  We consider the class of all the Hermite processes $(Z_{t}^{(q,H)})_{t\in
\lbrack 0,1]}$ of order $q\in \mathbf{N}^{\ast}$ and with Hurst parameter $%
H\in (\frac{1}{2},1)$. The process $Z^{(q,H)}$ is $H$-selfsimilar, it has
stationary increments and it exhibits long-range dependence identical to that
of fractional Brownian motion (fBm). For $q=1$, $Z^{(1,H)}$ is fBm, which is
Gaussian; for $q=2$, $Z^{(2,H)}$ is the Rosenblatt process, which lives in the
second Wiener chaos; for any $q>2$, $Z^{(q,H)}$ is a process in the $q$th
Wiener chaos. We study the variations of $Z^{(q,H)}$ for any $q$, by using
multiple Wiener -It\^{o} stochastic integrals and Malliavin calculus. We prove
a reproduction property for this class of processes in the sense that the terms
appearing in the chaotic decomposition of their variations give rise to other
Hermite processes of different orders and with different Hurst parameters. We
apply our results to construct a strongly consistent estimator for the
self-similarity parameter $H$ from discrete observations of $Z^{(q,H)}$; the
asymptotics of this estimator, after appropriate normalization, are proved to
be distributed like a Rosenblatt random variable (value at time $1$ of a
Rosenblatt process).with self-similarity parameter $1+2(H-1)/q$.
