Discussion Papers Department of Economics University of Copenhagen

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1 Discssion Papers Department of Economics University of Copenhagen No Discssion of The Forward Search: Theory and Data Analysis, by Anthony C. Atkinson, Marco Riani, and Andrea Ceroli Søren Johansen, Bent Nielsen Øster Farimagsgade 5, Bilding 26, DK-1353 Copenhagen K., Denmark Tel.: Fax: ISSN: (online)

2 Discssion of The Forward Search: Theory and Data Analysis by Anthony C. Atkinson, Marco Riani, and Andrea Ceroli Søren Johansen Department of Economics, University of Copenhagen and CREATES, University of Aarhs and Bent Nielsen y Department of Economics, University of Oxford Febrary 6, 2010 Abstract The Forward Search Algorithm is a statistical algorithm for obtaining robst estimators of regression coe cients in the presence of otliers. The algorithm selects a sccession of sbsets of observations from which the parameters are estimated. The present note shows how the theory of empirical processes can contribte to the nderstanding of how the sbsets are chosen and how the seqence of estimators is changing. Keywords: Empirical processes, Hber s skip, least trimmed sqares estimator, one step estimator, otlier robstness. JEL Classi cation: C2 Address: Department of Economics, University of Copenhagen, Øster Farimagsgade 5, DK-1353 Copenhagen K. Denmark, Soren.Johansen@econ.k.dk. The athor grateflly acknowledges spport from Center for Research in Econometric Analysis of Time Series, CREATES, fnded by the Danish National Research Fondation. y Address: N eld College, Oxford OX1 1NF, UK, bent.nielsen@n eld.ox.ac.k 1

3 1 Introdction The paper by Atkinson, Riani and Ceroli, henceforth ARC, is concerned with detection of otliers and nsspected strctres which is rather important in practice. This is done throgh a Forward Search Algorithm. The statistical analysis of sch algorithms poses many challenging problems, and we wold like to contribte to the theory of the algorithm in this discssion. We establish some reslts in a simple case for a single iteration of the algorithm sing empirical process theory. This wold then have to be extended to more models, and developed frther to nderstand the properties of the fll algorithm. The established reslts sggest that the heristic reslts from ARC cold be correct if the parameters were known, bt not when the parameters are estimated. A general reference for empirical process theory is the monograph by Kol (2002) which analyses weighted empirical processes. Some frther developments are made in Johansen and Nielsen (2009) which we exploit here. For simplicity we only consider a simple location-scale problem bt the reslts wold generalize to regressions and time-series regressions. The discssion is organized so that the algorithm is described in 2. Some potential reslts are described in 3. The analysis of a single step of the algorithm is then provided for the known parameter case in 4 and for the nknown parameter case in 5. We conclde in 6 and leave some proofs to an Appendix. 2 The algorithm We consider the regression problem y i = +" i ; i = 1; : : : ; n; where the i.i.d. " i follow a known distribtion fnction F and symmetric density f, with mean zero and variance 1. The distribtion is assmed to be continos and to satisfy some reglarity conditions for smoothness which are met by the standard normal distribtion, see Johansen and Nielsen (2009, Assmption A). The absolte standardized error j" i j has distribtion fnction G satisfying G() = P(j" i j ) = F() F( ); and = G 1 ( ) is the -qantile of G and its density is g() = 2f(): 2.1 Description of the algorithm We start with some initial robst location estimator ^ and an initial observation set of size m 0 : This set is constrcted by calclating absolte residals ^r i = jy i ^j; nding their order statistics ^r (i), and de ning the initial observation index set of size m 0 as the m 0 observations closest to ^; that is The algorithm then proceeds in the steps S (m 0) = fi : jy i ^j ^r (m0 )g: 2

