Core Rules 2e pag. 177
"...An initiative check is, effectively, a Reflex save, so roll 1d20 and add the character’s Refl ex save bonus (his base Reflex save bonus from classes, plus his Dexterity modifier, plus any other factors like feats that affect his Reflex save)..."
It's true that it's not specified which kind of bonus they are, but I feel that Improved Initiatice and Lighting Reflexes should add one with the other since they are kind of different bonuses (one adds only to Initiativem the other one to all Ref saves):
Pag.130:
On Improved Initiave it says: "You get a +4 bonus on initiative checks."
Pag.133
On Lighting Reflexes it says: You get a +2 bonus on all Reflex saving throws.