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STATISTICAL INFERENCE PART III

STATISTICAL INFERENCE PART III. EXPONENTIAL FAMILY, LOCATION AND SCALE PARAMETERS. EXPONENTIAL CLASS OF PDFS. X is a continuous (discrete) rv with pdf f(x; ),  . If the pdf can be written in the following form.

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STATISTICAL INFERENCE PART III

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  1. STATISTICAL INFERENCEPART III EXPONENTIAL FAMILY, LOCATION AND SCALE PARAMETERS

  2. EXPONENTIAL CLASS OF PDFS • X is a continuous (discrete) rv with pdf f(x;), . If the pdf can be written in the following form then, the pdf is a member of exponential class of pdfs of the continuous (discrete) type. (Here, k is the number of parameters)

  3. REGULAR CASE OF THE EXPONENTIAL CLASS OF PDFS • We have a regular case of the exponential class of pdfs of the continuous type if • Range of X does not depend on. • c() ≥ 0, w1(),…, wk() are real valued functions of  for . • h(x) ≥ 0, t1(x),…, tk(x) are real valued functions of x. If the range of X depends on , then it is called irregular exponential class or range-dependent exponential class.

  4. EXAMPLES X~Bin(n,p), where n is known. Is this pdf a member of exponential class of pdfs? Why? Binomial family is a member of exponential family of distributions.

  5. EXAMPLES X~Cauchy(1,). Is this pdf a member of exponential class of pdfs? Why? Cauchy is not a member of exponential family.

  6. REGULAR CASE OF THE EXPONENTIAL CLASS OF PDFS • Random Sample from Regular Exponential Class is a css for . If Y is an UE of , Yis the MVUE of .

  7. EXAMPLES Recall: X~Bin(n,p), where n is known. This family is a member of exponential family of distributions. is a CSS for p. is UE of p. is MVUE of p.

  8. EXAMPLES X~N(,2) where both  and 2 is unknown. Find a css for  and 2 .

  9. FISHER INFORMATION AND INFORMATION CRITERIA • X, f(x;), , xA (not depend on ). Definitions and notations:

  10. FISHER INFORMATION AND INFORMATION CRITERIA The Fisher Information in a random variable X: The Fisher Information in the random sample: Let’s prove the equalities above.

  11. FISHER INFORMATION AND INFORMATION CRITERIA

  12. FISHER INFORMATION AND INFORMATION CRITERIA

  13. FISHER INFORMATION AND INFORMATION CRITERIA The Fisher Information in a random variable X: The Fisher Information in the random sample: Proof of the last equality is available on Casella & Berger (1990), pg. 310-311.

  14. CRAMER-RAO LOWER BOUND (CRLB) • Let X1,X2,…,Xnbe sample random variables. • Range of X does not depend on . • Y=U(X1,X2,…,Xn): a statistic not containing . • Let E(Y)=m(). • Z=′(x1,x2,…,xn;) is a r.v. • E(Z)=0 and V(Z)=In() (from previous slides). • Let prove this!

  15. CRAMER-RAO LOWER BOUND (CRLB) • Cov(Y,Z)=E(YZ)-E(Y)E(Z)=E(YZ)

  16. CRAMER-RAO LOWER BOUND (CRLB) • E(Y.Z)=m’() • -1Corr(Y,Z)1 • 0 Corr(Y,Z)21  The Cramer-Rao Inequality (Information Inequality)

  17. CRAMER-RAO LOWER BOUND (CRLB) • CRLB is the lower bound for the variance of the unbiased estimator of m(). • When V(Y)=CRLB, Y is the MVUE of m(). • For a r.s., remember that In()=n I(), so,

  18. EFFICIENT ESTIMATOR • T is an efficient estimator (EE) of  if • T is UE of , and, • Var(T)=CRLB • Y is an efficient estimator (EE) of its expectation, m(), if its variance reaches the CRLB. • An EE of m() may not exist. • The EE of m(), if exists, is unique. • The EE of m() is the unique MVUE of m().

  19. ASYMPTOTIC EFFICIENT ESTIMATOR • Y is an asymptotic EE of m() if

  20. EXAMPLES A r.s. of size n from X~Poi(θ). • Find CRLB for any UE of θ. • Find UMVUE of θ. • Find an EE for θ. • Find CRLB for any UE of exp{-2θ}. Assume n=1, and show that is UMVUE of exp{-2θ}. Is this a reasonable estimator?

  21. EXAMPLE A r.s. of size n from X~Exp(). Find UMVUE of , if exists.

  22. LIMITING DISTRIBUTION OF MLEs • : MLE of  • X1,X2,…,Xnis a random sample.

  23. EE of m()= LIMITING DISTRIBUTION OF MLEs • Let be MLEs of 1, 2,…, m. • If Y is an EE of , then Z=a+bY is an EE of a+bm() where a and b are constants.

  24. 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 .

  25. 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.

  26. 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

  27. 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

  28. 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.

  29. 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 .

  30. 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.

  31. 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.

  32. 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

  33. 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.

  34. 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.

  35. 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.

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