Hypothesis: Therapy with PAP and supplemental oxygen will reverse neurocognitive deficits in one or more domains, reduce sleepiness, and improve QoL compared with CPAP alone in patients with moderate-to-severe OSA and concomitant moderate COPD. This aim will evaluate which specific deficits in neurocognitive function in patients with moderate-to-severe Overlap Syndrome are reversible and most sensitive to the effects of positive airway pressure (PAP) and oxygen vs CPAP alone.