Extremal behaviour and convergence rates for sample--based geometric quantiles and half space depths

Abstract

We consider the empirical versions of geometric quantile and halfspace depth, and study their extremal behaviour as a function of the sample size. The objective of this study is to establish connection between the rates of convergence and tail behaviour of the corresponding underlying distributions. The intricate interplay between the sample size and the parameter driving the extremal behaviour forms the main result of this analysis. In the process, we also fill certain gaps in the understanding of population versions of geometric quantile and halfspace depth.

Figures (15)

• Figure 1: Samples of size $1000$ are drawn from a Gaussian distribution with mean zero and diagonal covariance matrix $diag(1,100)$. Observe that the convexity present in the underlying sampling distribution is not reflected in the isoquantile contours.
• Figure 2: Representation of the Tukey contours for 6 different depths, considering a sample of $1000$ observations (black points) from a mean zero Gaussian distribution with covariance $\text{diag}(1,100)$.
• Figure 3: Comparing the convergence for different types of tail distributions, using the 2nd order characterisation of geometric quantiles. The $x$-scale is chosen for the $\alpha$-values to look equispaced. Left plot: Number of observations $n=10^5$ is fixed; quantiles are computed corresponding to the index $\alpha\in\{\alpha(k)=1-10^{-k},\,k=1,\cdots,10\}$ and the unit vector $u$ in the direction $(1,1)$. Right plot: Growing sample at each point ( i.e. for each $\alpha_n$); 10% regularly growing sample, from $10^4$ to $10^5$ simulated values ($\alpha_n=1-10^{-n/10^4}$ for $n=10^4,\cdots,10^5$).
• Figure 4: Slowly growing sample at each point ( i.e. for each $\alpha_n$), up to $10^5$ simulated values, choosing $\alpha_n$ satisfying Condition \ref{['eq:cdtionAlpha']} (e.g. $\alpha_n=1-\sqrt{2\log n /n}$), which gives values of $\alpha_n$ going roughly from $0.90$ to $0.99$ (with a partition of 10, i.e. 10 values of $\alpha_n$). Direction at which quantiles are computed is $(1,1)$.
• Figure 5: Plot of $y(\alpha_n)$ in terms of $\alpha_n=1-10^{-n\,10^{-4}}$, $n=10^4k$ for $k=1,2,\cdots,n$, for a bivariate centered Gaussian and Pareto($\delta$) distributions sharing the same covariance matrix. First row: $\delta=2.2$, 2nd row: $\delta=3.2$. On each row, the right plot does a zoom on the last 3 values of $\alpha_n$. The $y$-scale is chosen as $y^{1/2}$.

Theorems & Definitions (27)

• Definition 2.1: Geometric quantile Chaudhury1996
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Definition 2.7: Depth function
• Definition 2.8: Halfspace depth; Tukey1975
• Definition 2.9: Asymptotically elliptically symmetric
• Theorem 2.10
• Proposition 2.11
• Theorem 2.12