  A two-terminal distributed binary hypothesis testing problem over a noisy
channel is studied. The two terminals, called the observer and the decision
maker, each has access to $n$ independent and identically distributed samples,
denoted by $\mathbf{U}$ and $\mathbf{V}$, respectively. The observer
communicates to the decision maker over a discrete memoryless channel, and the
decision maker performs a binary hypothesis test on the joint probability
distribution of $(\mathbf{U},\mathbf{V})$ based on $\mathbf{V}$ and the noisy
information received from the observer. The trade-off between the exponents of
the type I and type II error probabilities is investigated. Two inner bounds
are obtained, one using a separation-based scheme that involves type-based
compression and unequal error-protection channel coding, and the other using a
joint scheme that incorporates type-based hybrid coding. The separation-based
scheme is shown to recover the inner bound obtained by Han and Kobayashi for
the special case of a rate-limited noiseless channel, and also the one obtained
by the authors previously for a corner point of the trade-off. Finally, we show
via an example that the joint scheme achieves a strictly tighter bound than the
separation-based scheme for some points of the error-exponents trade-off.
