32 estimating the cumulative distribution function
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1 32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio F. Let C be a set of possible values of θ which depeds o the data X 1,..., X. Sice we do t kow the true distributio F of the data, the probability P of somethig happeig depeds o F, ad to emphasize this, we will write P F. 1. C is a fiite sample 1 α cofidece set if if P F (θ C ) 1 α for all. F F 2. C is a F uiform asymptotic 1 α cofidece set if lim if if P F (θ C ) 1 α. F F 3. C is a F poitwise asymptotic 1 α cofidece set if for every F F, lim if P F (θ C ) 1 α. The cofidece sets are listed i order of preferece. Why do you thik that is? If θ is uivariate, the we typically use the term cofidece iterval istead of cofidece set. If θ is a fuctio (such as the desity), ad ot a poit (such as the mea), the the term cofidece iterval is ot used. If is some orm ad f is a estimate of f, the a cofidece ball for f is a cofidece set of the form C = {f F f f s }, where s may deped o the data. Suppose that f is defied o a set X. A pair of fuctios (l, u) is a 1 α cofidece bad of cofidece evelope if if P F (l() f() u() for all X ) 1 α. f F The above defiitios are take (almost eactly) out of Wasserma (2006). I have also see the term "cofidece bad" without the "if f F " part attached. If istead our pair of fuctios (l, u) satisfies P F (l() f() u()) 1 α for all X, the the term "poitwise cofidece iterval" is used. I should say that I do ot fid the above terms to be stadard i the literature, aside from the differetiatio betwee "bad" ad "iterval". However, I thik that it s icredibly importat to make it clear eactly what type of cofidece "set" you are workig with whe commuicatig with others (i.e. whe writig papers/reports). It s probably also good for you to uderstad what various results you are usig/ aimig for.
2 4.7 kolmogorov-smirov statistic 33 Eample 4.1. Let X 1,..., X be IID Beroulli(p) radom variables. cofidece set for p is A p ± z α/2 p (1 p ) is a asymptotic 1 α cofidece iterval for p. Note that the class F is defied by the problem: we re lookig at p (0, 1). Is the cofidece iterval F poitwise or F uiform? The aswer is that it is F poitwise. Although the radom variable p p p(1 p ) coverges weakly to a stadard ormal RV for ay value of p, to obtai a F uiform cofidece set you would eed this covergece to be uiform i p, ad we do ot ecessarily have this here. See the followig theorem for some further details here. Theorem 4.10 (Berry-Essée Boud). Let X 1,..., X be IID with fiite mea µ = E[X 1 ] ad variace σ 2 ad third momet E[ X 1 3 ] <. Let Z = (X µ)/σ. The sup P(Z z) Φ(z) 33 z 4 For the Beroulli, we have that E[ X 1 µ 3 ]. σ 3 E[ X 1 µ 3 ] σ 3 = p3 (1 p) + p(1 p) 3 p 3/2 (1 p) 3/2 = p2 + (1 p) 2 p(1 p), which does t quite do the trick to guaratee a uiform boud. 4.7 kolmogorov-smirov statistic The Kolmogorov-Smirov statistic is a oparametric approach used to test if the sample comes from the distributio F 0. Thus, let S = sup F () F 0 (). From Dosker s theorem, we kow that if F 0 has a desity the S sup U t, 0 t 1 ad (1) gives us the eact distributio of sup 0 t 1 U t. Let q α be a value such that The the test of P( sup U t > q α ) = α. 0 t 1 H 0 the data is distributed accordig to F 0, vs. H A the data is ot distributed accordig to F 0,
3 34 estimatig the cumulative distributio fuctio rejects if S > q α. Let s look at what the Gliveko-Catelli ad Dosker s Theorem tell us... Dosker s Theorem allows us to calculate the rejectio regio, thus esurig that this is a asymptotically correct α level test. Corollary The Kolmogorov-Smirov test is cosistet: that is, if the true distributio is G 0 F 0, the the probability of rejectig the ull hypothesis (i.e. the power) coverges to oe as. Proof. Let G 0 F 0 deote the true distributio of the data. The sup F 0 () G 0 () sup = S + sup F () F 0 () + sup F () G 0 () F () G 0 (). Now, by the Gliveko-Caelli theorem, we ca fid a 0 such that for all 0 S sup F 0 () G 0 () sup F () G 0 () q α, almost surely, ad hece for sufficietly large. P (S q α ) = eercises 1. Oe useful iequality is Theorem 4.12 (Hoeffdig s Iequality). Let Y 1,..., Y be idepedet observatios such that E[Y i ] = 0 ad a i Y i b i. Let ε > 0. The, for ay t > 0, P( Y i ε) e tε e t2 (b i a i ) 2 /2. Now, let X 1,..., X be IID Beroulli(p). a) Use Hoeffdig s iequality to show that P( X p > ε) 2e 2ε2. b) Use the above result to costruct a fiite sample cofidece iterval for p. Prove that it is i fact a fiite sample cofidece iterval. c) Compare the average legth of the cofidece iterval derived above ad that derived i Eample 4.1 usig a simulatio study.