4 1. Given an index set S (m) calclate estimators ^ (m) = m 1P y i2s (m) i ; (^ (m) ) 2 = m 1P i2s m (y i ^ (m) ) 2 ; residals ^r (m) i = jy i ^ (m) j; and their order statistics ^r (m) (i) for i = 1; : : : ; n: 2. Test whether the residal nearest to the observations in S (m) to an otlier. In ARC the test is based pon (1) test = min ^r (m) i62s (m) i =^ (m) ; does not correspond bt we sggest to se (2) test = ^r (m) (m+1) =^(m) : 3. In the next step we either contine with step 1 or stop the algorithm. (a) If the test based on the nearest residal does not reject, then de ne and retrn to 1. S (m+1) = fi : ^r (m) i ^r (m) (m+1) g (b) If the test rejects, then set ^m = m and de ne the terminal estimators ^ n = ^ ( ^m) ; ^ n = ^ ( ^m) and the observation set ^S n = S ( ^m) : An important problem is to determine distribtions of test statistic and estimators in the case of a sample withot otliers. ARC sggest that the initial estimator ^ cold be chosen as the least trimmed sqares estimator, see Rosseew (1984). This is constrcted by choosing some m > n=2 and nding ^ (LT S;n;m) = arg min P n i=1 (r() i ) 2 1 () (r i r () ); (m) r() i = jy i j: (2.1) 2.2 Comments on the choice of test statistic Comment 2.1 The motivation for the test statistic ^ (m) (2) test = ^r (m) (m+1) is that it will be the largest of the residals with index in S (m+1) : The rank of the observation ^r (m) (m+1) may, however, not enter S (m) : Comment 2.2 The index sets S (m+1) test statistic, (1) test or (2) test: are constrcted independently of the choice of Comment 2.3 In general the test statistics (1) test and (2) test will be di erent. Three reslts follow. 3

5 1. the statistic ^ (m) (1) test = min (m) i62s ^r (m) i sggested by ARC is not an order statistic of the residals r (m) i, becase S (m) (m 1) is based on the previos set of residals ^r i : 2. it holds (1) test (2) test: Indeed, if S (m) statistics are eqal. If S (m) mst inclde one of the ranks of r (m) (1) ; : : : ; r(m) (m) : is the ranks of r (m) (1) ; : : : ; r(m) (m) then these does not have this form then the complement of S (m) 3. The di erence between the two test statistics relates to the distance j^ (m 1) ^ (m) j and is therefore likely to be nimportant. As a nmerical example consider the data set ( 13; 8; 5; 5; 6; 7; 8): The initial estimator is chosen as the sample average ^ = y = 0: The absolte residals are then (^r i ) n i=1 = (13; 8; 5; 5; 6; 7; 8): An initial index set of size 3 is then the ranks S (3) = (3; 4; 5) pointing at the observations ( 5; 5; 6): Now, in the rst step of the algorithm compte the estimator and absolte residals Then (1) test is based on ^ (3) = 2; (^r (3) i ) n i=1 = (15; 10; 7; 3; 4; 5; 6): min ^r (3) i62s (3) i = min(15; 10; 5; 6) = 5; whereas (2) test is based on the order statistic ^r (3) (4) = 6: Regardless of the choice of test statistic the pdated index set is S (4) = (4; 5; 6; 7) pointing at the observations (5; 6; 7; 8): Note that so the sets S (m) (3; 4; 5) = S (3) 6 S (4) = (4; 5; 6; 7); are not in general increasing. 3 Some potential reslts In order that the Forward Search Algorithm can be applied with con dence it is important to derive the distribtions of the test statistics and the estimators. It wold also be of interest to see if the sets S (m) are monotone in m: 3.1 Consistency One wold like to have a reslt as the following 4

