Depth functions as measures of representativeness

Abstract

Data depth provides a natural means to rank multivariate vectors with respect to an underlying multivariate distribution. Most existing depth functions emphasize a centre-outward ordering of data points, which may not provide a useful geometric representation of certain distributional features, such as multimodality, of concern to some statistical applications. Such inadequacy motivates us to develop a device for ranking data points according to their “representativeness” rather than “centrality” with respect to an underlying distribution of interest. Derived essentially from a choice of goodness-of-fit test statistic, our device calls for a new interpretation of “depth” more akin to the concept of density than location. It copes particularly well with multivariate data exhibiting multimodality. In addition to providing depth values for individual data points, depth functions derived from goodness-of-fit tests also extend naturally to provide depth values for subsets of data points, a concept new to the data-depth literature.

Appendix

1.1 Proof of Proposition 1

For any $x\in \mathbb R$, let $r(x)$ satisfy $F(x+r(x))-F(x-r(x))=\gamma$. Differentiation of the latter condition twice with respect to $x$ gives $\{f(x-r(x))+f(x+r(x))\}r'(x)=f(x-r(x))-f(x+r(x))$ and $\{f(x-r(x))+f(x+r(x))\}r''(x)=(1-r'(x))^2f'(x-r(x))-(1+r'(x))^2f'(x+r(x))$. Thus $r(x)$ has a local minimum or local maximum at $x=x_0$ satisfying $r'(x_0)=0$, that is $f(x_0-r(x_0))=f(x_0+r(x_0))$, according as $r''(x_0)>0$ or $<0$ respectively. The proposition then follows by noting that $r''(x_0)=\{2f(x_0-r(x_0))\}^{-1}\{f'(x_0-r(x_0))-f'(x_0+r(x_0))\}$ and that $D(F,x)$ is a decreasing function of $r(x)$.

1.2 Proof of Proposition 2

The unimodality condition ensures that there exists some $w\in \mathbb R$ and $R>0$ such that $F(w+R)-F(w-R)=\gamma$ and $f(w+R)=f(w-R)=k$, say. Then necessarily $f$ increases at $w-R$, decreases at $w+R$ and $f(x)>k$ for all $x\in (w-R,w+R)$. Clearly the depth function (10) is locally maximised at $w$ by Proposition 1. Fix any $x_1>x_2>w$ and let $R_1, R_2>0$ satisfy $F(x_i+R_i)-F(x_i-R_i)=\gamma$, $i=1,2$. Note that $x_2-R_2>w-R$ and $x_2+R_2>w+R$, or otherwise $[x_2-R_2,x_2+R_2]$ either contains or is contained in $[w-R,w+R]$ strictly, contrary to the definition of $R_2$. It follows that $f(x)>f(x_2+R_2)$ for all $x\in [w-R,x_2+R_2)$. Consider two cases: (i) $x_1-R_2>x_2+R_2$, (ii) $x_1-R_2\le x_2+R_2$. Under (i), we have $f(x_1-R_2)<f(x)$ for all $x\in [x_2-R_2,x_2+R_2]$, so that $\int _{x_1-R_2}^{x_1+R_2}f(u)\,du<\int _{x_2-R_2}^{x_2+R_2}f(u)\,du=\gamma$, which implies that $R_1>R_2$. Under (ii), we have $f(x_2+R_2)<f(x)$ for all $x\in [x_2-R_2,x_1-R_2)$, so that

$$\begin{aligned} \int _{x_1-R_2}^{x_1+R_2}f(u)\,du-\int _{x_2-R_2}^{x_2+R_2}f(u)\,du=\int _{x_2+R_2}^{x_1+R_2}f(u)\,du-\int _{x_2-R_2}^{x_1-R_2}f(u)\,du<0. \end{aligned}$$

It follows that $F(x_1+R_2)-F(x_1-R_2)<\gamma$ and hence $R_1>R_2$. Thus we have, under either (i) or (ii), that the depth function at $x_1$ is strictly smaller than that at $x_2$, so that it decreases strictly on $(w,\infty )$. Similar arguments show that it increases strictly on $(-\infty ,w)$.