Sixth Edition. Chapter 7 Point Estimation of Parameters and Sampling Distributions Mean Squared Error of an 7-2 Sampling Distributions and

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3//06 Appled Statstcs ad Probablty for Egeers Sth Edto Douglas C. Motgomery George C. Ruger Chapter 7 Pot Estmato of Parameters ad Samplg Dstrbutos Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

1 3//06 Appled Statstcs ad Probablty for Egeers Sth Edto Douglas C. Motgomery George C. Ruger Chapter 7 Pot Estmato of Parameters ad Samplg Dstrbutos Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 7 Pot CHAPTER OUTLINE Estmato of Parameters ad Samplg Dstrbutos 7- Pot Estmato Mea Squared Error of a 7- Samplg Dstrbutos ad Estmator the Cetral Lmt Theorem 7-4 Methods of Pot Estmato 7-3 Geeral Cocepts of Pot 7-4. Method of Momets Estmato 7-4. Method of Mamum 7-3. Ubased Estmators Lkelhood 7-3. Varace of a Pot Bayesa Estmato of Estmator Parameters Stadard Error: Reportg a Pot Estmate Chapter 7 Ttle ad Outle Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

2 3//06 Learg Objectves for Chapter 7 After careful study of ths chapter, you should be able to do the followg:. Geeral cocepts of estmatg the parameters of a populato or a probablty dstrbuto.. Importat role of the ormal dstrbuto as a samplg dstrbuto. 3. The cetral lmt theorem. 4. Importat propertes of pot estmators, cludg bas, varaces, ad mea square error. 5. Costructg pot estmators usg the method of momets, ad the method of mamum lkelhood. 6. Compute ad epla the precso wth whch a parameter s estmated. 7. Costructg a pot estmator usg the Bayesa approach. Chapter 7 Learg Objectves Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 3 Pot Estmato A pot estmate s a reasoable value of a populato parameter. X, X,, X are radom varables. Fuctos of these radom varables, -bar ad s, are also radom varables called statstcs. Statstcs have ther uque dstrbutos whch are called samplg dstrbutos. Sec 7- Pot Estmato 4 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

3//06 Pot Estmator As a eample,suppose the radom varable X s ormally dstrbuted wth a ukow mea μ. The sample mea s a pot estmator of the ukow populato mea μ. That s, μ X.

3 3//06 Pot Estmator As a eample,suppose the radom varable X s ormally dstrbuted wth a ukow mea μ. The sample mea s a pot estmator of the ukow populato mea μ. That s, μ X. After the sample has bee selected, the umercal value s the pot estmate of μ. Thus f 5, 30, 9,ad 3, the pot estmate of μ s Sec 7- Pot Estmato 5 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Some Parameters & Ther Statstcs Parameter Measure Statstc μ Mea of a sgle populato -bar σ Varace of a sgle populato s σ Stadard devato of a sgle populato s p Proporto of a sgle populato p -hat μ - μ Dfferece meas of two populatos bar - bar p - p Dfferece proportos of two populatos p hat - p hat There could be choces for the pot estmator of a parameter. To estmate the mea of a populato, we could choose the: Sample mea. Sample meda. Average of the largest & smallest observatos the sample. Sec 7- Pot Estmato 6 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 3

4 3//06 Some Deftos The radom varables X, X,,X are a radom sample of sze f: a) The X s are depedet radom varables. b) Every X has the same probablty dstrbuto. A statstc s ay fucto of the observatos a radom sample. The probablty dstrbuto of a statstc s called a samplg dstrbuto. Sec 7- Samplg Dstrbutos ad the Cetral Lmt Theorem 7 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Cetral Lmt Theorem Sec 7- Samplg Dstrbutos ad the Cetral Lmt Theorem 8 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 4

5 3//06 Eample 7-: Cetral Lmt Theorem Suppose that a radom varable X has a cotuous uform dstrbuto:, 4 6 f 0, otherwse Fd the dstrbuto of the sample mea of a radom sample of sze = 40. By the CLT the dstrbuto X s ormal. ba 64 5 ba X 40 0 Sec 7- Samplg Dstrbutos ad the Cetral Lmt Theorem 9 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Fgure 7-5 The dstrbuto of X ad X for Eample 7-. Samplg Dstrbuto of a Dfferece Sample Meas If we have two depedet populatos wth meas μ ad μ, ad varaces σ ad σ, ad If X-bar ad X-bar are the sample meas of two depedet radom samples of szes ad from these populatos: The the samplg dstrbuto of: s appromately stadard ormal, f the codtos of the cetral lmt theorem apply. If the two populatos are ormal, the the samplg dstrbuto of Z s eactly stadard ormal. Sec 7- Samplg Dstrbutos ad the Cetral Lmt Theorem 0 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 5