6 Potential Theorem 3.1 If the initial estimator (^; ^ 2 ) is consistent for n! 1, and if m = n + O(1) for 0 < < 1; or if m = ^m; then it holds that ^ (m) and ^ (m) have a probability limit. Comment 3.1 We do not have a general proof of sch a reslt, bt some examples are given in 5. One can note here that the di erence between order statistics from i.i.d. observations are of the order of O P (n 1 ); so averaging order statistics which are that close to the initial (consistent estimator) will at most give a deviation from this of the order of O P (n 1 ); and hence shold not distrb consistency. If we cold nd the probability limit of ^ (m) and ^ (m) ; we cold correct the estimators to give consistent estimators, see ARC 3.3, and Theorem Monotonicity The next reslt is allded to in a nmber of places in ARC, even thogh it is realized that it does not hold withot some conditions. Potential Theorem 3.2 The sets S (m) are monotone S (m) S (m+1) ; m = 2; : : : ; ^m: Comment 3.2 As it stands it is nfortnately false, as seen in the example in Comment 2.3. If in general we have fond S (m) for some m; and want to add one point to ; call it y m+1 ; then the new average becomes S (m) y (m+1) = y (m) m (y m+1 y (m) ): Ths, if for instance y m+1 > y (m) ; the average is moved p by (y m+1 y (m) )=(m + 1) which is of the order of m 1 n 1 : The distance between order statistics is of the order of n 1 ; so if one of the observations in S (m) is close to the lower bondary of the band de ning S (m) ; then it cold easily happen that it falls otside when the band is moved p by (y m+1 y (m) )=(m + 1). Ths a point can leave the band when another is added even for large m: This event may have small probability, however, so the following reslt cold hold, bt we have no proof. Potential Theorem 3.3 The sets S (m) are monotone with probability tending to one, that is for m = n + O(1) as n! 1 then P(S (m) S (m+1) )! 1; or, perhaps, 1 n m P n m=m 0 1 (m) (S S (m+1) ) P! 1: 5

7 3.3 The test statistics In ARC it is sggested to nd the distribtion of the test statistics by simlation, and it is arged that it cold be costly measred by compter time. An approximation is sggested in 3.3 of ARC, sing the theory of order statistics, and in a previos paper (Atkinson and Riani, 2006) another approximation based on order statistics is sggested. One wold like to show that the test statistics are asymptotically normal. Potential Theorem 3.4 If the initial estimators (^; ^) are consistent for n! 1, and m = n + O(1); or m = ^m; and if (^ (m) ; ^ (m) ) is consistent, then the test statistic n 1=2 f(^ (m) ) is asymptotically normally distribted. 1^r (m) (m+1) g Comment We can prove this reslt nder sitable conditions and we have collected these contribtions in 5, where we otline a general strategy for nding these limit distribtions. 4 The case of known location and scale When the parameters are known, the residals are r i = jy i as r (1) r (m). Then j = j" i j, which we order S (m) = fi : j i j r (m) g; m = 2; 3; : : : ; ^m: In this case we clearly have S (m) S (m+1) so Theorem 3.2 is correct. The empirical distribtion fnction of j" i j = jy i j= is denoted G n () = n 1 nx i=1 1 (j"i j): The order statistics r (m) have the well known relation 1 r (m), m n G n() since G n ( 1 r (m) ) = m n ; (4.1) which transforms expressions in order statistics into expression involving the empirical distribtion fnction. Moreover, (1) test = (2) test = 1 r (m) and the distribtion is given by the expression (5) in ARC by applying (4.1) as P( 1 r (m) ) = Pf m n G n()g = P n n j=m G() j f1 G()g n j : (4.2) j 6

