  A quantile is defined as a value below which random draws from a given
distribution falls with a given probability. In a centralized setting where the
cumulative distribution function (CDF) is unknown, the empirical CDF (ECDF) can
be used to estimate such quantiles after aggregating the data. In a fully
distributed sensor network, however, it is challenging to estimate quantiles.
This is because each sensor node observes local measurement data with limited
storage and data transmission power which make it difficult to obtain the
global ECDF. This paper proposes consensus-based quantile estimation for such a
distributed network. The states of the proposed algorithm are recursively
updated with two-steps at each iteration: one is a local update based on the
measurement data and the current state, and the other is averaging the updated
states with neighboring nodes. We consider the realistic case of communication
links between nodes being corrupted by independent random noise. It is shown
that the estimated state sequence is asymptotically unbiased and converges
toward the sample quantile in the mean-square sense. The two step-size
sequences corresponding to the averaging and local update steps result in a
mixed-time scale algorithm with proper decay rates in order to achieve
convergence. We also provide applications to distributed estimation of trimmed
mean, computation of median, maximum, or minimum values and identification of
outliers through simulation.
