  In hypothesis testing problems, taking samples sequentially and stopping
opportunistically to make the inference greatly enhances the reliability. The
design of the stopping and inference policy, however, critically relies on the
knowledge of the underlying distribution of each hypothesis. When the knowledge
of distributions, say, $P_0$ and $P_1$ in the binary-hypothesis case, is
replaced by empirically observed statistics from the respective distributions,
the gain of sequentiality is less understood when subject to universality
constraints. In this work, the gap is mended by a unified study on
sequentiality in the universal binary classification problem. We propose a
unified framework where the universality constraints are set on the expected
stopping time as well as the type-I error exponent. The type-I error exponent
is required to achieve a pre-set distribution-dependent constraint
$\lambda(P_0,P_1)$ for all $P_0,P_1$. The framework is employed to investigate
a semi-sequential and a fully-sequential setup, so that fair comparison can be
made with the fixed-length setup. The optimal type-II error exponents in
different setups are characterized when the function $\lambda$ satisfies mild
continuity conditions. The benefit of sequentiality is shown by comparing the
semi-sequential, the fully-sequential, and the fixed-length cases in
representative examples of $\lambda$. Conditions under which sequentiality
eradicates the trade-off between error exponents are also derived.
