Myriam Elizabeth Saavedra López - Repositorio Digital USFQ ...

Regarding that min (d (x, X) , d (x, ∂W )) = min (d (x, X W ) , d (x, ∂W )) (Baddeley and Gill, 1997,
36), then it can indeed be observed that min (d (x, X) , d (x, ∂W ))
and I {d (x, X) ≤ d (x, ∂W )} for each x ∈ W . Then the set {x ∈ W : min (d (x, X) , d (x, ∂W )) ≥ r}
can be thought of as the set of points “at
risk of failure at distance r” and {x ∈ W : d (x, X) = r, d (x, ∂W ) ≥ r} are the “observed failures at
distance r”.
Geometrically, these two sets are the closures of:
{x ∈ W : d (x, W c ) > r} \ x ∈ R d : d (x, X) ≤ r and
∂ x ∈ R d : d (x, X) ≤ r {x ∈ W : d (x, W c ) > r} respectively.
The following panel illustrated the analogy with survival times:
Now, let X be an almost surely stationary point process and W be defined in the d-dimensional
Euclidean space, a fixed compact set. Based on the data X W , the Kaplan-Meier estimator ˆ F of
the Empty Space Function F of this point process, is defined by:
where ˆ Λ (r) is defined by:
ˆΛ (r) =
ˆ r
0
ˆF = 1 − r π 0
1 − dˆ
Λ (s) = 1 − exp −ˆ
Λ (r)
18
(23)
d c ∂ x ∈ R : d (x, X) ≤ s {x ∈ W : d (x, W ) > s} k−1
ds (24)
|{x ∈ W : d (x, W c ) > s} \ {x ∈ R d : d (x, X) ≤ s}| k
where |.| k−1 denotes k − 1 dimensional Hausdorff measure (surface area).