First-order Asymptotics Research Articles

Second-order asymptotics of tail distortion risk measure for portfolio loss in the multivariate regularly varying model

In order to conduct more precise quantitative risk management, we present the second-order asymptotics of tail distortion risk measure for the portfolio loss satisfying multivariate regular variation in terms of the notion of second-order regular variation as the confidence level tends to one. Furthermore, for the particular multivariate regularly varying case, the corresponding second-order asymptotics of tail distortion risk measure for portfolio loss is also given. The obtained second-order asymptotics makes the corresponding first-order asymptotics more accurate.

Limiting Properties of Random Graph Models with Vertex and Edge Weights

This paper provides an overview of results, concerning longest or heaviest paths, in the area of random directed graphs on the integers along with some extensions. We study first-order asymptotics of heaviest paths allowing weights both on edges and vertices and assuming that weights on edges are signed. We aim at an exposition that summarizes, simplifies, and extends proof ideas. We also study sparse graph asymptotics, showing convergence of the weighted random graphs to a certain weighted graph that can be constructed in terms of Poisson processes. We are motivated by numerous applications, ranging from ecology to parallel computing models. It is the latter set of applications that necessitates the introduction of vertex weights. Finally, we discuss some open problems and research directions.

FIRST-ORDER ASYMPTOTICS OF PATH-DEPENDENT DERIVATIVES IN MULTISCALE STOCHASTIC VOLATILITY ENVIRONMENT

In this paper, we extend the first-order asymptotics analysis of Fouque et al. to general path-dependent financial derivatives using Dupire’s functional Itô calculus. The main conclusion is that the market group parameters calibrated to vanilla options can be used to price to the same order exotic, path-dependent derivatives as well. Under general conditions, the first-order condition is represented by a conditional expectation that could be numerically evaluated. Moreover, if the path-dependence is not too severe, we are able to find path-dependent closed-form solutions equivalent to the first-order approximation of path-independent options derived in Fouque et al. Additionally, we exemplify the results with Asian options and options on quadratic variation.

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first- and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik, Kerov, Logan, and Shepp (for the first-order asymptotics) and Kerov (for the second-order asymptotics). This gives more evidence for the connection with the Gaussian β-ensemble, already suggested by a work of Matsumoto. Our main tool is a polynomiality result for the structure constants of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interesting in itself and we give several other applications of it.

Choosing a Set of Instruments among Many and Possibly Invalid Instruments

We consider the instrument selection problem in instrumental variable regression model when there is a large set of instruments with potential invalidity. Existing methods for instrument selection are based on a priori assumptions of an instrument's validity. Moreover, existing approaches based on consistent moment selection or first-order asymptotics can lead to a large finite sample bias with many instruments. We derive higher-order mean square error (MSE) approximation of a two-stage least squares (2SLS) estimator allowing locally invalid instruments. Based on the approximation to the higher-order MSE, we propose an invalidity-robust instrument selection criterion (IRC) that captures two sources of finite sample bias at the same time: bias from using many instruments and bias from invalid instruments. We show that our criterion is an asymptotically unbiased estimator of the higher-order MSE approximation. In various simulation experiments, our criterion performs well in terms of lower bias and better coverage properties of the estimator compare with other existing selection methods. We also apply our criterion to the empirical application in Card (1995).

Beyond first-order asymptotics for Cox regression

To go beyond standard first-order asymptotics for Cox regression, we develop parametric bootstrap and second-order methods. In general, computation of $P$-values beyond first order requires more model specification than is required for the likelihood function. It is problematic to specify a censoring mechanism to be taken very seriously in detail, and it appears that conditioning on censoring is not a viable alternative to that. We circumvent this matter by employing a reference censoring model, matching the extent and timing of observed censoring. Our primary proposal is a parametric bootstrap method utilizing this reference censoring model to simulate inferential repetitions of the experiment. It is shown that the most important part of improvement on first-order methods - that pertaining to fitting nuisance parameters - is insensitive to the assumed censoring model. This is supported by numerical comparisons of our proposal to parametric bootstrap methods based on usual random censoring models, which are far more unattractive to implement. As an alternative to our primary proposal, we provide a second-order method requiring less computing effort while providing more insight into the nature of improvement on first-order methods. However, the parametric bootstrap method is more transparent, and hence is our primary proposal. Indications are that first-order partial likelihood methods are usually adequate in practice, so we are not advocating routine use of the proposed methods. It is however useful to see how best to check on first-order approximations, or improve on them, when this is expressly desired.

