Halfspace depth and $\beta$-skeleton depth are two types of depth functions
in nonparametric data analysis. The halfspace depth of a query point $q\in
\mathbb{R}^d$ with respect to $S\subset\mathbb{R}^d$ is the minimum portion of
the elements of $S$ which are contained in a halfspace which passes through
$q$. For $\beta \geq 1$, the $\beta$-skeleton depth of $q$ with respect to $S$
is defined to be the total number of \emph{$\beta$-skeleton influence regions}
that contain $q$, where each of these influence regions is the intersection of
two hyperballs obtained from a pair of points in $S$. The $\beta$-skeleton
depth introduces a family of depth functions that contain \emph{spherical
depth} and \emph{lens depth} if $\beta=1$ and $\beta=2$, respectively. The main
results of this paper include approximating the planar halfspace depth and
$\beta$-skeleton depth using two different approximation methods. First, the
halfspace depth is approximated by the $\beta$-skeleton depth values. For this
method, two dissimilarity measures based on the concepts of \emph{fitting
function} and \emph{Hamming distance} are defined to train the halfspace depth
function by the $\beta$-skeleton depth values obtaining from a given data set.
The goodness of this approximation is measured by a function of error values.
Secondly, computing the planar $\beta$-skeleton depth is reduced to a
combination of some range counting problems. Using existing results on range
counting approximations, the planar $\beta$-skeleton depth of a query point is
approximated in $O(n\;poly(1/\varepsilon,\log n))$, $\beta\geq 1$. Regarding
the $\beta$-skeleton depth functions, it is also proved that this family of
depth functions converge when $\beta \to \infty$. Finally, some experimental
results are provided to support the proposed method of approximation and
convergence of $\beta$-skeleton depth functions.