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STATISTICAL INFERENCE PART II SOME PROPERTIES OF ESTIMATORS

STATISTICAL INFERENCE PART II SOME PROPERTIES OF ESTIMATORS. SOME PROPERTIES OF ESTIMATORS. θ: a parameter of interest; unknown Previously, we found good(?) estimator(s) for θ or its function g(θ). Goal: Check how good are these estimator(s). Or are they good at all?

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STATISTICAL INFERENCE PART II SOME PROPERTIES OF ESTIMATORS

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  1. STATISTICAL INFERENCEPART IISOME PROPERTIES OF ESTIMATORS

  2. SOME PROPERTIES OF ESTIMATORS • θ: a parameter of interest; unknown • Previously, we found good(?) estimator(s) for θ or its function g(θ). • Goal: • Check how good are these estimator(s). Or are they good at all? • If more than one good estimator is available, which one is better?

  3. Bias of for estimating  If is UE of , SOME PROPERTIES OF ESTIMATORS • UNBIASED ESTIMATOR (UE): An estimator is an UE of the unknown parameter , if Otherwise, it is a BiasedEstimator of .

  4. SOME PROPERTIES OF ESTIMATORS • ASYMPTOTICALLY UNBIASED ESTIMATOR (AUE): An estimator is an AUE of the unknown parameter , if

  5. SOME PROPERTIES OF ESTIMATORS • CONSISTENT ESTIMATOR (CE): An estimator which converges in probability to an unknown parameter  for all  is called a CE of . For large n, a CE tends to be closer to the unknown population parameter. • MLEs are generally CEs.

  6. EXAMPLE For a r.s. of size n, By WLLN,

  7. If is smaller, is a better estimator of . MEAN SQUARED ERROR (MSE) • The Mean Square Error (MSE) of an estimator for estimating  is

  8. MEAN SQUARED ERROR CONSISTENCY • Tn is called mean squared error consistent (or consistent in quadratic mean) if E{Tn}20 as n. Theorem: Tnis consistent in MSE iff i)Var(Tn)0 as n. • If E{Tn}20 as n, Tnis also a CE of .

  9. EXAMPLES X~Exp(), >0. For a r.s of size n, consider the following estimators of , and discuss their bias and consistency. Which estimator is better?

  10. SUFFICIENT STATISTICS • X, f(x;),  • X1, X2,…,Xn • Y=U(X1, X2,…,Xn ) is a statistic. • A sufficient statistic, Y, is a statistic which contains all the information for the estimation of .

  11. SUFFICIENT STATISTICS • Given the value of Y, the sample contains no further information for the estimation of . • Y is a sufficient statistic (ss) for  if the conditional distribution h(x1,x2,…,xn|y) does not depend on  for every given Y=y. • A ss for is not unique: • If Y is a ss for , then any 1-1 transformation of Y, say Y1=fn(Y) is also a ss for .

  12. SUFFICIENT STATISTICS • The conditional distribution of sample rvs given the value of y of Y, is defined as • If Y is a ss for , then Not depend on  for every given y. ss for  may include y or constant. • Also, the conditional range of Xi given y not depend on .

  13. SUFFICIENT STATISTICS EXAMPLE: X~Ber(p). For a r.s. of size n, show that is a ss for p.

  14. SUFFICIENT STATISTICS • Neyman’s Factorization Theorem:Y is a ss for  iff The likelihood function Does not depend on xi except through y Not depend on  (also in the range of xi.) where k1and k2 are non-negative functions.

  15. EXAMPLES 1. X~Ber(p). For a r.s. of size n, find a ss for p if exists.

  16. EXAMPLES 2. X~Beta(θ,2). For a r.s. of size n, find a ss for θ.

  17. SUFFICIENT STATISTICS • A ss may not exist. • Jointly ss Y1,Y2,…,Yk may be needed. Example: Example 10.2.5 in Bain and Engelhardt (page 342 in 2nd edition), X(1) and X(n) are jointly ss for  • If the MLE of  exists and unique and if a ss for  exists, then MLE is a function of a ss for .

  18. EXAMPLE X~N(,2). For a r.s. of size n, find jss for  and 2.

  19. MINIMAL SUFFICIENT STATISTICS • If is a ss for θ, then, is also a ss for θ. But, the first one does a better job in data reduction. A minimalss achieves the greatest possible reduction.

  20. MINIMAL SUFFICIENT STATISTICS • A ss T(X) is called minimal ss if, for any other ss T’(X),T(x) is a function of T’(x). • THEOREM: Let f(x;) be the pmf or pdf of a sampleX1, X2,…,Xn. Suppose there exist a function T(x) such that, for two sample points x1,x2,…,xn and y1,y2,…,yn, the ratio is constant with respect to  iff T(x)=T(y). Then, T(X) is a minimal sufficient statistic for .

