  Empirical distributions have their in-sample maxima as natural censoring. We
look at the "hidden tail", that is, the part of the distribution in excess of
the maximum for a sample size of $n$. Using extreme value theory, we examine
the properties of the hidden tail and calculate its moments of order $p$. The
method is useful in showing how large a bias one can expect, for a given $n$,
between the visible in-sample mean and the true statistical mean (or higher
moments), which is considerable for $\alpha$ close to 1. Among other
properties, we note that the "hidden" moment of order $0$, that is, the
exceedance probability for power law distributions, follows an exponential
distribution and has for expectation $\frac{1}{n}$ regardless of the
parametrization of the scale and tail index.
