone through various chemical pathways. Ethane is also an indirect greenhouse gas (global warming potential), influencing the atmospheric lifetime of methane through the consumption of the hydroxyl radical (OH). Understanding the development of trends and identifying trend reversals in atmospheric ethane is therefore crucial. Our dataset consists of four series of daily ethane columns obtained from ground-based FTIR measurements. As many other decadal time series, our data are characterized by autocorrelation, heteroskedasticity, and seasonal effects. Additionally, missing observations due to instrument failure or unfavorable measurement conditions are common in such series. The goal of this paper is therefore to analyze trends in atmospheric ethane with statistical tools that correctly address these data features. We present selected methods designed for the analysis of time trends and trend reversals. We consider bootstrap inference on broken linear trends and smoothly varying nonlinear trends. In particular, for the broken trend model, we propose a bootstrap method for inference on the break location and the corresponding changes in slope. For the smooth trend model we construct simultaneous confidence bands around the nonparametrically estimated trend. Our autoregressive wild bootstrap approach, combined with a seasonal filter, is able to handle all issues mentioned above.

In this paper we propose a general framework to analyze prediction in time series models and show how a wide class of popular time series models satisfies this framework. We postulate a set of high-level assumptions, and formally verify these assumptions for the aforementioned time series models. Our framework coincides with that of Beutner et al. (2019, arXiv:1710.00643) who establish the validity of conditional confidence intervals for predictions made in this framework. The current paper therefore complements the results in Beutner et al. (2019, arXiv:1710.00643) by providing practically relevant applications of their theory.

Previous articleNext article No AccessGeneral BiologyStatistical Intervals: A Guide for Practitioners and Researchers. Second Edition. Wiley Series in Probability and Statistics. By William Q. Meeker, Gerald J. Hahn, and Luis A. Escobar. Hoboken (New Jersey): Wiley. $110.00. xxxv + 592 p.; ill.; index. ISBN: 978-0-471-68717-7 (hc); 978-1-118-59516-9 (eb). 2017.Eric BeutnerEric BeutnerEconometrics, Vrije Universiteit, Amsterdam, The Netherlands Search for more articles by this author Econometrics, Vrije Universiteit, Amsterdam, The NetherlandsPDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The Quarterly Review of Biology Volume 94, Number 3September 2019 Published in association with Stony Brook University Article DOIhttps://doi.org/10.1086/705049 Views: 51Total views on this site For permission to reuse, please contact [email protected]PDF download Crossref reports no articles citing this article.

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.

We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector theta, say, exceeds the profile likelihood function evaluated at the parameter vector (theta) over bar, say, with probability p. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we will provide an example where the profile likelihood function is monotone with probability one regardless of the true data generating process. We also discuss the relation of our results to other semi-parametric models like the accelerated failure time model and Cox's proportional hazards model.

We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector θ, say, exceeds the profile likelihood function evaluated at the parameter vector $\tilde\theta$, say, with probability p. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we will provide an example where the profile likelihood function is monotone with probability one regardless of the true data generating process.

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.

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.