Page 284 - Applied Probability

12. Models of Recombination
272
is equivalent to the inequality

d
d
y
¯
¯
ln F ∞ (x + z) dx,
ln F ∞ (x) −
0 ≤
dx
0
¯
d
which is certainly true if dx ln F ∞ (x) is decreasing.
dx
A sufficient condition in turn for
d 2 d 1 − F(x)
¯
ln F ∞ (x) = −
¯
dx 2 dx µF ∞ (x)
f(x) [1 − F(x)] 2
= − (12.19)
¯
2 ¯
µF ∞ (x) µ F ∞ (x) 2
≤ 0
f(x)
to hold is that the hazard rate be increasing in x. If this is the
1−F (x)
case, then in view of equation (12.18), we can average the right-hand side
of the inequality
f(x) f(x + w)
≤
1 − F(x) 1 − F(x + w)
with respect to the probability density
1 − F(x + w) 1 − F(x + w)
=
∞ ¯
"
[1 − F(x + v)]dv µF ∞ (x)
0
to give the bound
f(x) 1 − F(x)
≤ .
¯
1 − F(x) µF ∞ (x)
This last bound implies the log-concavity condition (12.19).
In summary, increasing hazard rate leads to positive interference [14]. For
the particular case of the Poisson-skip process, we can assert considerably
∞
more. Let C now be the class of discrete skip distributions {s n } that
n=1
guarantee positive interference or no interference. Then C satisfies prop-
erties (a) and (b) enumerated for the count-location model. Furthermore,
properties (c) and (d) are replaced by
∞
(e) contains all distributions {s n} n=1 that are positive on some interval,
2
0 elsewhere, and log-concave in the sense that s ≥ s n−1 s n+1 for all
n
n,
(f) contains all distributions concentrated on a single integer or two ad-
jacent integers, all binomial, negative binomial, Poisson, and uniform
distributions, and all shifts of these distributions by positive integers.
Although the proofs of these assertions are not beyond us conceptually, we
refer interested readers to [18] for details.