Estimation of High Dimensional Mean Regression in the Absence of Symmetry and Light Tail Assumptions

Abstract

Data subject to heavy-tailed errors are commonly encountered in various scientific fields. To address this problem, procedures based on quantile regression and Least Absolute Deviation (LAD) regression have been developed in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions, especially when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultra-high dimensional setting with existence of only the second moment? To solve this problem, we propose a penalized Huber loss with diverging parameter to reduce biases created by the traditional Huber loss. Such a penalized robust approximate quadratic (RA-quadratic) loss will be called RA-Lasso. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, our results reveal that the RA-lasso estimator produces a consistent estimator at the same rate as the optimal rate under the light-tail situation. We further study the computational convergence of RA-Lasso and show that the composite gradient descent algorithm indeed produces a solution that admits the same optimal rate after sufficient iterations. As a byproduct, we also establish the concentration inequality for estimating population mean when there exists only the second moment. We compare RA-Lasso with other regularized robust estimators based on quantile regression and LAD regression. Extensive simulation studies demonstrate the satisfactory finite-sample performance of RA-Lasso.

Large covariance estimation through elliptical factor models

“…For example, to estimate the stock network, it is more meaningful to take out the common market factors from the return data and study the conditional independence relationships between idiosyncratic components; in genomics, the conditional independence graph after taking the confounding factors such as age and environment exposure are of better interest. The factors can also be interpreted as covariates to be adjusted before analyzing the correlatedness of the residual part (Fan et al, 2016). Rothman, Levina and Zhu (2010) and Cai et al (2013) adopted the same idea of adjusting the factors in predicting asset returns and analyzing genomics data, but they do not impose the pervasiveness condition, instead they need to impose the constraint of a sparse factor loading matrix B .…”

“…In addition, Hampel (1974); Rousseeuw and Croux (1993); Koenker (2005) considered the problem from a quantile perspective. In this section, we introduce two M-estimators proposed by Fan, Li and Wang (2016) and Catoni (2012). The methods allow asymmetric distributions, and thus are also useful for robust estimation of variances.…”

“…The result is very useful for high-dimensional problems. Fan, Li and Wang (2016) studied the same problem with a simpler Huber influence function. The ideas of Catoni (2012) and Fan, Li and Wang (2016) can also be used to estimate variances for heavy-tailed data.…”

“…Fan, Li and Wang (2016) studied the same problem with a simpler Huber influence function. The ideas of Catoni (2012) and Fan, Li and Wang (2016) can also be used to estimate variances for heavy-tailed data. Another approach for robust variance estimation is to use quantile estimator, such as median absolute deviation (Hampel, 1974) and the Q n estimator (Rousseeuw and Croux, 1993).…”

Robust Inference of Risks of Large Portfolios

“…In practice, we can use, for example, the average version to estimate the scaling factor. We could also replace the sample standard deviation estimator by the Catoni's M-estimator (Catoni, 2012) or less biased robust approximate quadratic (RA-quadratic) estimator (Fan et al, 2016), which are appealing alternatives in handling heavy-tailed data.…”