STATISTICAL INFERENCE PART IV LOCATION AND SCALE PARAMETERS 1

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 .

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. 3

4 LOCATION PARAMETER Let X 1,X 2,…,X n be a r.s. of a distribution with pdf (or pmf); f(x;  ); . An estimator t(x 1,…,x n ) is defined to be a location equivariant iff t(x 1 +c,…,x n +c)= t(x 1,…,x n ) +c for all values of x 1,…,x n and a constant c. t(x 1,…,x n ) is location invariant iff t(x 1 +c,…,x n +c)= t(x 1,…,x n ) for all values of x 1,…,x n and a constant c. Invariant = does not change

Example Is location invariant or equivariant estimator? Let t(x 1,…,x n ) =. Then, t(x 1 +c,…,x n +c)= (x 1 +c+…+x n +c)/n = (x 1 +…+x n +nc)/n = +c = t(x 1,…,x n ) +c  location equivariant 5

Example Is s² location invariant or equivariant estimator? Let t(x 1,…,x n ) = s²= Then, t(x 1 +c,…,x n +c)= =t(x 1,…,x n )  Location invariant 6 (x 1,…,x n ) and (x 1 +c,…,x n +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 .

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. 8

9 SCALE PARAMETER Let X 1,X 2,…,X n be a r.s. of a distribution with pdf (or pmf); f(x;  ); . An estimator t(x 1,…,x n ) is defined to be a scale equivariant iff t(cx 1,…,cx n )= ct(x 1,…,x n ) for all values of x 1,…,x n and a constant c>0. t(x 1,…,x n ) is scale invariant iff t(cx 1,…,cx n )= t(x 1,…,x n ) for all values of x 1,…,x n and a constant c>0.

Example Is scale invariant or equivariant estimator? Let t(x 1,…,x n ) =. Then, t(cx 1,…,cx n )= c(x 1 +…+x n )/n = c = ct(x 1,…,x n )  Scale equivariant 10

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 .

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. 12

13 LOCATION-SCALE PARAMETER Let X 1,X 2,…,X n be a r.s. of a distribution with pdf (or pmf); f(x;  ); . An estimator t(x 1,…,x n ) is defined to be a location-scale equivariant iff t(cx 1 +d,…,cx n +d)= ct(x 1,…,x n )+d for all values of x 1,…,x n and a constant c>0. t(x 1,…,x n ) is location-scale invariant iff t(cx 1 +d,…,cx n +d)= t(x 1,…,x n ) for all values of x 1,…,x n and a constant c>0.