Is there any relation between hoop stress and compressive (or tensile) stress?
Specifically, can hoop stress be derived from the aforementioned stresses?
Yes- Hoop stress can be either tensile or compressive, depending on the load (internal or external pressure). But your question is far too vague to get any more specific than that.
sotree ,
If I uderstand you well, you are asking of axial tensile strength and hoop(circumferential) stress in a cylendrical vessel under internal pressure, If it is so, a simple relationshinp exist-