First- and Second-order Asymptotics for the Tail Distortion Risk Measure of Extreme Risks

The tail distortion risk measure at level p was first introduced in Zhu and Li (2012), where the parameter p ∈ (0, 1) indicates the confidence level. They established first-order asymptotics for this risk measure, as p↑1, for the Fréchet case. In this article, we extend their work by establishing both first-order and second-order asymptotics for the Fréchet, Weibull, and Gumbel cases. Numerical studies are also carried out to examine the accuracy of both asymptotics.

Inference tests for tax progressivity and income redistribution: the Suits approach

The Suits index is often used in tax policy analysis to measure progressivity and to detect changes in progressivity over time and between different fiscal policies. The lack of attention that has been devoted to the inference issue to date is surprising. In this paper, the limiting distribution of the Suits index estimator is derived and simple plug-in formulae for the index estimator and its sampling variance are provided. By means of a simulation analysis, we show that inferences based on first-order asymptotics perform well for moderately large samples. Bootstrap-t – which uses the plug-in variance estimator – appears to perform well. The accuracy of the proposed inference tests is also illustrated by means of an empirical application that uses Spanish income tax data.

Improved mixing rates for infinite measure-preserving systems

AbstractIn this work, we introduce a new technique for operator renewal sequences associated with dynamical systems preserving an infinite measure that improves the results on mixing rates obtained by Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 1 (2012), 61–110]. Also, this technique allows us to offer a very simple proof of the key result of Melbourne and Terhesiu that provides first-order asymptotics of operator renewal sequences associated with dynamical systems with infinite measure. Moreover, combining techniques used in this work with techniques used by Melbourne and Terhesiu, we obtain first-order asymptotics of operator renewal sequences under some relaxed assumption on the first return map.

Alternatives to the usual likelihood ratio test in mixed linear models

The small-sample performance of alternatives to the usual likelihood ratio test in mixed linear models is investigated. Specifically, the following tests for fixed effects are considered: (i) a bootstrap-based test, (ii) the Bartlett-corrected usual test, and (iii) an adjusted profile likelihood ratio test. The last test is derived using an approximation to the modified profile likelihood proposed by Barndorff-Nielsen, based on the work of Severini. Bootstrap resampling is performed to numerically construct a Bartlett correction factor for the usual test statistic, and also to obtain a critical value that does not rely on first-order asymptotics. The numerical evidence presented in the paper slightly favors the Bartlett-corrected usual test. An application to real longitudinal data is presented.

Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries

The paper is concerned with finite Hermitian Toeplitz matrices whose entries in the first row grow like a polynomial. Such matrices cannot be viewed as truncations of an infinite Toeplitz matrix which is generated by an integrable function or a nice measure. The main results describe the first-order asymptotics of the extreme eigenvalues as the matrix dimension goes to infinity and also deliver unexpected barriers for the eigenvalues. One purpose of the paper is to popularize once more that questions on the eigenvalues of matrices can be answered in a very elegant way by passing to integral operators. This idea was introduced by Harold Widom about fifty years ago. In this way one can also give an alternative proof to results by William F. Trench on Hermitian Toeplitz matrices with increasing entries.

TheRPackagemetaLikfor Likelihood Inference in Meta-Analysis

Meta-analysis is a statistical method for combining information from different studies about the same issue of interest. Meta-analysis is widely diffuse in medical investigation and more recently it received a growing interest also in social disciplines. Typical applications involve a small number of studies, thus making ordinary inferential methods based on first-order asymptotics unreliable. More accurate results can be obtained by exploiting the theory of higher-order asymptotics. This paper describes the <b>metaLik</b> package which provides an R implementation of higher-order likelihood methods in meta-analysis. The extension to meta-regression is included. Two real data examples are used to illustrate the capabilities of the package.

Optimal Alphabets and Binary Labelings for BICM at Low SNR

Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binary code results in a higher capacity than the binary reflected gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e. constellations that make BICM achieve the Shannon limit $-1.59 \tr{dB}$. It is shown that the $\Eb/N_0$ required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC; if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.

