5. Best Unbiased Estimators
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1 Best Ubiased Estimators 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > Best Ubiased Estimators Basic Theory Cosider agai the basic statistical model, i which we have a radom experimet that results i a observable radom variable X takig values i a set S. Oce agai, the experimet is typically to sample objects from a populatio ad record oe or more measuremets for each item. I this case, the observable radom variable has the form X = (X 1, X 2,..., X ) where X i is the vector of measuremets for the i th item. Suppose that θ is a real parameter of the distributio of X, takig values i a parameter space Θ R. Let f θ deote the probability desity fuctio of X for θ Θ. Note that the expected value, variace, ad covariace operators also deped o θ, although we will sometimes suppress this to keep the otatio from becomig too uwieldy. Suppose ow that λ = λ(θ) is a parameter of iterest that is derived from θ. I this sectio we will cosider the geeral problem of fidig the best estimator of λ amog a give class of ubiased estimators. Recall that if U is a ubiased estimator of λ, the var θ (U) is the mea square error. Thus, if U ad V are ubiased estimators of λ ad var θ (U) var θ (V) for all θ Θ The U is a uiformly better estimator tha V. O the other had, it may be the case that U has smaller variace for some values of θ while V has smaller variace for other values of θ. If U is uiformly better tha ay other ubiased estimator of λ, the U is a Uiformly Miimum Variace Ubiased Estimator (UMVUE) of λ. The Cramér-Rao Lower Boud We will show that uder mild coditios, there is a lower boud o the variace of ay ubiased estimator of the parameter λ. Thus, if we ca fid a estimator that achieves this lower boud for all θ Θ, the the estimator must be a UM VUE of λ. The assumptio that we must make is that if h : S R with θ ( h(x) ) < for θ Θ the d θ (h(x)) = θ ( h(x) dl( f θ(x)), θ Θ
2 Best Ubiased Estimators 2 of 7 7/16/2009 6:13 AM 1. Show that this coditio is equivalet to the assumptio that the derivative operator d ca be iterchaged with the expected value operator θ. Geerally speakig, the fudametal assumptio will be satisfied if f θ (x) is differetiable as a fuctio of θ, with a derivative that is joitly cotiuous i x ad θ, ad if the support set {x S : f θ (x) > 0} does ot deped o θ. 2. Show that θ dl (f θ( X)) = 0 for θ Θ. Hit: Use the basic coditio with h(x) = 1 for x S. 3. Show that cov θ ( h(x), dl( f θ(x)) = d θ(h(x)) a. b. First ote that the covariace is simply the expected value of the product of the variables, sice the secod variable has mea 0 by the Exercise 2. Use the basic coditio. 4. Prove the followig result. Hit: The variable has mea 0. dl( f θ (X)) var θ ( = dl( f θ (X)) θ ( ( 5. Fially, use the Cauchy-Scharwtz iequality to establish the Cramér-Rao lower boud, amed for Harold Cramér ad CR Rao: ( d 2 θ(h(x)) θ ( dl( f 2 ) 6. Suppose ow that λ(θ) is a parameter of iterest ad h(x) is a ubiased estimator of λ(θ). Use the Cramér-Rao lower boud to show that θ ( dl( f 2 ) 7. Show that equality holds i Exercise 6, ad hece h(x) is a UM VUE, if ad oly if there exists a fuctio u(θ) such that (with probability 1) a. b. h(x) = λ(θ) + u(θ) dl( f θ(x)) Equality holds i the Cauchy-Schwartz iequality if ad oly if the radom variables are liear trasformatios of each other. Recall also that dl (f θ( X)) has mea 0. 2 )
3 Best Ubiased Estimators 3 of 7 7/16/2009 6:13 AM The quatity θ ( dl 2 (f θ( X)) that occurs i the deomiator of the lower bouds of Exercise 5 ad ) Exercise 6 is called the Fisher iformatio umber of X, amed after Sir Roald Fisher. The followig exercises gives a alterate versio for the expressio i Exercise 6 that is usually computatioally better. 8. Show that if the appropriate derivatives exist ad if the appropriate iterchages are permissible the dl( f θ (X)) θ( ( 2) = d 2 θ ( 2 l( f θ(x)) ) 9. Combie Exercise 6 ad Exercise 8 to show that if λ(θ) is a parameter of iterest ad h(x) is a ubiased estimator of λ(θ) the Radom Samples θ d2 2 l( f Suppose ow that X = (X 1, X 2,..., X ) is a radom sample of size from the distributio of a radom variable X havig probability desity fuctio g θ. 10. Prove the followig special case of the Cramér-Rao lower boud. Hit: The joit probability desity fuctio is the product of the margial probability desity fuctios. ( d 2 θ(h(x)) θ ( dl(g 2 ) 11. Suppose ow that λ(θ) is a parameter of iterest ad h(x) is a ubiased estimator of λ(θ). Use Exercise 10 to show that θ ( dl(g 2 ) From Exercise 11, ote that the Cramér-Rao lower boud varies iversely with the sample size. 12. I the settig of the previous exercise, show the followig result (assume that the appropriate derivatives exist ad the appropriate iterchages are permissible): θ d2 2 l(g