3//06 Eample 7-3: Arcraft Ege Lfe The effectve lfe of a compoet used jet-turbe arcraft ege s a radom varable wth mea 5000 ad SD 40 hours ad s close to a ormal dstrbuto.

What s the probablty that the dfferece the two sample meas s at least 5 hours? Fgure 7-6 The samplg dstrbuto of X X Eample 7-3.

6 3//06 Eample 7-3: Arcraft Ege Lfe The effectve lfe of a compoet used jet-turbe arcraft ege s a radom varable wth mea 5000 ad SD 40 hours ad s close to a ormal dstrbuto. The ege maufacturer troduces a mprovemet to the Maufacturg process for ths compoet that chages the parameters to 5050 ad 30. Radom samples of sze 6 ad 5 are selected. What s the probablty that the dfferece the two sample meas s at least 5 hours? Fgure 7-6 The samplg dstrbuto of X X Eample 7-3. Process Old () New () Dff (-) -bar = 5,000 5, s = = 6 5 Calculatos s / = z = -.4 P(bar -bar > 5) = P(Z > z) = = - NORMSDIST(z) Sec 7- Samplg Dstrbutos ad the Cetral Lmt Theorem Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Ubased Estmators Defed Sec 7-3. Ubased Estmators Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 6

7 3//06 Eample 7-4: Sample Mea & Varace Are Ubased- X s a radom varable wth mea μ ad varace σ. Let X, X,,X be a radom sample of sze. Show that the sample mea (X-bar) s a ubased estmator of μ. Sec 7-3. Ubased Estmators 3 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Eample 7-4: Sample Mea & Varace Are Ubased- Show that the sample varace (S ) s a ubased estmator of σ. Sec 7-3. Ubased Estmators 4 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 7

3//06 Mmum Varace Ubased Estmators If we cosder all ubased estmators of θ, the oe wth the smallest varace s called the mmum varace ubased estmator (MVUE).

8 3//06 Mmum Varace Ubased Estmators If we cosder all ubased estmators of θ, the oe wth the smallest varace s called the mmum varace ubased estmator (MVUE). If X, X,, X s a radom sample of sze from a ormal dstrbuto wth mea μ ad varace σ, the the sample X-bar s the MVUE for μ. Sec 7-3. Varace of a Pot Estmate 5 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Stadard Error of a Estmator Sec Stadard Error Reportg a Pot Estmate 6 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 8

3//06 Eample 7-5: Thermal Coductvty These observatos are 0 measuremets of thermal coductvty of Armco ro. Sce σ s ot kow, we use s to calculate the stadard error. Sce the stadard error s 0.

9 3//06 Eample 7-5: Thermal Coductvty These observatos are 0 measuremets of thermal coductvty of Armco ro. Sce σ s ot kow, we use s to calculate the stadard error. Sce the stadard error s 0.% of the mea, the mea estmate s farly precse. We ca be very cofdet that the true populato mea s 4.94 ± (0.0898) or betwee ad = Mea 0.84 = Std dev (s ) = Std error Sec Stadard Error Reportg a Pot Estmate 7 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Mea Squared Error Cocluso: The mea squared error (MSE) of the estmator s equal to the varace of the estmator plus the bas squared. Sec Mea Squared Error of a Estmator 8 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 9

3//06 Relatve Effcecy The MSE s a mportat crtero for comparg two estmators. If the relatve effcecy s less tha, we coclude that the st estmator s superor tha the d estmator. Sec 7-3.