Statistical Inference with Generalized Gini Indices of Inequality, Poverty, and Welfare

This article considers statistical inference for consistent estimators of generalized Gini indices of inequality, poverty, and welfare. Our method does not require grouping the population into a fixed number of quantiles. The empirical indices are shown to be asymptotically normally distributed using functional limit theory. Easily computed asymptotic variance expressions are obtained using influence functions. Inference based on first-order asymptotics is then compared with the grouped method and various bootstrap methods in simulations and with U.S. income data. The bootstrap-t method based on our asymptotic theory is found to have superior size and power properties in small samples.

Universality Limits for Exponential Weights

We establish universality in the bulk for fixed exponential weights on the whole real line. Our methods involve first-order asymptotics for orthogonal polynomials and localization techniques. In particular, we allow exponential weights such as |x|2βg2(x)exp (−2Q(x)), where β>−1/2, Q is convex and Q′′ satisfies some regularity conditions, while g is positive, and has a uniformly continuous and slowly growing or decaying logarithm.

Width and mode of the profile for some random trees of logarithmic height

We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quad trees, plane-oriented ordered trees and other varieties of increasing trees. 1. Introduction. Most random trees in the discrete probability literature have height either of order √ n or of order log n (n being the tree size); see [1]. For simplicity, we call these trees square-root trees and log trees, respectively. Profiles (number of nodes at each level of the tree) of random square-root trees have a rich connection to diverse structures in combinatorics and in probability, and have been extensively studied. In contrast, profiles of random log trees, arising mostly from data structures and computer algorithms, were less addressed and only quite recently were their limit distributions, drastically different from those of square-root trees, better understood; see [3, 12, 13, 21, 27] and the references therein. We study in this paper the asymptotics of width, which is defined to be the size of the most abundant level, and its close connection to the profile. There are many results on first-order asymptotics of profiles for standard log trees, such as binary search trees, random recursive trees, m-ary search trees and quad trees. In some cases, quite accurate asymptotic expressions are known for the expected profile. There is already a paucity of results with regard to

The block bootstrap test of Hausman's exogeneity in the presence of serial correlation

The conventional Hausman's exogeneity test is corrected for models with serial correlation of unknown form. Using simulations we find that the corrected Hausman test based on the block bootstrap outperforms the corrected test based on the first-order asymptotics.

First order asymptotics for the travelling waves in the Gross-Pitaevskii equation

In a previous paper [7], we investigated the asymptotic behaviour of subsonic travelling waves of finite energy for the Gross-Pitaevskii equation in every dimension $N \geq 2$. In particular, we gave their first-order asymptotics in case they were axisymmetric. In the present paper, we compute their first-order asymptotics at infinity in the general case.

Evaluation of methods for interval estimation of model outputs, with application to survival models

When a published statistical model is also distributed as computer software, it will usually be desirable to present the outputs as interval, as well as point, estimates. The present paper compares three methods for approximate interval estimation about a model output, for use when the model form does not permit an exact interval estimate. The methods considered are first-order asymptotics, using second derivatives of the log-likelihood to estimate variance information; higher-order asymptotics based on the signed-root transformation; and the non-parametric bootstrap. The signed-root method is Bayesian, and uses an approximation for posterior moments that has not previously been tested in a real-world application. Use of the three methods is illustrated with reference to a software project arising in medical decision-making, the UKPDS Risk Engine. Intervals from the first-order and signed-root methods are near- identical, and typically 1% wider to 7% narrower than those from the non-parametric bootstrap. The asymptotic methods are markedly faster than the bootstrap method.

Bootstrap critical values for tests based on the smoothed maximum score estimator

The smoothed maximum score estimator of the coefficient vector of a binary response model is consistent and asymptotically normal under weak distributional assumptions. However, the differences between the true and nominal levels of tests based on smoothed maximum score estimates can be very large in finite samples when first-order asymptotics are used to obtain critical values. This paper gives conditions under which the differences between the true and nominal levels can be reduced by using the bootstrap to obtain critical values. A set of Monte Carlo experiments illustrates the numerical performance of the bootstrap.