Page 148 - Elements of Distribution Theory

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134 Parametric Families of Distributions

consider a random variable Y with a normal distribution with mean µ and standard deviation
σ, where −∞ <µ< ∞ and σ> 0. The model function of the model is
1 1 2
p(y; θ) = √ exp − (y − µ) , −∞ < y < ∞,
σ (2π) 2σ 2
where θ = (µ, σ) ∈ R × R .
+

A parameterization of a model is not unique; for example, given a model with parameter
θ, the model may also be parameterized by any one-to-one function of θ. Selection of a
parameterization is arbitrary, although in certain cases a particular parameterization may
be more useful if, for example, it simplifies the interpretation of the results.

Example 5.5 (Poisson distribution). Let X denote a random variable with a Poisson dis-
tribution with mean λ> 0; the model function is therefore given by

x
p(x; λ) = λ exp(−λ)/x!, x = 0, 1,....
Let P denote the set of all such Poisson distributions with mean λ> 0.
If X represents the number of “arrivals” observed in a given unit of time, then λ represents
the mean arrival rate measured in arrivals per unit of time. We could also parameterize the
model in terms of θ = 1/λ, which has the interpretation as the mean time between arrivals.
The set P could be described as the set of all Poisson distributions with θ> 0. The model
function in terms of θ is given by

f (x; θ) = θ −x exp(−1/θ)/x!, x = 0, 1,....

A statistical analysis could be based on either parameterization.

Identifiability
One requirement of a parameterization is that it be identifiable; that is, there must be exactly
one value of θ ∈ corresponding to each element of P. Stated another way, a parame-
terization P(·; θ), θ ∈ ,of P is identifiable if θ 1 = θ 2 implies that P(·; θ 1 ) = P(·; θ 2 ). The
condition for identifiability may also be expressed in terms of the model function; however,
when doing so, it is important to keep in mind that absolutely continuous distributions
whose density functions are equal almost everywhere are, in fact, the same distribution.
For instance, if two density functions differ at only a finite number of points, the densities
represent the same distribution.

Example 5.6 (Binomial distribution with a random index). Let X denote a random vari-
able with a binomial distribution with parameters n and η,0 <η < 1, and suppose that n is
itself a random variable with a Poisson distribution with mean λ> 0. Take θ = (η, λ) with
+
= (0, 1) × R .
Consider a model for X.Given n, X has a binomial distribution so that
n x n−x

Pr(X = x|n) = η (1 − η) , x = 0, 1,..., n.
x