  21. EXAMPLE • X~N(,2) where 2 is known. For a r.s. of size n, find minimal ss for . Note: A minimal ss is also not unique. Any 1-to-1 function is also a minimal ss.

  22. RAO-BLACKWELL THEOREM • Let X1, X2,…,Xn have joint pdf or pmf f(x1,x2,…,xn;) and let S=(S1,S2,…,Sk) be a vector of jss for . If T is an UE of () and (T)=E(TS), then • (T) is an UE of() . • (T) is a fn of S, so it is also jss for . • Var((T) ) Var(T) for all . • (T) is a uniformly better unbiased estimator of () .

  23. RAO-BLACKWELL THEOREM • Notes: • (T)=E(TS) is at least as good as T. • For finding the best UE, it is enough to consider UEs that are functions of a ss, because all such estimators are at least as good as the rest of the UEs.

  24. Example • Hogg & Craig, Exercise 10.10 • X1,X2~Exp(θ) • Find joint p.d.f. of ss Y1=X1+X2 for θ and Y2=X2. • Show that Y2 is UE of θ with variance θ². • Find φ(y1)=E(Y2|Y1) and variance of φ(Y1).

  25. ANCILLARY STATISTIC • A statistic S(X) whose distribution does not depend on the parameter  is called an ancillary statistic. • Unlike a ss, an ancillary statistic contains no information about .

  26. Example • Example 6.1.8 in Casella & Berger, page 257: Let Xi~Unif(θ,θ+1) for i=1,2,…,n Then, range R=X(n)-X(1) is an ancillary statistic because its pdf does not depend on θ.

  27. COMPLETENESS • Let {f(x; ), } be a family of pdfs (or pmfs) and U(x) be an arbitrary function of x not depending on . If requires that the function itself equal to 0 for all possible values of x; then we say that this family is a complete family of pdfs (or pmfs). i.e., the only unbiased estimator of 0 is 0 itself.

  28. EXAMPLES 1. Show that the family {Bin(n=2,); 0<<1} is complete.

  29. EXAMPLES 2. X~Uniform(,). Show that the family {f(x;), >0}is not complete.

  30. COMPLETE AND SUFFICIENT STATISTICS (css) • Y is a complete and sufficient statistic (css) for  if Y is a ss for  and the family is complete. The pdf of Y. 1) Y is a ss for . 2) u(Y) is an arbitrary function of Y. E(u(Y))=0 for all  implies that u(y)=0 for all possible Y=y.

  31. (n-1)S2/ 2 ~ By Basu theorem, and S2are independent. BASU THEOREM • If T(X) is a complete and minimal sufficient statistic, then T(X) is independent of every ancillary statistic. • Example:X~N(,2). S2 Ancillary statistic for 

  32. BASU THEOREM • Example: • Let T=X1+ X2 and U=X1 - X2 • We know that T is a complete minimal ss. • U~N(0, 2)  distribution free of   T and U are independent by Basu’s Theorem X1, X2~N(,2), independent, 2 known.

  33. THE MINIMUM VARIANCE UNBIASED ESTIMATOR • Rao-Blackwell Theorem: If T is an unbiased estimator of , and S is a ss for , then (T)=E(TS)is • an UE of , i.e.,E[(T)]=E[E(TS)]= and • the MVUE of .

  34. LEHMANN-SCHEFFE THEOREM • Let Y be a cssfor . If there is a function Y which is an UE of , then the function is the unique Minimum Variance Unbiased Estimator (UMVUE) of . • Y css for . • T(y)=fn(y) and E[T(Y)]=. • T(Y) is the UMVUE of . • So, it is the best estimator of .

  35. THE MINIMUM VARIANCE UNBIASED ESTIMATOR • Let Y be a cssfor . Since Y is complete, there could be only a unique function of Y which is an UE of . • Let U1(Y) and U2(Y) be two function of Y. Since they are UE’s, E(U1(Y)U2(Y))=0 imply W(Y)=U1(Y)U2(Y)=0 for all possible values of Y. Therefore, U1(Y)=U2(Y) for all Y.

  36. Example • Let X1,X2,…,Xn ~Poi(μ). Find UMVUE of μ. • Solution steps: • Show that is css for μ. • Find a statistics (such as S*) that is UE of μ and a function of S. • Then, S* is UMVUE of μ by Lehmann-Scheffe Thm.

  37. Note • The estimator found by Rao-Blackwell Thm may not be unique. But, the estimator found by Lehmann-Scheffe Thm is unique.

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