Page 150 - Elements of Distribution Theory

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052184472Xc05 CUNY148/Severini May 24, 2005 17:53

136 Parametric Families of Distributions

function is then given by
n
1 1 2 n
p(x; θ) = n exp − (x j − βt j ) , x = (x 1 ,..., x n ) ∈ R ;
n 2
σ (2π) 2 2σ
j=1
here θ = (β, σ) and = R × R .
+
Likelihood ratios
Likelihood ratios play an important role in statistical inference. Consider a parametric model
for a random variable X with model function p(·; θ), θ ∈ .A function of X of the form
p(X; θ 1 )
,
p(X; θ 0 )
where θ 0 ,θ 1 ∈ ,is called a likelihood ratio. For cases in which the true parameter value
θ is unknown, the ratio p(x; θ 1 )/p(x; θ 0 ) may be used as a measure of the strength of the
evidence supporting θ = θ 1 versus θ = θ 0 , based on the observation of X = x.
Note that
p(X; θ 1 )

E ; θ 0 = 1
p(X; θ 0 )
for all θ 1 ∈ .To see this, suppose that X has an absolutely continuous distribution with
density function p(·; θ 0 ). Then
p(X; θ 1 ) p(x; θ 1 )

∞
E ; θ 0 = p(x; θ 0 ) dx
p(X; θ 0 ) −∞ p(x; θ 0 )

∞
= p(x; θ 1 ) dx = 1.
−∞
A similar result holds for frequency functions.
Another important property of likelihood ratios is given in the following example.

Example 5.8 (Martingale property of likelihood ratios). Consider a sequence of real-
valued random variables Y 1 , Y 2 ,... ;in particular, we are interested in the case in which
Y 1 , Y 2 ,... are not independent, although the following analysis also applies in the case
of independence. Suppose that, for each n = 1, 2,..., distribution of Y 1 , Y 2 ,..., Y n is
absolutely continuous with density function p n (·; θ) where θ is a parameter taking values
in a set .We assume that for each θ ∈ the density functions p n (·; θ), n = 1, 2,..., are
consistent in the following sense. Fix n.For any m < n the marginal density of (Y 1 ,..., Y m )
based on p n (·; θ)is equal to p m (·; θ). That is,

∞ ∞
p m (y 1 ,..., y m ; θ) = ··· p n (y 1 ,..., y m , y m+1 ,..., y n ; θ) dy m+1 ··· dy n ,θ ∈ .
−∞ −∞
Let θ 0 and θ 1 denote distinct elements of and define
p n (Y 1 ,..., Y n ; θ 1 )
X n = , n = 1, 2,...
p n (Y 1 ,..., Y n ; θ 0 )
where Y 1 , Y 2 ,... are distributed according to the distribution with parameter θ 0 . Note that
the event p n (Y 1 ,..., Y n ; θ 0 ) = 0 has probability 0 and, hence, may be ignored.