
Patterns of Several Types of Points 601
This is not related to the functions G
i j
(r) in a simple way. The corresponding version of the J-
function is
J
i•
(r) =
1 −G
i•
1 −F(r)
. (14.49)
Under the null hyp othesis of inde pendent components (MPP4) ,
G
i•
(r) = 1 −(1 −G
ii
(r))
∏
j6=i
(1 −F
j
(r)) (14.50)
J
i•
(r) = J
ii
(r). (14.51)
Under completely random labelling (MPP2), a typical point of type i is just a typical point of X
(•)
,
so that
G
i•
(r) = G
••
(r) (14.52)
J
i•
(r) = J
••
(r), (14.53)
the G- and J-func tions of the unmarked process X
(•)
.
The functions G
i•
(r) and J
i•
(r) are computed by Gdot and Jdot, respectively. The command
alltypes can be used with them, and produces ...