# Lecturer/Senior Lecturer Seminars - Garth Tarr

Title: Quantile based estimation of scale dependence

Abstract: This talk outlines an intuitive, robust and highly efficient scale estimator, $P_{n}$, derived from the difference of two $U$-quantile statistics based on the same kernel as the Hodges-Lehmann estimate of location, $h(x,y) = (x+y)/2$ (Tarr, Müller and Weber, 2012). Through the device proposed by Gnanadesikan and Kettenring (1972) and discussed further in Ma and Genton (2000), we extend the robust scale estimator, $P_{n}$, to robust covariance and autocovariance estimators.

The asymptotic results for $P_{n}$ in the iid setting follow from nesting $P_{n}$ inside the class of generalised $L$-statistics ($GL$-statistics; Serfling, 1984) which encompasses $U$-statistics, $U$-quantile statistics and $L$-statistics and as such contains numerous well established scale estimators. The interquartile range, the difference of two quantiles, fits into the family of $GL$-statistics; as does the standard deviation which can be written as the square root of a $U$-statistic with kernel $h(x, y) = (x-y)^{2}$. The trimmed and Winsorised variance also fall into the class of $GL$-statistics. The robust scale estimator, $Q_{n}$ as introduced by Rousseeuw and Croux (1993), can be represented in terms of a $U$-quantile statistic corresponding to the kernel $h(x, y) = |x-y|$.

The primary advantage of $P_{n}$ is its high efficiency at the Gaussian distribution whilst maintaining desirable robustness and efficiency properties at heavy tailed and contaminated distributions.

This work has potential applications in many fields.  Robust estimators are particularly important given the increasing prevalence of automated statistical methods, for example in the financial industry.  Efficiency results and asymptotics will be discussed under both short and long range dependent processes (Tarr, Weber and Müller, 2015).  A link with robust estimation of sparse precision matrices will also be briefly outlined (Tarr, Müller and Weber, 2016).

Gnanadesikan R and Kettenring JR, (1972). Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 28(1), 81–124.

Ma Y and Genton MG, (2000). Highly robust estimation of the autocovariance function. Journal of Time Series Analysis, 21(6), 663–684.

Rousseeuw PJ and Croux C, (1993). Alternatives to the median absolute deviation. Journal of the American Statistical Association, 88(424), 1273–1283.

Serfling R, (1984). Generalized L-, M-, and R-statistics. The Annals of Statistics, 12(1), 76–86.

Tarr G, Müller S and Weber NC, (2012). A robust scale estimator based on pairwise means. Journal of Nonparametric Statistics, 24(1), 187-199. DOI: 10.1080/10485252.2011.621424

Tarr G, Müller S and Weber NC, (2016). Robust estimation of precision matrices under cellwise contamination. Computational Statistics & Data Analysis, 93, 404-420. DOI: 10.1016/j.csda.2015.02.005

Tarr G, Weber NC and Müller S, (2015). The difference of symmetric quantiles under long range dependence. Statistics & Probability Letters, 98, 144-150. DOI: 10.1016/j.spl.2014.12.022