  A binary renewal process is a stochastic process $\{X_n\}$ taking values in
$\{0,1\}$ where the lengths of the runs of 1's between successive zeros are
independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the
future behavior, and the problem of universal estimators is to do so without
any prior knowledge of the distribution. We prove a variety of results of this
type, including universal estimates for the expected time to renewal as well as
estimates for the conditional distribution of the time to renewal. Some of our
results require a moment condition on the time to renewal and we show by an
explicit construction how some moment condition is necessary.
