Theory of Statistics - George Mason University

132 1 Probability Theory
Example 1.34 likelihood ratios
Let f and g be probability densities. Let X1, X2, . . . be an iid sequence of
random variables whose range is within the intersection of the domains of f
and g. Let
n
Yn = g(Xi)/f(Xi). (1.278)
i=1
(This is called a “likelihood ratio” and has applications in statistics. Note
that f(x) and g(x) are likelihoods, as defined in equation (1.18) on page 20,
although the “parameters” are the functions themselves.) Now suppose that
f is the PDF of the Xi. Then {Yn : n = 1, 2, 3, . ..} is a martingale with
respect to {Xn : n = 1, 2, 3, . ..}.
The martingale in Example 1.34 has some remarkable properties. Robbins
(1970) showed that for any ɛ > 1,
Pr(Yn ≥ ɛ for some n ≥ 1) ≤ 1/ɛ. (1.279)
Robbins’s proof of (1.279) is straightforward. Let N be the first n ≥ 1 such
that n i=1 g(Xi) ≥ ɛ n i=1 f(Xi), with N = ∞ if no such n occurs. Also, let
gn(t) = n i=1 g(ti) and fn(t) = n i=1 f(ti).
Pr(Yn ≥ ɛ for some n ≥ 1) = Pr(N < ∞)
∞
= I{n}(N)fn(t)dt
i=1
≤ 1
ɛ
≤ 1
ɛ .
∞
I{n}(N)gn(t)dt
Another important property of the martingale in Example 1.34 is
Yn
You are asked to show this in Exercise 1.83.
i=1
a.s.
→ 0. (1.280)
Example 1.35 Wiener process
If {W(t) : t ∈ [0, ∞[} is a Wiener process, then W 2 (t) − t is a martingale.
(Exercise.)
Example 1.36 A martingale that is not Markovian and a Markov
process that is not a martingale
The Markov property is based on conditional independence of distributions
and the martingale property is based on equality of expectations. Thus it is
easy to construct a martingale that is not a Markov chain beginning with
Theory of Statistics c○2000–2013 James E. Gentle