8 Ths, the distribtion sggested as an approximation in ARC wold in fact be the exact distribtion if the parameters were known. In Atkinson and Riani (2006) another approximation to the distribtion of the test statistic is sggested. It is arged that if m is proportional to n; then we are essentially working in a trncated distribtion where 100 % have been inclded in the sample. Ths, it is sggested to approximate the distribtion of the test statistic (j) test = 1 r (m) by the distribtion of the largest of m observations, z i say, where z i are drawn from the trncated distribtion G()= for 0. By the same methodology as in 3.3 of ARC we then get the approximate reslt P( 1 r (m) ) P(max im z i ) = 1 G() m = h 1 G() i n ; (4.3) which is sggested as an approximation, which is clearly di erent from (4.2). Yet another way of arging is that when we have already sed r (1) ; : : : ; r (m) to constrct S (m) and the estimator ^ (m) ; then signi cance of r (m+1) cold be evalated in the conditional distribtion given r (1) ; : : : ; r (m) ; and that is given by P( 1 r (m+1) jr (1) ; : : : ; r (m) ) = n m 1 G() ; 1 r (m): 1 G( 1 r (m) ) This distribtion can be interpreted as the distribtion of the smallest observation among n m observations r i = jy i j with density 1 g( 1 r)=f1 G( 1 r (m) )g for r r (m) and cold be sed for a conditional test of the next residal given what has already been sed. 5 The case of nknown location and scale We assme we have n 1=2 -consistent estimators (^; ^ 2 ) = (; 2 ) + O P (n 1=2 ) and de ne ^r i = jy i ^j; with order statistics ^r (i) : We de ne ~r = ^r (m) and nx ~ = m 1 y i 1 (jyi ^j~r); (5.1) i=1 nx ~ 2 = m 1 (y i ~) 2 1 (jyi ^j~r): (5.2) i=1 We nd stochastic expansions and limit distribtions of ~r and the one-step estimators (~; ~ 2 ), which are based pon the m observations closest to the initial estimator ^. When applied to the rst iteration of the algorithm then ^; ^ 2 ; ^r (m) represent the initial estimator ^ and residal ^r (m) along with some sitable variance estimator so ~ = ^ (m0) and ~ = ^ (m0) : When applied to a later iteration of the algorithm then ^; ^; ^r (m) represent ^ (m 1) ; ^ (m 1) (m 1) ; ^r (m) so ~ = ^ (m) and ~ = ^ (m). 7

9 5.1 The asymptotic expansions We rst nd an expansion of the test statistic (2) test = ^ 1 ~r which can be applied to nd the asymptotic distribtion of the test for di erent choices of estimators (^; ^ 2 ), and then give expansions of the one-step estimators ~ and ~ 2 ; which show how the estimator is changed from initial estimators (^; ^ 2 ) to one-step estimators (~; ~ 2 ). The proofs are given in the Appendix. Theorem 5.1 If m = n + O(1) and if (^ ; ^ 2 2 ) 2 O P (n 1=2 ) then n 1=2 (^ 1 ~r ) = 1 2f( ) n 1=2P n i=1 f1 (j" i j ) g 2 n1=2 ( 2^ 2 1) + o P (1): Theorem 5.2 If m = n+o(1) and if (^ ; ^ 2 2 ) 2 O P (n 1=2 ) then the estimator ~ de ned in (5.1) satis es n 1=2 (~ ) = n 1=2P n i=1 " i1 (j"i j< ) + 2 f( ) n 1=2 (^ ) + o P (1): Theorem 5.3 If m = n+o(1) and if (^ ; ^ 2 2 ) 2 O P (n 1=2 ) then the estimator ~ 2 de ned in (5.2) satis es ~ 2! P 1 2 ; where 2 = R 2 f()d: We therefore de ne ~ corrected 2 = ~2 = 2 : It holds n 1=2 f 2 ~ 2 corrected 1g = ( 2 ) 1 n 1=2P n i=1 ("2 i 1 2 )1 (j"i j< ) ( 2 ) 1 ( )n 1=2P n i=1 f1 (j" i j ) g + o P (1): Comment 5.1 Note that these reslts are all derived for symmetric distribtions. If this assmption is dropped, a bias term will appear in some asymptotic distribtions and terms inclding n 1=2 (^ ) will appear in Theorems 5.1, Examples We illstrate these reslts by nding the asymptotic distribtion of the test statistic for di erent choices of initial estimators (^; ^ 2 ). First, for comparison we give the reslt for the test statistic for known parameters, where we re-discover a classical reslt on the asymptotic distribtion of order statistics, see for instance David (1981, Theorem 9.2, p. 255). Corollary 5.1 If m = n + O(1) and if ^ = ; ^ = then n 1=2 ( 1 ~r ) D! N[0; (1 ) f2f( )g 2 ]: 8