10 3//06 Relatve Effcecy The MSE s a mportat crtero for comparg two estmators. If the relatve effcecy s less tha, we coclude that the st estmator s superor tha the d estmator. Sec Mea Squared Error of a Estmator 9 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Optmal Estmator A based estmator ca be preferred tha a ubased estmator f t has a smaller MSE. Based estmators are occasoally used lear regresso. A estmator whose MSE s smaller tha that of ay other estmator s called a optmal estmator. Fgure 7-8 A based estmator has a smaller varace tha the ubased estmator. that Sec Mea Squared Error of a Estmator 0 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 0

11 3//06 Momets Defed Let X, X,,X be a radom sample from the probablty dstrbuto f(), where f() ca be ether a: Dscrete probablty mass fucto, or Cotuous probablty desty fucto The k th populato momet (or dstrbuto momet) s E(X k ), k =,,. th k The k sample momet s / X, k,,... If k = (called the frst momet), the: Populato momet s μ. Sample momet s -bar. The sample mea s the momet estmator of the populato mea. Sec 7-4. Method of Momets Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Momet Estmators Sec 7-4. Method of Momets Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

12 3//06 Eample 7-8: Normal Dstrbuto Momet Estmators Suppose that X, X,, X s a radom sample from a ormal dstrbuto wth parameter μ ad σ where E(X) = μ ad E(X ) = μ + σ. X X X X X ad X X X X X X Sec 7-4. Method of Momets 3 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. (based) Eample 7-9: Gamma Dstrbuto Momet Estmators- Suppose that X, X,, X s a radom sample from a gamma dstrbuto wth parameter r ad λ where E(X) = r/ λ ad E(X ) = r(r+)/ λ. r E X r E X rr E X r E X X s the mea s the varace or ad ow solvg for r ad : / X X X X / X X Sec 7-4. Method of Momets 4 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

13 3//06 Eample 7-9: Gamma Dstrbuto Momet Estmators- Usg the tme to falure data the table. We ca estmate the parameters of the gamma dstrbuto. -bar =.646 ΣX = X.646 r / X X X / X X Sec 7-4. Method of Momets 5 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Mamum Lkelhood Estmators Suppose that X s a radom varable wth probablty dstrbuto f(;θ), where θ s a sgle ukow parameter. Let,,, be the observed values a radom sample of sze. The the lkelhood fucto of the sample s: L(θ) = f( ;θ) f( ; θ) f( ; θ) Note that the lkelhood fucto s ow a fucto of oly the ukow parameter θ. The mamum lkelhood estmator (MLE) of θ s the value of θ that mamzes the lkelhood fucto L(θ). Sec 7-4. Method of Mamum Lkelhood 6 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 3

14 3//06 Eample 7-0: Beroull Dstrbuto MLE Let X be a Beroull radom varable. The probablty mass fucto s f(;p) = p (-p) -, = 0, where P s the parameter to be estmated. The lkelhood fucto of a radom sample of sze s:... L p p p p p p p l L p l p l p d l L p dp p p p p p p Sec 7-4. Method of Mamum Lkelhood 7 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Eample 7-: Normal Dstrbuto MLE for μ Let X be a ormal radom varable wth ukow mea μ ad kow varace σ. The lkelhood fucto of a radom sample of sze s: L e l L l dl L d e Sec 7-4. Method of Mamum Lkelhood 8 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 4

15 3//06 5 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Eample 7-: Epoetal Dstrbuto MLE Let X be a epoetal radom varable wth parameter λ. The lkelhood fucto of a radom sample of sze s: Sec 7-4. Method of Mamum Lkelhood 9 l l l L e e L d L d Equatg the above to zero we get (same as momet estmator) X Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Eample 7-3: Normal Dstrbuto MLEs for μ & σ Let X be a ormal radom varable wth both ukow mea μ ad varace σ. The lkelhood fucto of a radom sample of sze s: Sec 7-4. Method of Mamum Lkelhood 30 4, l, l l, 0 l, 0 ad L e e L L L X X

Sec 7-4. Method of Mamum Lkelhood 3 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

16 3//06 Propertes of a MLE Notes: Mathematcal statstcas wll ofte prefer MLEs because of these propertes. Propertes () ad () state that MLEs are MVUEs. To use MLEs, the dstrbuto of the populato must be kow or assumed. Sec 7-4. Method of Mamum Lkelhood 3 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Ivarace Property Ths property s llustrated Eample 7-3. Sec 7-4. Method of Mamum Lkelhood 3 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 6

