Chapter 8: Statistical Analysis of Simulated Data

Transcription

1 Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by

2 Marquette Uversty MSCS600 Ageda 8. The Sample Mea ad Sample Varace 8. Iterval Estmates of a Populato Mea 8.3 The Bootstrappg Techque for Estmatg Mea Square Errors

3 Marquette Uversty MSCS The Sample Mea ad Sample Varace Suppose we have,, d all from f(). Let E ad var,.e. same mea ad varace. Wth the arthmetc mea beg, we kow that E E E 3

4 Marquette Uversty MSCS The Sample Mea ad Sample Varace If the expected value of a statstc s equal to the parameter t s estmatg, t s sad to be a ubased estmator. To determe the worth of as a estmator for θ, we look at expected squared dfferece E ( ) var( ) var var 4

5 Marquette Uversty MSCS The Sample Mea ad Sample Varace By Chebyshev s equalty c P c But usg the CLT whe s large, c P P{ Z c} [ ( c)] c.96 (.96) where Φ s the ormal dstrbuto cumulatve dstrbuto fucto

6 Marquette Uversty MSCS The Sample Mea ad Sample Varace If we defe S to be s ( ) we kow that t s ubased because ( ) S ~ ( ), Ad because the mea of above s (-), therefore. ( ) S E ( ) E S E S 6

7 Marquette Uversty MSCS The Sample Mea ad Sample Varace I a smulato, we ofte geerate a extremely large umber of radom varates (.e. 0 6 ). It would be great f we kew whe we had eough. Assume we are terested estmatg the value of θ=e[ ]. Oe stoppg rule s to specfy a stadard devato d for. The, cotue geeratg radom varates utl S/ <d. Whe s small the followg s recommeded. 7

8 Marquette Uversty MSCS The Sample Mea ad Sample Varace Method for Determg Whe to Stop Geeratg New Data. Choose a acceptable value of d for the stadard devato of the estmator.. Geerate at least 00 data values. 3. Cotue to geerate addtoal data values, stoppg whe you have geerated k values ad S/ k<d, where S s the sample stadard devato based o those k values. k 4. The estmate of θ s gve by. k 8

9 Marquette Uversty MSCS Iterval Estmates of a Populato Mea Assume we have,, d all from the same dstrbuto f(). We use as a pot estmator for the populato mea θ. We ca also geerate a terval estmator for θ. We kow that E [ ] ad Va r[ ]. We use the fact that whe s large, has a approxmate ormal dstrbuto,.e. ~ N(, / ). 9

10 Marquette Uversty MSCS Iterval Estmates of a Populato Mea What ths mples s that z / has a approxmate stadard ormal dstrbuto! P(.96 z.96)= 0.95 Or more geerally,.95 P( z z z )=. -z z

11 Marquette Uversty MSCS Iterval Estmates of a Populato Mea z z z x z z z x P( z z z )=

12 Marquette Uversty MSCS Iterval Estmates of a Populato Mea P( z z z )= z z z z z z x x

13 Marquette Uversty MSCS Iterval Estmates of a Populato Mea P( z z z )= Thus, a (-α) 00% cofdece terval for θ s - z σ + z P - z whch f α=0.05, a 95% cofdece terval for θ s σ σ σ σ σ + z 3

14 Marquette Uversty MSCS Iterval Estmates of a Populato Mea Usg smlar logc, t s also true that whe σ s ukow, a (-α) 00% cofdece terval for θ s - z s + z whch f α=0.05, a 95% cofdece terval for θ s stll s s whe s large. s t z s / / 4

15 Marquette Uversty MSCS Iterval Estmates of a Populato Mea For Beroull radom varates, where wth probablty p 0 wth probablty p we have the same scearo. Usg smlar logc, a (-α) 00% cofdece terval for p s P - z ( ) / p + z ( ) / whe s large. 5

16 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors Assume that,, are d from dstrbuto fucto F. If θ s a parameter of terested ad g(,, ) a estmator, we would lke to estmate the value of MSE F E g F ( ) F[( (,..., ) ( )) ] we ca usually estmate t aalytcally f F s kow. 6

17 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors But whe F s ot kow, all we have s,,. As we kow we ca estmate F by ECDF umber of : Fe ( x) F e should be close to F especally f s large ad F e coverges to F as. x. 7

18 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors Ths also mples that θ(f e ) should be close to θ, especally f s large, ad coverges to t as. The MSE(F) should be approxmately equal to MSE F E g F ( e) F [( (,..., ) ( e)) ] e called the bootstrap approxmato to the MSE. 8

19 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors Let s exame the bootstrap approxmato to the MSE whe we do t eed t. Assume θ=μ ad g(,, ) =. The we kow that MSE E[( ) ] /. whch we would estmate by S /. To estmate the MSE va bootstrap, we have to calculate MSE F E g F ( e) F [( (,..., ) ( e)) ] e 9

20 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors If we thk of,, as a populato of values, the the vector (x,,x ), where each elemet s draw from,, wth replacemet ca take o possble values. The MSE s the approxmately MSE F ( ) e [( g(,..., ) ( Fe )) ] j {,..., },,..., j 0

21 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors MSE F ( ) e [( g(,..., ) ( Fe )) ] j {,..., },,..., But ths requres summg terms, a dautg task. If =0, the there are terms! j To get aroud ths, we use smulato ad approxmate the emprcal MSE.

22 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors From,,, geerate samples of sze wth replacemet,..., () () () (),..., Y [( g(,..., ) ( F )] () () e () () [( (,..., ) ( e)] Y g F,..., ( r) ( r) Y [( g(,..., ) ( F )] ( r) ( r) r e Y, Y,..., Yr MSE( F ) e r r Y

23 Marquette Uversty MSCS The Bootstrappg Techque for Estmatg Mea Square Errors From,,, geerate r samples of sze wth replacemet,..., () () () (),..., Y [ s s (,..., )] () [ () (,..., )] Y s s,..., ( r) ( r) Y [ s s (,..., )] r ( r) Y, Y,..., Yr MSE s ( Fe ) r r Y 3

24 Marquette Uversty MSCS600 Homework 5: Chapter 8: # 6, 7,, 3, 5. * Geerate =5 radom umbers from a ormal dstrbuto wth µ=00 ad σ=5. Compute ad s. Compute a bootstrap estmate of var(s ). Usg pecl ad paper, start wth (-)s /σ havg a χ dstrbuto, state the dstrbuto of s, the state the E(s ) ad var(s ). Compare pecl ad paper to Bootstrap. x 4