4 4.8 eercises Let X 1,..., X N(µ, 1) with IID samplig. Let C = X ± z α/2 s /, where s 2 = (X i X ) 2. 1 Is C a fiite sample, F-poitwise asymptotic, or F-uiform asymptotic cofidece iterval? What happes if we remove the coditio σ 2 = 1? 3. My applied stats professor disliked the Kolmogorov-Smirov statistic because, as he claimed, it had little power. Eplai what you thik he meas by this, especially i light of the results i Corollary Desig a simulatio which tests this idea. 4. Usig the results of this chapter, create a fiite sample ad a F uiform asymptotic cofidece bad for the ukow true distributio fuctio F. Make sure to justify why the type of the cofidece bad holds. 5. Usig the results from Sectio4.2 we ca derive poitwise cofidece itervals for the true cdf F as F ()(1 F ()) F () ± z α/2. Compare the coverage of these bads ad their width with those you developed i the above questio usig simulatios.
5 E S T I M AT I N G S TAT I S T I C A L F U N C T I O N A L S O F T H E C D F the plug-i estimator A statistical fuctioal T(F) is ay fuctio of F. Eamples iclude the mea, the variace, or the media. The (empirical) plug-i estimator of θ = T(F) is defied by θ = T(F ). The fuctioal T(F) is liear if it takes the form T(F) = a()df(), for some fuctio a. Note that i this case we have that T(F ) = a()df () = 1 Which of the followig eamples are liear? a(x i ). Eample 5.1. Some importat/popular statistical fuctioals are: 1. The mea T(F) = df(). 2. The r th -momet, T(F) = r df() (r is a iteger). 3. The variace, T(F) = 2 2 df() ( df()) = 1 2 ( y) 2 df()df(y). Oe ca show that the plug-i estimator of the variace is the T (F) = 1 (X i X ) 2, which is the biased versio of variace estimator (asymptotically, it s still OK). 4. The geeralized iverse of F is the fuctio F 1 (u) = if { F() u}. We eed the more geeral defiitio sice the cdf of a discrete radom variable is ot ivertible, ad our empirical cdf is i this category. Note that if F is the cdf of the Beroulli(q) radom variable, the F 1 (u) = u {0} 0 u (0, 1 q] 1 u (1 q, 1]. 37
6 38 estimatig statistical fuctioals of the cdf For ay p [0, 1], we call T(F) = F 1 (p) the p th quatile. Its estimate, F (p) is called the p th sample 1 quatile. 5. The skewess is κ = E[(X µ)3 ] σ 2, ad writte as a fuctioal it is equal to T(F) = ( µ) 3 df() { ( µ) 2 df()} 3/2, otig that µ is a fuctioal of F as well! 6. The correlatio is aother eample, but it s a little differet sice F is ow a bivariate distributio. Let T 1 (F) = df((, y)) T 2 (F) = ydf((, y)) T 3 (F) = ydf((, y)) T 4 (F) = 2 df((, y)) T 5 (F) = y 2 df((, y)). The the correlatio ca be writte as the fuctioal T(F) = T 3 (F) T 1 (F)T 2 (F) (T 4 (F) T 2 1 (F))(T 5(F) T 2 2 (F)). Oe ca show that T(F ) = (X i X )(Y i Y ) (X i X ) 2 (Y i Y ) 2. (3) Note that by Theorem 4.1, the bivariate empirical distributio is the distributio which places mass 1/ o each observed pair (X i, Y i ). 7. The Hodges-Lehma fuctioal T(F) = (1/2)(F F) 1 (1/2), where deotes a covolutio. 8. The Ma-Whitey fuctioal T(F, G) = FdG. Assumig that F, G are idepedet, oe ca show that this is P F,G (X Y) = y df()dg(y). 9. Let S(X 1,..., X k+1 ) deote the k-simple of the poits X 1,..., X k+1. Recall that a k-simple of the poits 0,..., k is the set {θ θ k k θ i 0, θ i = 1}. k i=0
7 5.2 eercises 39 Thus, the simple of two poits i R 2 is simply the lie joiig the two poits, while the simple of three poits i R 2 is the triagle formed by the three poits. I geeral, a simple is the cove hull of the k + 1 vertices (poits). The simplicial depth fuctio is the fuctioal defied for a distributio F o R k where the X i are IID F. T(F) T (F) = P F ( S(X 1,..., X k+1 )), The depth fuctio (i this case the simplicial depth) measures the cetrality of a poit w.r.t. a probability distributio F. The correspodig empirical depth fuctio will measure the cetrality of a poit w.r.t the observed data set. 5.2 eercises 1. Suppose that the geeralized iverse is defied istead by F 1 (u) = sup { F() < u}. Fid the iverse of the cdf of the Beroulli(q) radom variable usig this alterative defiitio. 2. Show that (3) holds.
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