17 3//06 Eample 7-4: Ivarace For the ormal dstrbuto, the MLEs were: Sec 7-4. Method of Mamum Lkelhood 33 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Complcatos of the MLE Method The method of mamum lkelhood s a ecellet techque, however there are two complcatos:. It may ot be easy to mamze the lkelhood fucto because the dervatve fucto set to zero may be dffcult to solve algebracally.. It may ot always be possble to use calculus methods drectly to determe the mamum of L(ѳ). The followg eample llustrate ths. Sec 7-4. Method of Mamum Lkelhood 34 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 7

18 3//06 Eample 7-6: Gamma Dstrbuto MLE- Let X, X,, X be a radom sample from a gamma dstrbuto. The log of the lkelhood fucto s: r r e l Lr, l r Equatg the above dervatve to zero we get r l r l l r l L r, ' l l r l L r, r r r Sec 7-4. Method of Mamum Lkelhood 35 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Eample 7-6: Gamma Dstrbuto MLE- Fgure 7- Log lkelhood for the gamma dstrbuto usg the falure tme data. (a) Log lkelhood surface. (b) Cotour plot. Sec 7-4. Method of Mamum Lkelhood 36 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 8

19 3//06 Bayesa Estmato of Parameters- The momet ad lkelhood methods terpret probabltes as relatve frequeces ad are called objectve frequeces. The radom varable X has a probablty dstrbuto of parameter θ called f( θ). Addtoal formato about θ s that t ca be summarzed as f(θ), the pror dstrbuto, wth mea μ 0 ad varace σ 0. Probabltes assocated wth f(θ) are subjectve probabltes. The jot dstrbuto s f(,,, θ). The posteror dstrbuto s f(θ,,, ) s our degree of belef regardg θ after observg the sample data Bayesa Estmato of Parameters 37 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Bayesa Estmato of Parameters- Now the jot probablty dstrbuto of the sample s f(,,,, θ) = f(,,, θ) f(θ) The margal dstrbuto s: f,,...,,θ, for θ dscrete θ f,,..., f,,...,,θ dθ, for θ cotuous The desred posteror dstrbuto s: f θ,,..., f,,...,,θ f,,..., The Bayesa estmator of θ s θ, the mea of the posteror dstrbuto Bayesa Estmato of Parameters 38 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 9

20 3//06 Eample 7-6: Bayes Estmator for the mea of a Normal Dstrbuto - Let X, X,, X be a radom sample from a ormal dstrbuto ukow mea μ ad kow varace σ. Assume that the pror dstrbuto for μ s: f μ e e 0 0 The jot probablty dstrbuto of the sample s: f,,..., e e ( ) Bayesa Estmato of Parameters 39 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Eample 7-7: Bayes Estmator for the mea of a Normal Dstrbuto- Now the jot probablty dstrbuto of the sample ad μ s: f,,...,, f,,..., f μ e e / / h (,,...,,, 0, 0 ) 0 0 Upo completg the square the epoet, f,,...,, e Bayesa Estmato of Parameters 40 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. ( / ) 0 0 / h (,,...,,, 0, 0 ) 0 / 0 / 0 / where h (,,...,,,, )s a fucto of the observed values ad the parameters, ad. 0 0 Sce f,,...,, does ot deped o f,,..., e ( / ) 0 0 / h 3 (,,...,,, 0, 0 ) 0 / 0 / 0

21 3//06 Eample 7-7: Bayes Estmator for the mea of a Normal Dstrbuto-3 whch s recogzed as a ormal probablty desty fucto wth posteror mea ad posteror varace V Bayesa Estmato of Parameters 4 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. Eample 7-7: Bayes Estmator for the mea of a Normal Dstrbuto-4 To llustrate: The parameters are: μ 0 = 0, σ 0 = Sample: = 0, -bar = 0.75, σ = Bayesa Estmato of Parameters 4 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

22 3//06 Importat Terms & Cocepts of Chapter 7 Bayes estmator Bas parameter estmato Cetral lmt theorem Estmator vs. estmate Lkelhood fucto Mamum lkelhood estmator Mea square error of a estmator Mmum varace ubased estmator Momet estmator Normal dstrbuto as the samplg dstrbuto of the: sample mea dfferece two sample meas Parameter estmato Pot estmator Populato or dstrbuto momets Posteror dstrbuto Pror dstrbuto Sample momets Samplg dstrbuto A estmator has a: Stadard error Estmated stadard error Statstc Statstcal ferece Ubased estmator Chapter 7 Summary 43 Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved.

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