Stat 5101 Lecture Notes - School of Statistics

3.1. PARAMETRIC FAMILIES OF DISTRIBUTIONS 85variables. We write the density of the random vector X =(X 1 ,...,X n )asf λ (x)=n∏i=1λe −λxi( )n∑= λ n exp −λ x i , x i > 0, i =1,...,n.i=1or, according to taste, we might write the left hand side as f λ (x 1 ,...,x n )orf(x|λ)orf(x 1 ,...,x n |λ).Vector ParameterSimilarly, when we have a vector parameter θ =(θ 1 ,...,θ m ), we write thedensity as f θ (x) orf(x|θ). And, as usual, we are sloppy about whether thereis really one vector parameter or several scalar parameters θ 1 , ..., θ m . Whenwe are thinking in the latter mode, we write f θ1,...,θ m(x) orf(x|θ 1 ,...,θ m ).Example 3.1.3 (The Gamma Distribution).We want to write the gamma distribution (Section B.2.3 in Appendix B) in thenotation of parametric families. The parameter is θ =(α, λ). We write thedensity asf θ (x) =f α,λ (x) =λαΓ(α) xα−1 e −λx , x > 0or if we prefer the other notation we write the left hand side as f(x | θ) orf(x|α, λ).The parameter space of this probability model isΘ={(α, λ) ∈ R 2 : α>0,λ>0}that is, the first quadrant with boundary points excluded.Vector Variable and Vector ParameterAnd, of course, the two preceeding cases can be combined. If we have a vectorrandom variable X =(X 1 ,...,X n ) and a vector parameter θ =(θ 1 ,...,θ m ),we can write write the density as any ofaccording to taste.f θ (x)f(x | θ)f θ1,...,θ m(x 1 ,...,x n )f(x 1 ,...,x n |θ 1 ,...,θ m )