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S  0  ``  >*  0     * Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. >ZDf B H  0޽h ? ̙33 Default Design h`d(  d d 08P`  `Chapter 6 Point Estimation*0(2 0<H d 0޽h ? 3380___PPT10.^; 0h;(  h h 6D8c"`  K6.1 General Concepts of Point Estimation(L0(20H<H h 0޽h ? 3380___PPT10. Pۘ @?(    0N: APoint Estimator 2(  0P  .A point estimator of a parameter is a single number that can be regarded as a sensible value for . A point estimator can be obtained by selecting a suitable statistic and computing its value from the given sample data.0 2$$$`  c $A ??P `  c $A ?? P g  H  0޽h ? ̙33- P m(     0\j DUnbiased Estimator 2(  0` @  QA point estimator is said to be an unbiased estimator of if for every possible value of . If is not biased, the difference is called the bias of . J 2$$$$ $K{`   c $A ??p` w  `   c $A ??p `   c $A  ??   `   c $A  ??b  `   c $A  ??0 P g    `   c $A ??  `   c $A ?? H  0޽h ? ̙33/ `\o(  \x \ <A) ??`0 4  XB \ 0D>P  XB \ 0D>P  ^ \ 6^~[D    \ 0 ` XB ` 0D> ` ` ` c $A ??   ` ` c $A. ??@   RB `@ s *D  RB `@ s *D H ` 0޽h ? ̙33 ?7(    0tP DUnbiased Estimator 2(  0ԓp@B  #wWhen X is a binomial rv with parameters n and p, the sample proportion is an unbiased estimator of p.nx 2$$"$$$$F$$$``  c $A ??p`   H  0޽h ? ̙33 A(    0tpp0* R Principle of Unbiased Estimation! 2!(  0ԥp  UWhen choosing among several different estimators of , select one that is unbiased.V 2V$`  c $A ??   H  0޽h ? ̙33 3+(    0P DUnbiased Estimator 2(  0 Let X1, X2,& ,Xn be a random sample from a distribution with mean and variance . Then the estimator~m 2$$ $$$ $$$ $^$ ]  0  Kis an unbiased estimator. 2$`  c $A ??Jv `  c $A ??`J `  c $A ??  H  0޽h ? ̙33 `X (     0hP DUnbiased Estimator 2(  0  If X1, X2,& ,Xn is a random sample from a distribution with mean , then is an unbiased estimator of . If in addition the distribution is continuous and symmetric, then and any trimmed mean are also unbiased estimators of .~ 2$$ $$$ $$$ $$ `   c $A ?? v `   c $A #??p3< #`   c $A  ??p J   `   c $A !??P  !`   c $A &?? p  o  &H  0޽h ? ̙33V (    0t@@ c1Principle of Minimum Variance Unbiased Estimation2 22([  0  Among all estimators of that are unbiased, choose the one that has the minimum variance. The resulting is called the minimum variance unbiased estimator (MVUE) of .0 2~$*$$`  c $A '??  '`  c $A (??p 0 G  (`  c $A )??pB  )H  0޽h ? ̙33 80 T(  T$ T 0lP pGraphs of the pdf s of two different unbiased estimators9 29(%x T <A3 ??0  `  T c $A, ??  `  T c $A+ ?? PVP  `  T c $A. ??`   RB  T@ s *DP  RB  T@ s *D  H T 0޽h ? ̙336 $v(  $ $ 0 PMVUE for a Normal Distribution 2( $ 0d < 2$ B $ 0-"  &Let X1, X2,& ,Xn be a random sample from a normal distribution with parameters and . Then the estimator is the MVUE for .~ 2$$ $$$ $$$ $$ ` $ c $A *?? *` $ c $A +??0F  +` $ c $A 2??m  2` $ c $A 5??@`4 b  5H $ 0޽h ? ̙33'  Xg(  X X 0d - c1A biased estimator that is preferable to the MVUE2 22(x X <A4 ??`  ` X c $A, ?? . : ` X c $A( ??0  `  X c $A6 ??`  RB  X@ s *D RB  X@ s *D  H X 0޽h ? ̙33 D=(  D D 0|-0: CThe Estimator for 2(` D c $A# ?? B`  D 0H-`cD<4___PPT9 If the random sample comes from a normal distribution, then is the best estimator since it has minimum variance among all unbiased estimators.  2$ @`` D c $A$ ??0@ H D 0޽h ? ̙33 6.H(  H H 0%-0: CThe Estimator for 2(` H c $A# ?? B` C H 0X*-`  g2. If the random sample comes from a Cauchy distribution, then is good (the MVUE is not known). h 2h$,&9 @`` H c $A$ ??0@ ` H c $A" ??P  _  ` H c $A% ??80  H H 0޽h ? ̙336  Lv(  L L 08-0: CThe Estimator for 2(` L c $A# ?? B` [ L 0=-`\  3. If the underlying distribution is uniform, the best estimator is this estimator is influenced by outlying observations, but the lack of tails makes this impossible. 2$ @`` L c $A$ ??0@ ` L c $A& ??P3  H L 0޽h ? ̙33 0PW(  P P 0`K-0: CThe Estimator for 2(` P c $A# ?? B` < P 0,P-`  r4. The trimmed mean works reasonably well in all three situations but is not the best for any of them. s 2s$U @`` P c $A$ ??0@ ` P c $A' ??0 7 H P 0޽h ? ̙33 @(P(  ( ( 0^-P: @Standard Error 2( ( 0hb-P@0 p The standard error of an estimator is its standard deviation . If the standard error itself involves unknown parameters whose values can be estimated, substitution into yields the estimated standard error of the estimator, denoted J  2$$$$$` ( c $A >?? `w >` ( c $A 9?? ? 9` ( c $A G??  G` ( c $A A??P F  AH ( 0޽h ? ̙33? Pl?(  l l 6o-8c"`  O6.2 Methods of Point Estimation(P0(20L<H l 0޽h ? 3380___PPT10. Pۘ `,F(  , , 0Lz-@@ :Moments  2 ( , 0x-|  <jLet X1, X2,& ,Xn be a random sample from a pmf or pdf f (x). For k = 1, 2,& the kth population moment, or kth moment of the distribution f (x) is The kth sample moment is : 2$$ $$$ $$$ $&$$$$$$$$$!$$$$$$$t 2` , c $A J??pp06  J` , c $A K?? p  KH , 0޽h ? ̙33 php 0(  0 0 0-z DMoment Estimators  2(^ 0 0x-p^ &Let X1, X2,& ,Xn be a random sample from a distribution with pmf or pdf ~ 2$$ $$$ $$$ $$6 ,Q 0 0-`pN  Eare parameters whose values are unknown. Then the moment estimators DF 23$$$$A 0 0-0   are obtained by equating the first m sample moments to the corresponding first m population moments and solving forB 23$$+$$$$` 0 c $A N?? Nx  0 <A O??0 0  Ox  0 <A R?? P $ RH 0 0޽h ? ̙33  4>(  4 4 0X-z FLikelihood Function  2( 4 0-0p Let X1, X2,& ,Xn have joint pmf or pdf ~s 2$$ $$$ $$$ $d$6  Q 4 0-`@ Rwhere parameters& 2$$ 4 0H- < 2$ ` 4 c $A z??02} zx 4 <A w??` 0E wx 4 <A |??P J  |n  4 0d-  8have unknown values. 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