10 Proof of Corollary 5.1. In the expansion of Theorem 5.1 the second term drops ot since ^ = : Then apply the Central Limit Theorem to the rst term. Alternatively, initial estimators cold be chosen as fll sample estimators ^ = n P 1 n i=1 y i and ^ 2 = n P 1 n i=1 (y i y) 2. This changes the limit distribtion. Corollary 5.2 If m = n + O(1) and if (^; ^ 2 ) are the fll sample estimators then n 1=2 (^ 1 ~r ) D! N[0; (1 ) f2f( )g f1 + 2 f( ) g]: Proof of Corollary 5.2. Apply Theorem 5.1 by inserting the expansion to get that n 1=2 (^ 1 ~r n 1=2 ( 2^ 2 ) eqals 1 2f( ) n 1=2P n i=1 f1 (j" i j ) g 1) = n 1=2P n i=1 ("2 i 1) + o P (1) This is asymptotically normal with a variance as indicated. 2 n 1=2P n i=1 ("2 i 1) + o P (1): Comment 5.2 In general we get a di erent limit distribtion for the test statistic when the variance is estimated. Note the crios reslt that for the standard normal distribtion we nd 2 = Z " 2 f(")d" = Z f(")d" ["f(")] "= "= = 2 f( ): so the asymptotic variance in Corollary 5.2 becomes f2f( )g 2 (1 ) 2 =2; which is less than the variance we get for known parameters. Finally, we shall see what happens when we choose ^ as the least trimmed sqares estimator ^ (LT S;n;m) de ned in (2.1). A stochastic expansion of ^ (LT S;n;m) is given by Víšek (2006, Theorem 1, p. 215) as n 1=2 (^ (LT S;n;m) ) = 2 f( ) n 1=2P n i=1 " i1 (j"i j ) + o P (1): (5.3) Corollary 5.3 If m = n + O(1) and ^ = ^ (LT S;n;m) we nd the expansion n 1=2 (~ ) = so that the limit distribtion is n 1=2 (~ 2 f( ) n 1=2P n i=1 " i1 (j"i j ) + o P (1); ) D! N[0; 2 2 f 2 f( )g 2 ]: If F is standard normal then 2 = 2 f( ) so the variance is 2 = 2 : 9

11 Proof of Corollary 5.3. Insert Víšek s expansion (5.3) in Theorem 5.2 to get n 1=2 (~ ) = f1 + 2 f( ) 2 f( ) gn 1=2P n i=1 " i1 (j"i j ) + o P (1) = 2 f( ) n 1=2P n i=1 " i1 (j"i j ) + o P (1): This converges to a normal distribtion with the variance as indicated. Comment 5.3 Ths, if the initial estimator ^ has the expansion of the least trimmed sqares estimator, then so does the one-step estimator ~: In this sense the least trimmed sqares estimator is a " xed point" in the mapping from the initial estimator to the one-step estimator. ARC do indeed sggest, possibly for other bene cial reasons, to start with the least trimmed sqares estimator. A similar reslt holds if the initial estimator is the Hber skip, which has the same expansion and limit distribtion as the least trimmed sqares estimator, see Johansen and Nielsen (2009). 6 Some nal comments 6.1 The simlation method The above theoretical reslts indicate that the idea of jdging signi cance sing the exact theory of order statistics, seems to be ne if parameters are known. Bt if parameters are estimated the (asymptotic) distribtions change, depending on the choice of initial estimator. Ths it wold be very helpfl with some simlations of the algorithm, as it is sed, to check if asymptotic distribtions can describe the variation of estimators and tests. We have seen that di erent initial estimators give di erent (limit) distribtions. There may also be a problem for very large ; where a di erent asymptotic theory may be needed. 6.2 Generalizations The reslts of Johansen and Nielsen (2009) cover models with general xed or random regressors, as well as time series regressions, both stationary and non-stationary. Tools to stdy the general theory of empirical processes for residals of sch models are otlined in Engle and Nielsen (2009). So it is possible to extend the theory of the Forward Search Algorithm mch beyond what we have indicated in this discssion. 6.3 Algorithms for time series In 7 of ARC it is sggested However, many things remain to be developed in the application of the Forward Search in the time series context, sch as... the constrction of an algorithm which can atomatically distingish among the di erent types of otliers and level shifts. 10

