Page 146 - Elements of Distribution Theory

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5

Parametric Families of Distributions

5.1 Introduction
Statistical inference proceeds by modeling data as the observed values of certain random
variables; these observed values are then used to draw conclusions about the process that
generated the data. Let Y denote a random variable with probability distribution P. The
function P is typically unknown and the goal is to draw conclusions about P on the basis of
observing Y = y. The starting point for such an analysis is generally the specification of a
model for the data. A model consists of a set of possible distributions P such that we are
willing to proceed as if P is an element of P.
Thus, in addition to the properties of the individual distributions in P, the properties of
the family itself are of interest; it is these properties that we consider in this chapter.

5.2 Parameters and Identifiability
Consider a family P of probability distributions. A parameterization of P is a mapping
from a parameter space to the set P so that P may be represented

P ={P(·; θ): θ ∈ }.

Hence, corresponding to any statement regarding the elements P of P is an equivalent
statement regarding the elements θ of .

Example 5.1 (Normal distributions). Let P denote the set of all normal distributions with
finite mean and nonnegative variance. For θ = (µ, σ), let P(·; θ)be the normal distribution
with mean µ and standard deviation σ and take = R × R . Then P may be written
+
P ={P(·; θ): θ ∈ }.

Let P 0 denote the subset of P consisting of those normal distributions with mean 0. Then
P 0 consists of those elements of P of the form P(·; θ) with θ = (0,σ), σ> 0.

Example 5.2 (Distributions with median 0). Let P denote the set of all probability distri-
butions on the real line such that 0 is a median of the distribution. An element P of P is

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