CHAPTER 7 Bads for distributios This chapter cosiders cofidece bads for a distributio fuctio ad some related fuctios. Chapter 5.8 describes bads for kerel desity estimates. For X R,the cumulative distributio fuctio is F(x) =F ((,x]) = Pr(X x) take as a fuctio of x. A cofidece bad for F(x) is a pair of fuctios L(x) ad H(x) for which Pr(L(x) F(x) H(x), x R) =1 α (7.1) uder idepedet samplig of X i F. The radomess i (7.1) arises from the fact that L ad U deped o X 1,...,X,although this is suppressed from the otatio. Some exact cofidece bads are available,others are asymptotic. If the iequalities i (7.1) were imposed oly at B poits x,the result could be described as a B-dimesioal hyper-rectagular cofidece regio. Bads are essetially ifiite dimesioal hyper-rectagles. As such,they are do ot ecessarily correspod to tests with the greatest power. Ellipsoids or other shapes are ofte better. Bads have the advatage that they ca be easily plotted. Bads are also of iterest for some related fuctios. The quatile fuctio Q(u) is defied through Q(u) =F 1 (u) if{x u F(x)}, 0 <u<1. (7.2) The defiitio (7.2) makes Q uique eve whe F(x) =F(x )=ufor x x. For idepedet real-valued data X 1,...,X F ad Y 1,...,Y m G,the QQ plot is formed by plottig a estimate of QQ(x) =G 1 (F(x)).Ifthesample QQ plot lies far from the 45 lie QQ(x) =x,the the distributios F ad G differ. For three or more samples from distributios F 1,...,F k,we ca select oe of the distributios,say F 1,as a baselie,ad defie a k 1 dimesioal quatilequatile fuctio by (F 1 2 (F 1(x)),...,F 1 k (F 1(x))),over x. The survival fuctio is S(t) =F ((t, )) = 1 F ((,t]). It is widely used i medical applicatios,as is the cumulative hazard fuctio t df (u) Λ(t) = 0 F ((,u)). These are discussed i Chapter 6.5.
7.1 The ECDF The empirical CDF is the value ˆF(x) =#{X i x}/,take as a fuctio of x. The 95% Kolmogorov-Smirov bads for F are of the form ˆF(x) ± D 0.95,where D 1 α is defied i terms of the radom variable D sup ˆF(x) F(x), (7.3) <x< by Pr(D D 1 α )=1 α. Such bads ca have exact coverage levels for fiite,because the distributio of D for X i F is the same for ay cotiuous distributio F.IfF is ot cotiuous,the Kolmogorov-Smirov bads have greater tha the omial coverage level. To see why the distributio of D does ot deped o F,write the order statistics of the sample as X (1) X (2)... X (),ad itroduce radom variables U i = F(X i ).TheU i are idepedet observatios from the U(0, 1) distributio,ad have order statistics U (i) = F(X (i) ). For cotiuous F the supremum i (7.3) occurs either immediately to the left or right of a observatio X (i), so ( ) D = max max i 1 1 i F(X (i)), i F(X (i)) ( ) = max max i 1 1 i U (i), i U (i). For ay cotiuous F, D ca be expressed i terms of the order statistics of a uiform sample,ad so D 1 α ca be calculated for oe distributio,such as F = U(0, 1),ad the applied to ay cotiuous distributio. The hypothesis that X i have CDF F is rejected at level α whe F is ot cotaied withi the bads at all t. Where the upper bad goes above 1 it is replaced by 1,ad similarly the lower bad is replaced by 0 where it goes below 0. The Kolmogorov-Smirov bads are widely used,but they are ot particularly sesitive i the tails. To address this problem,weighted Kolmogorov-Smirov bads,of the form D ψ = ψ(f(x)) ˆF(x) F(x), sup <x< have bee proposed. For example,the choice ψ(z) =(z (1 z)) 1/2 (7.4) weights each poit x i iverse proportio to the stadard deviatio of ˆF(x),ad so puts more weight o the tail regios. The radom variable ˆF(x) has the biomial distributio with parameters ad p = F(x). Kolmogorov-Smirov bads are based o the most extreme discrepacy betwee the observed ad expected biomial radom variables. The weighted versio with weights (7.4) takes accout of the uequal variaces of
