  Estimating ranks, quantiles, and distributions over streaming data is a
central task in data analysis and monitoring. Given a stream of $n$ items from
a data universe equipped with a total order, the task is to compute a sketch
(data structure) of size polylogarithmic in $n$. Given the sketch and a query
item $y$, one should be able to approximate its rank in the stream, i.e., the
number of stream elements smaller than or equal to $y$.
  Most works to date focused on additive $\varepsilon n$ error approximation,
culminating in the KLL sketch that achieved optimal asymptotic behavior. This
paper investigates multiplicative $(1\pm\varepsilon)$-error approximations to
the rank. Practical motivation for multiplicative error stems from demands to
understand the tails of distributions, and hence for sketches to be more
accurate near extreme values.
  The most space-efficient algorithms due to prior work store either
$O(\log(\varepsilon^2 n)/\varepsilon^2)$ or $O(\log^3(\varepsilon
n)/\varepsilon)$ universe items. We present a randomized sketch storing
$O(\log^{1.5}(\varepsilon n)/\varepsilon)$ items that can
$(1\pm\varepsilon)$-approximate the rank of each universe item with high
constant probability; this space bound is within an $O(\sqrt{\log(\varepsilon
n)})$ factor of optimal. Our algorithm does not require prior knowledge of the
stream length and is fully mergeable, rendering it suitable for parallel and
distributed computing environments.