4 Best Ubiased Estimators 4 of 7 7/16/2009 6:13 AM Examples ad Special Cases We will apply the results above to several parametric families of distributios. First we eed to recall some stadard otatio. Suppose that X = (X 2, X 2,..., X ) is a radom sample of size from the distributio of a real-valued radom variable X with mea μ. The sample mea is M = 1 i =1 X i The special ad stadard versios of the sample variace are, respectively, The Beroulli Distributio W 2 = 1 i =1 (X i μ) 2, S 2 = 1 1 i =1 (X i M) 2 Suppose that X = (X 1, X 2,..., X ) is a radom sample of size from the Beroulli distributio with ukow success parameter p ( 0, 1). The basic assumptio is satisfied. 13. Show that 1 p (1 p) is the Cramér-Rao lower boud for the variace of ubiased estimators of p. 14. Show that the sample mea M (which is the proportio of successes) attais the lower boud i the previous exercise ad hece is a UM VUE of p. The Poisso Distributio Suppose that X = (X 1, X 2,..., X ) is a radom sample of size from the Poisso distributio with ukow parameter a ( 0, ). The basic assumptio is satisfied. 15. Show that a is the Cramér-Rao lower boud for the variace of ubiased estimators of a. 16. Show that the sample mea M attais the lower boud i the previous exercise ad hece is a UMVUE of a. The Normal Distributio Suppose that X = (X 1, X 2,..., X ) is a radom sample of size from the ormal distributio with mea μ R ad variace σ 2 ( 0, ). The basic assumptio is satisfied with respect to both of these parameters. Recall also that the fourth cetral momet is ((X μ) 4 ) = 3 σ Show that σ 2 is the Cramér-Rao lower boud for the variace of ubiased estimators of μ. 18. Show that the sample mea M attais the lower boud i the previous exercise ad hece is a
5 Best Ubiased Estimators 5 of 7 7/16/2009 6:13 AM UMVUE of μ. 19. Show that 2 σ 4 is the Cramér-Rao lower boud for the variace of ubiased estimators of σ Show (or recall) that the sample variace S 2 has variace 2 σ 4 boud i the previous exercise. 1 ad hece does ot attai the lower 21. Show that if μ is kow, the the special sample variace W 2 attais the lower boud i Exercise 19 ad hece is a UMVUE of σ Show that if μ is ukow, o ubiased estimator of σ 2 attais the Cramér-Rao lower boud i Exercise 19. Hit: Use the result i Exercise 7. The Gamma Distributio Suppose that X = (X 1, X 2,..., X ) is a radom sample of size from the gamma distributio kow shape parameter k ad ukow scale parameter b ( 0, ). The basic assumptio is satisfied with respect to b. 23. Show that b 2 is the Cramér-Rao lower boud for the variace of ubiased estimators of b. k 24. Show that M k attais the lower boud i the previous exercise ad hece is a UMVUE of b. The Beta Distributio Suppose that X = (X 1, X 2,..., X ) is a radom sample of size from the beta distributio with left parameter a > 0 ad right parameter b = 1. The basic assumptio is satisfied with respect to a. 25. Show or recall that the mea ad variace of the distributio are a. b. μ = σ 2 = a a +1 a (a +1) 2 (a +2) 26. Show that the Cramér-Rao lower boud for the variace of ubiased estimators of μ is a 2 (a +1) Show that the sample mea M does ot achieve the Cramér-Rao lower boud i the previous exercise, ad hece is ot a UMVUE of μ. The Uiform Distributio Suppose that X = (X 1, X 2,..., X ) is a radom sample of size from the uiform distributio o [ 0, a] where a > 0 is the ukow parameter.
6 Best Ubiased Estimators 6 of 7 7/16/2009 6:13 AM 28. Show that the basic assumptio is ot satisfied. 29. Show that the Cramér-Rao lower boud for the variace of ubiased estimators of a is a 2. Of course, the Cramér-Rao Theorem does ot apply, by the previous exercise. 30. Show (or recall) that V = +1 smaller tha the Cramér-Rao boud i the previous exercise. max {X 1, X 2,..., X } is ubiased ad has variace a 2, which is ( +2) The reaso that the basic assumptio is ot satisfied is that the support set {x R : f a (x) > 0} depeds o the parameter a. Best Liear Ubiased Estimators We ow cosider a somewhat specialized problem, but oe that fits the geeral theme of this sectio. Suppose that X = (X 1, X 2,..., X ) is a sequece of observable real-valued radom variables that are ucorrelated ad have the same ukow mea μ, but possibly differet stadard deviatios. Let σ = (σ 1, σ 2,..., σ ) where σ i = sd(x i ) for i {1, 2,..., }. We will cosider estimators of μ that are liear fuctios of the outcome variables. Specifically, we will cosider estimators of the followig form, where the vector of coefficiets c = (c 1, c 2,..., c ) is to be determied: 31. Show that Y is ubiased if ad oly if i =1 Y = i =1 32. Compute the variace of Y i terms of c ad σ. c i = Use the method of Lagrage multipliers (amed after Joseph-Louis Lagrage) to show that the variace is miimized, subject to the ubiased costrait, whe 2 1 / σ j c j = i =1 c i X i, j {1, 2,..., } 1 / σ 2 i This exercise shows how to costruct the Best Liear Ubiased Estimator (BLUE) of μ, assumig that the vector of stadard deviatios σ is kow. Suppose ow that σ i = σ for i {1, 2,..., } so that the outcome variables have the same stadard deviatio. I particular, this would be the case if the outcome variables form a radom sample of size from a distributio with mea μ ad stadard deviatio σ. 34. Show that i this case the variace is miimized whe c i = 1 mea. for each i ad hece Y = M, the sample
7 Best Ubiased Estimators 7 of 7 7/16/2009 6:13 AM This exercise shows that the sample mea M is the best liear ubiased estimator of μ whe the stadard deviatios are the same, ad that moreover, we do ot eed to kow the value of the stadard deviatio. Virtual Laboratories > 7. Poit Estimatio > Cotets Applets Data Sets Biographies Exteral Resources Key words Feedback
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