
92 Estimation of the rate of mortality
and smaller, using more and more intricate methods to extract “local
information” in the neighborhood of x.
Let us present a general way of doing this. Starting with a function
K(z), such that
Z
K(z)dz = 1, (8.8)
transfer it into a kernel
1
∆
K
x −y
∆
.
This kernel is now a function of two variables, x and y, and for any x,
as a function of y it is more and more “concentrated” around x as ∆
becomes smaller.
Now consider the integral
e
f
n
(x) =
1
∆
Z
K
x −y
∆
d
b
F
n
(y) =
1
n∆
n
∑
i=1
K
x −T
i
∆
as an estimator of a density f (x) at point x. For example, if K(z) =
I
{−1≤z≤0}
, then
1
∆
K
x −y
∆
=
1
∆
I
{x≤y≤x+∆}
and
e
f
n
(x) =
b
f
n
(x),
and if K(z) = I
{−