Bi-log-concave distribution functions

Nonparametric statistics for distribution functions F or densities f=F' under qualitative shape constraints provides an interesting alternative to classical parametric or entirely nonparametric approaches. We contribute to this area by considering a new shape constraint: F is said to be bi-log-concave, if both log(F) and log(1 - F) are concave. Many commonly considered distributions are compatible with this constraint. For instance, any c.d.f. F with log-concave density f = F' is bi-log-concave. But in contrast to the latter constraint, bi-log-concavity allows for multimodal densities. We provide various characterizations. It is shown that combining any nonparametric confidence band for F with the new shape-constraint leads to substantial improvements, particularly in the tails. To pinpoint this, we show that these confidence bands imply non-trivial confidence bounds for arbitrary moments and the moment generating function of F

Strong Approximation of Empirical Copula Processes by Gaussian Processes

We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm

Random walks - a sequential approach

In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its as- sociated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results we consider local nonpara- metric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Further, we provide a theoretical result about the optimal kernel for a given alternative

Inconsistency of the MLE for the joint distribution of interval censored survival times and continuous marks

This paper considers the nonparametric maximum likelihood estimator (MLE) for the joint distribution function of an interval censored survival time and a continuous mark variable. We provide a new explicit formula for the MLE in this problem. We use this formula and the mark specific cumulative hazard function of Huang and Louis (1998) to obtain the almost sure limit of the MLE. This result leads to necessary and sufficient conditions for consistency of the MLE which imply that the MLE is inconsistent in general. We show that the inconsistency can be repaired by discretizing the marks. Our theoretical results are supported by simulations.Comment: 27 pages, 4 figure

An Edgeworth expansion for finite population L-statistics

In this paper, we consider the one-term Edgeworth expansion for finite population L-statistics. We provide an explicit formula for the Edgeworth correction term and give sufficient conditions for the validity of the expansion which are expressed in terms of the weight function that defines the statistics and moment conditions.Comment: 14 pages. Minor revisions. Some explanatory comments and a numerical example were added. Lith. Math. J. (to appear

Semiparametric theory and empirical processes in causal inference

In this paper we review important aspects of semiparametric theory and empirical processes that arise in causal inference problems. We begin with a brief introduction to the general problem of causal inference, and go on to discuss estimation and inference for causal effects under semiparametric models, which allow parts of the data-generating process to be unrestricted if they are not of particular interest (i.e., nuisance functions). These models are very useful in causal problems because the outcome process is often complex and difficult to model, and there may only be information available about the treatment process (at best). Semiparametric theory gives a framework for benchmarking efficiency and constructing estimators in such settings. In the second part of the paper we discuss empirical process theory, which provides powerful tools for understanding the asymptotic behavior of semiparametric estimators that depend on flexible nonparametric estimators of nuisance functions. These tools are crucial for incorporating machine learning and other modern methods into causal inference analyses. We conclude by examining related extensions and future directions for work in semiparametric causal inference

Robustness Regions for Measures of Risk Aggregation

One of risk measures key purposes is to consistently rank and distinguish between different risk profiles. From a practical perspective, a risk measure should also be robust, that is, insensitive to small perturbations in input assumptions. It is known in the literature [14, 39], that strong assumptions on the risk measures ability to distinguish between risks may lead to a lack of robustness. We address the trade-off between robustness and consistent risk ranking by specifying the regions in the space of distribution functions, where law-invariant convex risk measures are indeed robust. Examples include the set of random variables with bounded second moment and those that are less volatile (in convex order) than random variables in a given uniformly integrable set. Typically, a risk measure is evaluated on the output of an aggregation function defined on a set of random input vectors. Extending the definition of robustness to this setting, we find that law-invariant convex risk measures are robust for any aggregation function that satisfies a linear growth condition in the tail, provided that the set of possible marginals is uniformly integrable. Thus, we obtain that all law-invariant convex risk measures possess the aggregation-robustness property introduced by [26] further studied by [40]. This is in contrast to the widely-used, non-convex, risk measure Value-at-Risk, whose robustness in a risk aggregation context requires restricting the possible dependence structures of the input vectors

On the asymptotic normality of kernel estimators of the long run covariance of functional time series

We consider the asymptotic normality in L2 of kernel estimators of the long run covariance of stationary functional time series. Our results are established assuming a weakly dependent Bernoulli shift structure for the underlying observations, which contains most stationary functional time series models, under mild conditions. As a corollary, we obtain joint asymptotics for functional principal components computed from empirical long run covariance operators, showing that they have the favorable property of being asymptotically independent

Probing Loop Quantum Gravity with Evaporating Black Holes

This letter aims at showing that the observation of evaporating black holes should allow distinguishing between the usual Hawking behavior and Loop Quantum Gravity (LQG) expectations. We present a full Monte-Carlo simulation of the evaporation in LQG and statistical tests that discriminate between competing models. We conclude that contrarily to what was commonly thought, the discreteness of the area in LQG leads to characteristic features that qualify evaporating black holes as objects that could reveal quantum gravity footprints.Comment: 5 pages, 3 figures. Version accpeted by Phys. Rev. Let