Stat 5101 Lecture Notes - School of Statistics

Chapter 4Parametric Families ofDistributionsThe first thing the reader should do before reading the rest of this chapteris go back and review Section 3.1, since that establishes the basic notation forparametric families of distributions.4.1 Location-Scale FamiliesConsider a probability density f of a real-valued random variable X. By thetheorem on linear changes of variables (Theorem 7 of Chapter 3 in Lindgren),for any real number µ and any positive real number σ, the random variableY = µ + σX has the densityf µ,σ (y) = 1 σ f ( y−µσThis generates a two-parameter family of densities called the location-scale familygenerated by the reference density f. The parameter µ is called the locationparameter, and the parameter σ is called the scale parameter.We could choose any distribution in the family as the reference distributionwith density f. This gives a different parameterization of the family, but thesame family. Suppose we choose f α,β as the reference density. The family itgenerates has densitiesf µ,σ (y) = 1 ( ) y − µσ f α,β .σ= 1 ( [ ])1 y − µσβ f − αβ σ= 1 ( )y − µ − σασβ f σβ111).