STATISTICAL INFERENCE PART IV

1. STATISTICAL INFERENCEPART IV LOCATION AND SCALE PARAMETERS

2. LOCATION PARAMETER • Let f(x) be any pdf. The family of pdfs f(x) indexed by parameter  is called the location family with standard pdf f(x) and  is the location parameter for the family. • Equivalently,  is a location parameter for f(x) iff the distribution of X does not depend on .

3. Example • If X~N(θ,1), then X-θ~N(0,1)  distribution is independent of θ.  θ is a location parameter. • If X~N(0,θ), then X-θ~N(-θ,θ)  distribution is NOT independent of θ.  θ is NOT a location parameter.

4. LOCATION PARAMETER • Let X1,X2,…,Xnbe a r.s. of a distribution with pdf (or pmf); f(x; ); . An estimator t(x1,…,xn) is defined to be a location equivariantiff t(x1+c,…,xn+c)= t(x1,…,xn) +c for all values of x1,…,xnand a constant c. • t(x1,…,xn) is location invariantiff t(x1+c,…,xn+c)= t(x1,…,xn) for all values of x1,…,xnand a constant c. Invariant = does not change

5. Example • Is location invariant or equivariant estimator? • Let t(x1,…,xn) = . Then, t(x1+c,…,xn+c)= (x1+c+…+xn+c)/n = (x1+…+xn+nc)/n = +c = t(x1,…,xn) +c  location equivariant

6. Example • Is s² location invariant or equivariant estimator? • Let t(x1,…,xn) = s²= • Then, t(x1+c,…,xn+c)= =t(x1,…,xn) Location invariant (x1,…,xn) and (x1+c,…,xn+c) are located at different points on real line, but spread among the sample values is same for both samples.

7. SCALE PARAMETER • Let f(x) be any pdf. The family of pdfs f(x/)/ for >0, indexed by parameter , is called the scale family with standard pdf f(x) and  is the scale parameter for the family. • Equivalently, is a scale parameter for f(x) iff the distribution of X/ does not depend on .

8. Example • Let X~Exp(θ). Let Y=X/θ. • You can show that f(y)=exp(-y) for y>0 • Distribution is free of θ • θ is scale parameter.

9. SCALE PARAMETER • Let X1,X2,…,Xnbe a r.s. of a distribution with pdf (or pmf); f(x; ); . An estimator t(x1,…,xn) is defined to be a scale equivariantiff t(cx1,…,cxn)= ct(x1,…,xn) for all values of x1,…,xnand a constant c>0. • t(x1,…,xn) is scale invariantiff t(cx1,…,cxn)= t(x1,…,xn) for all values of x1,…,xnand a constant c>0.

10. Example • Is scale invariant or equivariant estimator? • Let t(x1,…,xn) = . Then, t(cx1,…,cxn)= c(x1+…+xn)/n = c = ct(x1,…,xn)  Scale equivariant

11. LOATION-SCALE PARAMETER • Let f(x) be any pdf. The family of pdfs f((x) /)/ for >0, indexed by parameter (,), is called the location-scale family with standard pdf f(x) and  is a location parameter and  is the scale parameter for the family. • Equivalently,  is a location parameter and is a scale parameter for f(x) iff the distribution of (X)/ does not depend on  and.

12. Example 1. X~N(μ,σ²). Then, Y=(X- μ)/σ ~ N(0,1) • Distribution is independent of μ and σ² • μ and σ² are location-scale paramaters 2. X~Cauchy(θ,β). You can show that the p.d.f. of Y=(X- β)/ θ is f(y) = 1/(π(1+y²))  β and θ are location-and-scale parameters.

13. LOCATION-SCALE PARAMETER • Let X1,X2,…,Xnbe a r.s. of a distribution with pdf (or pmf); f(x; ); . An estimator t(x1,…,xn) is defined to be a location-scale equivariant iff t(cx1+d,…,cxn+d)= ct(x1,…,xn)+d for all values of x1,…,xnand a constant c>0. • t(x1,…,xn) is location-scale invariant iff t(cx1+d,…,cxn+d)= t(x1,…,xn) for all values of x1,…,xnand a constant c>0.