Page 168 - Probability and Statistical Inference

3. Multivariate Random Variables 145

In the contexts of both the one-parameter and multi-parameter exponential
families, there are such notions referred to as the natural parameterization,
the natural parameter space, and the natural exponential family. A serious
discussion of these topics needs substantial mathematical depth beyond the
assumed prerequisites. Elaborate discussions of intricate issues and related
references are included in Chapter 2 of both Lehmann (1983, 1986) and
Lehmann and Casella (1998), as well as Barndorff-Nielson (1978). Let us
again consider some examples.
Example 3.8.7 Let X be distributed as N(µ, σ ), with k = 2, θθ θθ θ = (µ, σ) ∈
2
ℜ × ℜ where µ and σ are both treated as parameters. Then, the correspond-
+
ing pdf has the form given in (3.8.4) where x ∈ ℜ, θ = µ, θ = σ, R (x) = x,
1 2 1
R (x) = x , and , g(x) = 1, ,
2
2
and . !
Example 3.8.8 Let X be distributed as Gamma(α, β) where both α(> 0)
and β (> 0) are treated as parameters. The pdf of X is given by (1.7.20) so
α
that f(x; α, β) = {β Γ(α)} exp(–x/β)x , where we have k = 2, θθ θθ θ = (α, β) ∈
–1
α–1
ℜ × ℜ , x ∈ ℜ . We leave it as an exercise to verify that this pdf is also of the
+
+
+
form given in (3.8.4). !
The regularity conditions stated in (3.8.5) may sound too mathematical.
But, the major consolation is that many standard and useful distributions in
statistics belong to the exponential family and that the mathematical condi-
tions stated in (3.8.5) are routinely satisfied.
3.9 Some Standard Probability Inequalities

In this section, we develop some inequalities which are frequently encoun-
tered in statistics. The introduction to each inequality is followed by a few
examples. These inequalities apply to both discrete and continuous random
variables.

3.9.1 Markov and Bernstein-Chernoff Inequalities

Theorem 3.9.1 (Markov Inequality) Suppose that W is a real valued
random variable such that P(W = 0) = 1 and E(W) is finite. Then, for any
fixed δ (> 0), one has:

Proof Suppose that the event A stands for the set [W ≥ δ], and then