Robust Estimation and Inference for Expected Shortfall Regression with Many Regressors

Expected Shortfall (ES), also known as superquantile or Conditional Value-at-Risk, has been recognized as an important measure in risk analysis and stochastic optimization, and is also finding applications beyond these areas. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a recently proposed joint regression framework that simultaneously models the quantile and the ES of a response variable given a set of covariates, for which the state-of-the-art approach is based on minimizing a joint loss function that is non-differentiable and non-convex. This inevitably raises numerical challenges and limits its applicability for analyzing large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity with respect to nuisance parameters, we propose a statistically robust (to highly skewed and heavy-tailed data) and computationally efficient two-step procedure for fitting joint quantile and ES regression models. With increasing covariate dimensions, we establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors, which lay the foundation for statistical inference. Finally, we demonstrate through numerical experiments and two data applications that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression

Higher order asymptotic theory for semiparametric averaged derivatives.

This thesis investigates higher order asymptotic properties of a semiparametric averaged derivative estimator. Classical parametric models assume that we know the distribution function of random variables of interest up to finite dimensional parameters, while nonparametric models do not assume this knowledge. Parametric estimators typically enjoy - consistency and asymptotic normality under certain conditions, while nonparametric estimators converge to the true functionals of interest slower than parametric ones. Semiparametric estimators, a compromise between the two, have been intensively studied since the 1970s. Some of them have been shown to have the same convergence rate as parametric estimators despite involving nonparametric functional estimates. Semiparametric methods often suit econometrics because economic theory typically does not provide the whole information on economic variables which parametric methods require, and a sample of very large size is rarely available in econometrics. This thesis treats a semiparametric averaged derivative estimator of single index models. Its first order asymptotic theory has been studied since late 1980s. It has been shown to be n-consistent and asymptotically normally distributed under certain regularity conditions despite involving a nonparametric density estimate. However its higher order properties could be affected by the property of nonparametric estimates. We obtain valid Edgeworth expansions for both normalized and studentized estimators, and moreover show the bootstrap distribution approximates the exact distribution of the estimator asymptotically as well as the Edgeworth expansion for the normalized statistics. We propose optimal bandwidth choices which minimize the normal approximation error using the expansion. We also examine the finite sample performance of the Edgeworth expansions by a Monte Carlo study

Asymptotic properties of Bernstein estimators on the simplex

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when $d = 2$. The simplex is an important case as it is the natural domain of compositional data. In this paper, we make an effort to prove several asymptotic results (bias, variance, mean squared error (MSE), mean integrated squared error (MISE), asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the $d$-dimensional simplex. Our results generalize the ones in Leblanc (2012) and Babu et al. (2002), who treated the case $d = 1$, and significantly extend those found in Tenbusch (1994). In particular, our rates of convergence for the MSE and MISE are optimal.Comment: 22 pages, 1 figur

Studentization in Edgworth Expansions for Estimates of Semiparametric Index Models - (Now published in C Hsiao, K Morimune and J Powell (eds): Nonlinear Statistical Modeling (Festschrift for Takeshi Amemiya), (Cambridge University Press, 2001), pp.197-240.)

We establish valid theoretical and empirical Edgeworth expansions for density-weighted averaged derivative estimates of semiparametric index models.Edgeworth expansions, semiparametric estimates, averaged derivatives

Partially linear models

In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an up-to-date presentation of the state of the art of partially linear regression techniques. The emphasis of this monograph is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically efficient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models. We hope that this monograph will serve as a useful reference for theoretical and applied statisticians and to graduate students and others who are interested in the area of partially linear regression. While advanced mathematical ideas have been valuable in some of the theoretical development, the methodological power of partially linear regression can be demonstrated and discussed without advanced mathematics. This monograph can be divided into three parts: part one–Chapter 1 through Chapter 4; part two–Chapter 5; and part three–Chapter 6. In the first part, we discuss various estimators for partially linear regression models, establish theo- retical results for the estimators, propose estimation procedures, and implement the proposed estimation procedures through real and simulated examples. The second part is of more theoretical interest. In this part, we construct several adaptive and efficient estimates for the parametric component. We show that the LS estimator of the parametric component can be modified to have both Bahadur asymptotic efficiency and second order asymptotic efficiency. In the third part, we consider partially linear time series models. First, we propose a test procedure to determine whether a partially linear model can be used to fit a given set of data. Asymptotic test criteria and power investigations are presented. Second, we propose a Cross-Validation (CV) based criterion to select the optimum linear subset from a partially linear regression and estab- lish a CV selection criterion for the bandwidth involved in the nonparametric kernel estimation. The CV selection criterion can be applied to the case where the observations fitted by the partially linear model (1.1.1) are independent and identically distributed (i.i.d.). Due to this reason, we have not provided a sepa- rate chapter to discuss the selection problem for the i.i.d. case. Third, we provide recent developments in nonparametric and semiparametric time series regression. This work of the authors was supported partially by the Sonderforschungs- bereich373“QuantifikationundSimulationO ̈konomischerProzesse”.Thesecond author was also supported by the National Natural Science Foundation of China and an Alexander von Humboldt Fellowship at the Humboldt University, while the third author was also supported by the Australian Research Council. The second and third authors would like to thank their teachers: Professors Raymond Car- roll, Guijing Chen, Xiru Chen, Ping Cheng and Lincheng Zhao for their valuable inspiration on the two authors’ research efforts. We would like to express our sin- cere thanks to our colleagues and collaborators for many helpful discussions and stimulating collaborations, in particular, Vo Anh, Shengyan Hong, Enno Mam- men, Howell Tong, Axel Werwatz and Rodney Wolff. For various ways in which they helped us, we would like to thank Adrian Baddeley, Rong Chen, Anthony Pettitt, Maxwell King, Michael Schimek, George Seber, Alastair Scott, Naisyin Wang, Qiwei Yao, Lijian Yang and Lixing Zhu. The authors are grateful to everyone who has encouraged and supported us to finish this undertaking. Any remaining errors are ours

Regenerative block empirical likelihood for Markov chains

Empirical likelihood is a powerful semi-parametric method increasingly investigated in the literature. However, most authors essentially focus on an i.i.d. setting. In the case of dependent data, the classical empirical likelihood method cannot be directly applied on the data but rather on blocks of consecutive data catching the dependence structure. Generalization of empirical likelihood based on the construction of blocks of increasing nonrandom length have been proposed for time series satisfying mixing conditions. Following some recent developments in the bootstrap literature, we propose a generalization for a large class of Markov chains, based on small blocks of various lengths. Our approach makes use of the regenerative structure of Markov chains, which allows us to construct blocks which are almost independent (independent in the atomic case). We obtain the asymptotic validity of the method for positive recurrent Markov chains and present some simulation results