arXiv subjects

Hailin Sang

Publications and source records attributed to Hailin Sang.

29 records · Page 2Linked to original sources

arXiv2016

Gini Covariance Matrix and its Affine Equivariant Version

We propose a new covariance matrix called Gini covariance matrix (GCM), which is a natural generalization of univariate Gini mean difference (GMD) to the multivariate case. The extension is based on the covariance representation of GMD by applying the multivariate spatial rank function. We study properties of GCM, especially in the elliptical distribution family. In order to gain the affine equivariance property for GCM, we utilize the transformation-retransformation (TR) technique and obtain an affine equivariant version GCM that turns out to be a symmetrized M-functional. The influence function of those two GCM's are obtained and their estimation has been presented. Asymptotic results of estimators have been established. A closely related scatter Kotz functional and its estimator are also explored. Finally, asymptotical efficiency and finite sample efficiency of the TR version GCM are compared with those of sample covariance matrix, Tyler-M estimator and other scatter estimators under different distributions.

stat.ME↗

arXiv2016

Adjusting for Misclassification: A Three-Phase Sampling Approach

The United States Department of Agriculture's National Agricultural Statistics Service (NASS) conducts the June Agricultural Survey (JAS) annually. Substantial misclassification occurs during the pre-screening process and from field-estimating farm status for non-response and inaccessible records, resulting in a biased estimate of the number of US farms from the JAS. Here the Annual Land Utilization Survey (ALUS) is proposed as a follow-on survey to the JAS to adjust the estimates of the number of US farms and other important variables. A three-phase survey design-based estimator is developed for the JAS-ALUS with non-response adjustment for the second phase (ALUS). A design-unbiased estimator of the variance is provided in explicit form.

stat.ME↗

arXiv2016

Symmetric Gini Covariance and Correlation

Standard Gini covariance and Gini correlation play important roles in measuring the dependence of random variables with heavy tails. However, the asymmetry brings a substantial difficulty in interpretation. In this paper, we propose a symmetric Gini-type covariance and a symmetric Gini correlation ($ρ_g$) based on the joint rank function. The proposed correlation $ρ_g$ is more robust than the Pearson correlation but less robust than the Kendall's $τ$ correlation. We establish the relationship between $ρ_g$ and the linear correlation $ρ$ for a class of random vectors in the family of elliptical distributions, which allows us to estimate $ρ$ based on estimation of $ρ_g$. The asymptotic normality of the resulting estimators of $ρ$ are studied through two approaches: one from influence function and the other from U-statistics and the delta method. We compare asymptotic efficiencies of linear correlation estimators based on the symmetric Gini, regular Gini, Pearson and Kendall's $τ$ under various distributions. In addition to reasonably balancing between robustness and efficiency, the proposed measure $ρ_g$ demonstrates superior finite sample performance, which makes it attractive in applications.

stat.ME↗

arXiv2016

Memory properties of transformations of linear processes

In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (2002) studied the polynomial transformations of Gaussian FARIMA(0,d,0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (1996, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p,d,q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.

math.ST↗

arXiv2015

On kernel estimators of density for reversible Markov chains

In this paper we investigate the kernel estimator of the density for a stationary reversible Markov chain. The proofs are based on a new central limit theorem for a triangular array of reversible Markov chains obtained under conditions imposed to covariances, which has interest in itself.

math.PR↗

arXiv2013

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive (AR) processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator (PCMLE) satisfies a strong consistency, $O_P(N^{-1/2})$ consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator (LASSO) and smoothly clipped average deviation (SCAD), are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations (see \cite{Resnick}). A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.

stat.ME↗

arXiv2013

Exact Moderate and Large Deviations for Linear Processes

Large and moderate deviation probabilities play an important role in many applied areas, such as insurance and risk analysis. This paper studies the exact moderate and large deviation asymptotics in non-logarithmic form for linear processes with independent innovations. The linear processes we analyze are general and therefore they include the long memory case. We give an asymptotic representation for probability of the tail of the normalized sums and specify the zones in which it can be approximated either by a standard normal distribution or by the marginal distribution of the innovation process. The results are then applied to regression estimates, moving averages, fractionally integrated processes, linear processes with regularly varying exponents and functions of linear processes. We also consider the computation of value at risk and expected shortfall, fundamental quantities in risk theory and finance.

math.ST↗

arXiv2011

Asymptotic Properties of Self-Normalized Linear Processes with Long Memory

In this paper we study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moment. For the sake of applications we derive the self-normalized version of this theorem. The study is motivated by models arising in economical applications where often the linear processes have long memory, and the innovations have heavy tails.

stat.ME↗

arXiv2011

Central Limit Theorem for Linear Processes with Infinite Variance

This paper addresses the following classical question: giving a sequence of identically distributed random variables in the domain of attraction of a normal law, does the associated linear process satisfy the central limit theorem? We study the question for several classes of dependent random variables. For independent and identically distributed random variables we show that the central limit theorem for the linear process is equivalent to the fact that the variables are in the domain of attraction of a normal law, answering in this way an open problem in the literature. The study is also motivated by models arising in economic applications where often the innovations have infinite variance, coefficients are not absolutely summable, and the innovations are dependent.

math.PR↗

arXiv2010

On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm

It is shown that the variable bandwidth density estimator proposed by McKay (1993a and b) following earlier findings by Abramson (1982) approximates density functions in $C^4(\mathbb R^d)$ at the minimax rate in the supremum norm over bounded sets where the preliminary density estimates on which they are based are bounded away from zero. A somewhat more complicated estimator proposed by Jones McKay and Hu (1994) to approximate densities in $C^6(\mathbb R)$ is also shown to attain minimax rates in sup norm over the same kind of sets. These estimators are strict probability densities.

math.ST↗

arXiv2010

Uniform asymptotics for kernel density estimators with variable bandwidths

It is shown that the Hall, Hu and Marron [Hall, P., Hu, T., and Marron J.S. (1995), Improved Variable Window Kernel Estimates of Probability Densities, {\it Annals of Statistics}, 23, 1--10] modification of Abramson's [Abramson, I. (1982), On Bandwidth Variation in Kernel Estimates - A Square-root Law, {\it Annals of Statistics}, 10, 1217--1223] variable bandwidth kernel density estimator satisfies the optimal asymptotic properties for estimating densities with four uniformly continuous derivatives, uniformly on bounded sets where the preliminary estimator of the density is bounded away from zero.

math.ST↗