those biomial radom variables. Empirical likelihood bads may be costructed usig the most extreme biomial likelihood at ay x. Empirical likelihood for F(x) at a sigle poit x was preseted i Chapter 3.6. For 0 <p<1,ad <q<,defie { +1 R(p, q) = max w i i=1 i=0 +1 w i Z i (p, q) =0,w i 0, i=0 w i =1 with Z i (p, q) =1 Xi q p,takig X 0 = ad X +1 =,so that Z 0 = 1 p ad Z +1 = p. A asymptotic cofidece iterval for F(x) is {p 2 log R(p, x) χ 2 (1) }. To get a cofidece bad for F,we cosider the distributio of the most extreme poitwise likelihood,via Let c 1 α E = satisfy Pr(E c 1 α sup log R(F(x),x). <x< )=1 α. The the bad (L(x),H(x)) with } L(x) = mi { p log R(p, x) c 1 α H(x) = max { p log R(p, x) c 1 α is a 100(1 α)% cofidece bad for F(x). First we cosider costructig L ad H give c 1 α,the we cosider how to fid c 1 α. } }, 7.2 Exact calibratio of ECDF bads It is computatioally easy to obtai a exact calibratio for empirical likelihood bads. The reaso is that for ay set of umbers a 1,...,a ad b 1,...,b,there is a recursive algorithm to compute Pr ( a i U (i) b i, i =1,..., ). See the discussio of Noé s recursio i Chapter 7.4. Noé s recursio also applies to weighted Kolmogorov-Smirov cofidece bads. From equatio (3.15) i Chapter 3.6, 1 log R(p, x) =ˆp log(ˆp/p)+(1 ˆp) log((1 ˆp)/(1 p)), (7.5) where ˆp =ˆp(x) =#{X i x}/ = F ((,x]),ad p = F(x). For fixed ˆp, equatio (7.5) is a covex fuctio of p with a miimum of 0 at p =ˆp. Thus L(x) ad H(x) ca be easily foud by safeguarded searches,like those described i Chapter 2.9,startig i the itervals (0, ˆp) ad (ˆp, 1),respectively. Covexity i p of (7.5) implies that log R(p, x) c 1 α if ad oly if L(x) p H(x).The bads L(x) ad H(x) are piecewise costat fuctios,takig jumps at the observed values X (i). Therefore,it is oly ecessary to compute them at +1 differet poits. Let L i ad H i be the values of L(x) ad H(x),respectively,
o the ope iterval (X (i),x (i+1) ),for i = 0,...,,with X (0) = ad X (+1) =. Havig foud either the L i or the H i,the other oes ca be foud by symmetry through L i =1 H i. Note that L(X (i) ) = mi(l i 1,L i )=L i 1 ad H(X (i) ) = max(h i 1,H i ),for 1 i. Therefore, H(x) is cotiuous from the right ad L(x) is cotiuous from the left. To calibrate the curves we eed to fid c 1 α. The extreme value of E must take place at or just to the left of a order statistic X (i). Thus E = max 1 i max ( log R ( F(X (i) ),X (i) ), log R ( F(X (i) ),X (i) )). Suppose that F is cotiuous. The R(p, q) with X i F is the same as R(p, F(q)) o data U i = F(X i ). Thus we may write ( E = max max log R (U (i), i ) (, log R U (i), i )) 1 i ( ( = max max log R U (i), i 1 ) (, log R U (i), i )). 1 i Now E c 1 α is equivalet to a i L i 1 U (i) H (i) b i, i =1,...,. It follows that Noé s algorithm ca be employed to fid the coverage probability for ay value of c 1 α. A oe-dimesioal umerical search ca the be employed to fid the value of c 1 α. Critical values c 1 α ca be precomputed ad tabulated. It may be more coveiet to store them as afuctio of. The fuctio values i Table 7.1 give very accurate coverage for the stadard coverage levels 0.95 ad 0.99,for sample sizes up to 1000. 7.3 Asymptotics of bads The cofidece bads of the previous sectio were costructed without employig ay asymptotics. This was made possible by Noé s recursio. These bads have good power properties. Suppose that X i have a cotiuous distributio F. The the empirical likelihood cofidece bad of level 1 α has better asymptotic power for rejectig a alterative F F tha a weighted Kolmogorov-Smirov bad of level 1 α. This holds simultaeously for all weighted Kolmogorov- Smirov bads ad all alteratives F F. Such uiversal optimality is surprisig because F might oly differ from F i a arrow iterval,ad a weighted Kolmogorov-Smirov bad might be costructed to be particularly sesitive to departures from F i just that oe iterval. See Chapter 7.4. The power cosid-
