  Component-wise MCMC algorithms, including Gibbs and conditional
Metropolis-Hastings samplers, are commonly used for sampling from multivariate
probability distributions. A long-standing question regarding Gibbs algorithms
is whether a deterministic-scan (systematic-scan) sampler converges faster than
its random-scan counterpart. We answer this question when the samplers involve
two components by establishing an exact quantitative relationship between the
$L^2$ convergence rates of the two samplers. The relationship shows that the
deterministic-scan sampler converges faster. We also establish qualitative
relations among the convergence rates of two-component Gibbs samplers and some
conditional Metropolis-Hastings variants. For instance, it is shown that if
some two-component conditional Metropolis-Hastings samplers are geometrically
ergodic, then so are the associated Gibbs samplers.
