how that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameterregular exponential family, stating what the canonical parameter isequal to in terms of θ.
Question
how that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameterregular exponential family, stating what the canonical parameter isequal to in terms of θ.
Solution
The two-parameter regular exponential family is a class of probability distributions that can be expressed in the form:
f(x; θ, τ) = exp{ (xθ - b(θ))/a(τ) + c(x, τ) }
where θ is the canonical parameter, τ is a dispersion parameter, and a(τ), b(θ), and c(x, τ) are known functions.
Let's consider a random sample X1, ..., Xn from a distribution with probability density function (p.d.f.) f(x; θ). The joint p.d.f. of X1, ..., Xn is given by the product of the individual p.d.f.s, since the X's are independent:
f(x1, ..., xn; θ) = ∏_{i=1}^{n} f(xi; θ)
If each f(xi; θ) belongs to the two-parameter regular exponential family, then we can write:
f(xi; θ) = exp{ (xiθ - b(θ))/a(τ) + c(xi, τ) }
Substituting this into the expression for the joint p.d.f., we get:
f(x1, ..., xn; θ) = ∏_{i=1}^{n} exp{ (xiθ - b(θ))/a(τ) + c(xi, τ) }
This can be simplified to:
f(x1, ..., xn; θ) = exp{ (∑{i=1}^{n} xiθ - nb(θ))/a(τ) + ∑{i=1}^{n} c(xi, τ) }
This is again in the form of a two-parameter regular exponential family distribution, with canonical parameter θ. The functions a(τ), b(θ), and c(x, τ) may be different from those in the individual p.d.f.s, but the form of the distribution is preserved.
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