Coverage 95% to95.01% Sample size =1: 2.9957 Sample sizes 1 < 100: 3.0123 + 0.4835log() 0.00957 log() 2 0.001488 log() 3 Sample sizes 100 < 1000: 3.0806 + 0.4894 log() 0.02086 log() 2 Coverage 99% to99.01% Sample size =1: 4.60517 Sample sizes 1 < 100: 4.626 + 0.541 log() 0.0242 log() 2 Sample sizes 100 < 1000: 4.71 + 0.512 log() 0.219 log() 2 Table 7.1 Show are approximate critical values c 1 α, for empirical likelihood cofidece bads for the CDF from Owe (1995). The omial coverage level is 1 α, either 0.95 or 0.99. The actual coverage level is betwee the omial level, ad the omial plus 0.0001. The sample sizes are from = 1to = 1000. ered is of large deviatios type. Further large deviatios results are described i Chapter 13.5. The empirical likelihood cofidece bads are based o the distributio of the most extreme of 2 biomial p-values,arisig from a upper ad a lower boud at each of poits. These p-values are strogly correlated with each other because they are based o the same data. It is iterestig to compare the critical value of the likelihood used i settig bads with the fiite degrees of freedom case. Figure 7.1 plots c 0.95 versus for 1 1000. The effective degrees of freedom correspodig to c are defied to be d such that Pr(χ 2 (d) 2c )=0.95. The factor of 2 eters because i parametric settigs the test statistic is mius twice a log likelihood where c 0.95 was developed for a egative log likelihood. Chisquareds o fractioal degrees of freedom are Gamma distributios. For =1,the effective degrees of freedom are d =2. The effective degrees of freedom icrease very slowly with,to d =3at =7,to d =4at =62,ad to d =5at some >1000. The effective degrees of freedom would be slightly differet at a cofidece level other tha 0.95. The effective degrees of freedom are very early liear i c. The case =1is iterestig. It ivolves just oe quatile. As for oe quatile a χ 2 (1) limit is appropriate. The effect of =1istead of = is to chage the degrees of freedom from 1 to 2.
Critical Likelihood Effective DF 3.0 3.5 4.0 4.5 5.0 5.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 200 400 600 800 1000 Sample size 1 5 10 50 500 Sample size Figure 7.1 The left plot shows the critical likelihood threshold for exact 95% empirical likelihood cofidece bads for the distributio fuctio. The sample sizes rage from 1 to. A critical likelihood of c correspods to a effective degrees of freedom of d where Pr(χ 2 (d) 2c) =0.95. The right plot shows effective degrees of freedom versus sample size. The two quatities have early the same depedece o sample size. This is early liear o a log scale as show i the right plot. 7.4 Bibliographic otes Exact cofidece bads for the CDF based o empirical likelihood were published by Owe (1995). Hollader,McKeague & Yag (1997) fid asymptotic cofidece bads for the survival fuctio, 1 F,from right-cesored data. The weights (7.4) were proposed by Aderso & Darlig (1952). The better kow Aderso-Darlig statistic is based o a itegral over x,ot a extreme as preseted here. It correspods to a ifiite dimesioal ellipsoidal regio istead of a ifiite dimesioal hyper-rectagle. The recursive algorithm for fidig the probability that the ECDF from a U(0, 1) sample stays withi a give bad is due to Noé (1972). It takes O() space, ad appears to be umerically stable for 1000. Noé s algorithm is give i Shorack & Weller (1986). The fact that the bads described here give a test with better asymptotic power tha ay weighted Kolmogorov-Smirov test at ay alterative to U(0, 1) was proved by Berk & Joes (1979) usig the otio of relative optimality discussed i Berk & Joes (1978). Qi & Lawless (1994) show that the error i estimatig a distributio fuctio is smaller if side iformatio is used. Zhag (1996a) ad Zhag (1999) describe cofidece bads for the distributio fuctio,give some side iformatio expressed through estimatig equatios.
Switzer (1976) computes a cofidece bad for the QQ fuctio by ivertig Smirov s two sample rak test. Cofidece bads for the quatile fuctio are give by Zhag (1997),by resamplig from the NPMLE. Li,Hollader,Mc- Keague & Yag (1996) preset cofidece bads for the quatile fuctio from cesored data. Eimahl & McKeague (1999) create empirical likelihood-based cofidece tubes for QQ plot relatig samples from two or more populatios.