1. Introduction - Econometrics at Illinois - University of Illinois at ...

Roger Koenker and Zhijie Xiao 9Let X 1 ;::: ;X n be iid from F 0 ,soY i = F 0 (X i ); i =1;::: ;n are iid uniform,U[0; 1]: The empirical distribution functionG n (t) =F n (F ,10(t)) = n ,1 nXi=1I(Y i t):viewed as a process, is a submartingale. We have an associated ltration F Gn =fF Gnt :0 t 1g and the order statistics Y (1) ;::: ;Y (n) are Markov times withrespect to F Gn ,thatisfY (i) tg = fF n (t) i=ng 2F Gnt .The process G n (t) isMarkov; Khmaladze notes that for t 0,with G n (0)=0,thus(3.2)nG n (t) = n[G n (t +t) , G n (t)]This suggests the decompositionBinomial(n(1 , G n (t)); t=(1 , t))E(G n (t)jF GntG n (t) =Z t0)= 1 , G n(t)t:1 , t1 , G n (s)ds + m n (t):1 , sThat m n (t) is a martingale then follows from the fact that, from (3.2),E(m n (t)jF Gns )=m n (s)and integrability ofm n (t) follows from the inequalityZ t01 , G n (s)ds ,log(1 , Y (n) );1 , swhich implies a nite mean for the compensator, or predictable component. Substitutingfor G n (t) in(3.2)wehave the classical Doob-Meyer decomposition of theempirical process v nZ tv n (s)v n (t) =w n (t) ,01 , s dswhere v n (t) = p n(G n (t) , t) and the normalized process w n (t) = p nm n (t) convergesweakly to a standard Brownian motion process, w 0 (t); by the argument ofKhmaladze(1981, x2.6).3.3. The Parametric Empirical Process. To extend this approach to the generalparametric empirical process, we nowletg(t) =(t; g(t) > ) > =(t; g 1 (t);:::g m (t)) >be a (m +1)-vector of real-valued functions on [0; 1]. Suppose that the functions_g(t) =dg(t)=dt are linearly independent in a neighborhood of 1 soC(t) Z 1t_g(s)_g(s) > ds