12 While it cold prove very sefl to develop extensions of the Forward Search to time series one shold bear in mind that the algorithm Atometrics by Doornik (2009) bilding on the work of Hoover and Perez (1999) and the PcGets algorithm of Hendry and Krolzig (2005) has been developed for this prpose. In any case, it is a bit srprising to see in ARC that the reslts sggested for regression analysis be applied to ozone data where the residals are likely to be atocorrelated. A Proofs of main theorems The proofs exploit Theorem 1.17 of Johansen and Nielsen (2009); see also their eqation For (^a; ^b) = O P (n 1=2 ) and ` = 0; 1; 2; it was shown nder reglarity conditions that n 1=2P n i=1 "ìf1 (j"i ^bja+^a) 1 (j"i ja)g = a` 1 n 1=2 (^a `+1 + ^b ` ) + o P (1); (A.1) where, for a symmetric density 2j = 0 and 2j+1 = 22j+1 f( ) for j = 0; 1; : : : Proof of Theorem 5.1. In order to nd the asymptotic distribtion of the test statistic ^ 1 ~r we expand as n 1=2 ( ~r^ ) = n 1=2 ^ f( ~r ) ( ^ 1)g: Since ^ 2 2 = O P (n 1=2 ) then 1^ 1 = ( 2^ 2 1)=2 + o P (n 1=2 ) and ^ 1 = 1 + o P (1): It follows that n 1=2 ( ~r^ ) = n 1=2 f( ~r ) 2 ( ^2 2 1)gf1 + o P (1)g + o P (1): (A.2) The rst term in (A.2), is now shown to be asymptotically normal n 1=2 ( ~r ) D! Nf0; (1 ) 2f( ) g: (A.3) The qantile ~r satis es b G n (~r=) = m=n where b G n (r) = n 1 P n i=1 1 (j" i 1 (^ )j<r) is an empirical distribtion fnction; see (4.1). Ths it holds P n def = Pfn 1=2 ( ~r ) zg = Pf ~r + n 1=2 zg where G n = n 1=2 f b G n ( + n 1=2 z) = Pf m n b G n ( + n 1=2 z)g = P(0 G n ); m n g = n 1=2P n i=1 [1 (j" i 1 (^ )j< +n 1=2 z) 11 m n ]:

13 This can be expanded as G n = n 1=2P n i=1 f1 (j" i j< ) g + n 1=2P n i=1 [1 (j" i 1 (^ )j< +n 1=2 z) 1 (j"i j< )] + n 1=2 ( m n ): (A.4) Here, the rst term is asymptotically Nf0; (1 )g: Applying (A.1) with ` = 0; a = ; ^a = n 1=2 z; ^b = 1 (^ ) the second term is 2f( )z + o P (1): The third term vanishes by assmption. Ths, G n is asymptotically Nf2f( )z; (1 )g: In particlar, it follows that P n def = Pfn 1=2 ( ~r ) zg! f 2f( )z p (1 ) g: In trn, ~r is asymptotically normal as stated in (A.3). Next, an expansion for ~r is derived. Since b G n (~r=) = m=n then n 1=2 f b G n ( ~r ) g = n1=2 ( m n ) = o P (1); by the assmption to m: Expanding the left hand term as in (A.4) then gives n 1=2P n i=1 f1 (j" i j< ) g + n 1=2P n i=1 [1 (j" i 1 (^ )j< 1^r (m) ) 1 (j"i j< )] = o P (1): Applying (A.1) with ` = 0; a = ; ^a = 1 ~r ; ^b = 1 (^ ) then shows n 1=2P n i=1 f1 (j" i j< ) g + 2f( )n 1=2 ( 1 ~r ) = o P (1): Solving this eqation for n 1=2 ( 1 ~r into (A.2). ) gives the desired expansion when inserted Proof of Theorem 5.2. The estimator ~ satis es ~ = m 1P n i=1 (y i )1 (jyi ^j~r) = n m n 1P n i=1 (y i )1 (j"i 1 (^ )j< 1 ~r): Adding and sbtracting 1 (j"i j< ) and sing y i = + " i it holds m n 1 n 1=2 (~ ) = n 1=2P n i=1 " i1 (j"i j< ) + n 1=2P n i=1 " if1 (j"i 1 (^ )j< 1 ~r) 1 (j"i j< )g: Here, m 1 n! by assmption. The rst term on the right converges in distribtion by the central limit theorem. Applying (A.1) with ` = 1; a = ; ^a = 1 ~r ; ^b = 1 (^ ) the second term is 2 f( ) 1 n 1=2 (^ ) + o P (1): Inserting these reslts gives the desired expansion. 12

14 Proof of Theorem 5.3. The estimator ~ 2 satis es 2 ~ 2 Using y i ^ = " i (^ ) then 1 2 = m 1P n i=1 f 2 (y i ^) g1 (jyi ^j~r): 2 (y i ^) = (" 2 i 1 2 ) + 2 (^ ) (^ )" i ; while 1 (jyi ^j~r) = 1 (j"i 1 (^ )j< 1 ~r) as before. We de ne the fnctions h 2 () = ; h 1 () = "; and h 0 () = 1; and, for ` = 0; 1; 2; so that S` = n 1=2P n i=1 h`(" i )1 (j"i 1 (^ )j< 1 ~r) = n 1=2P n i=1 h`(" i )1 (j"i j< ) + R`; R` = n 1=2P n i=1 h`(" i )[1 (j"i 1 (^ )j< 1 ~r) 1 (j"i j< )]; m n n1=2 f 2 ~ g = S (^ ) 2 S (^ )S 1 : (A.5) For S 2 the rst term converges in distribtion by the central limit theorem. Applying (A.1) with ` = 2 and ` = 0, a = ; ^a = 1 ~r ; ^b = 1 (^ ); the second term is R 2 = 2f( )( )n 1=2 ( 1 ~r ) + o P (1): It follows that S 2 = n 1=2P n i=1 ("2 i 2 )1 (j"i j< ) + 2f( )( 4 2 )n 1=2 ( 1 ~r ) + o P (1): Inserting the expression for ~r from Theorem 5.1 with ^ = then shows S 2 = n 1=2P n i=1 ("2 i 2 )1 (j"i j< ) ( 4 2 )n 1=2P n i=1 f1 (j" i j ) g + o P (1): The terms S 0 and S 1 are O P (1) by a similar argment. These terms, are however, pre-mltiplied by vanishing terms in that = O P (n 1=2 ) by assmption. Inserting these reslts in (A.5) noting that m 1 n! by assmption shows n 1=2 f 2 ~ g = S 2 + o P (1): With ~ 2 corrected = ~2 = 2 the left hand side becomes and the desired reslt follows. References 2 n 1=2 f 2 ~ 2 corrected 1g Atkinson, A.C. and Riani, M. (2006) Distribtion theory and simlations for tests of otliers in regression. Jornal of Comptational and Graphical Statistics 15,

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