NONPARAMETRIC ESTIMATION FOR MIDDLE-CENSORED DATA

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1 Nonparameric Saisics, 2003, Vol. 15(2), pp NONPARAMETRIC ESTIMATION FOR MIDDLE-CENSORED DATA S. RAO JAMMALAMADAKA a, * and VASUDEVAN MANGALAM b a Deparmen of Saisics and Applied Probabiliy, Universiy of California, Sana Barbara, CA 93106, USA; b Deparmen of Mahemaics, Universii Brunei Darussalam, Brunei (Received 2 November 1999; In final form 22 Augus 2000) This paper provides he self-consisen esimaor (SCE) and he nonparameric maximum likelihood esimaor (NPMLE) for middle-censored daa, in which a daa value becomes unobservable if i falls wihin a random inerval. We provide an algorihm o find he SCE and show ha he NPMLE saisfies he self-consisency equaion. We find a sufficien condiion for he SCE o be concenraed on he uncensored observaions. In addiion, we find sufficien condiions for he consisency of he SCE and prove ha consisency holds for he special case when one of he ends is a consan. Some simulaion resuls and an illusraive example, using Danish melanoma daa se, are provided. Keywords: Survival funcion; Middle-censoring; Self-consisency; Nonparameric maximum likelihood esimaion AMS 1991 Subjec Classificaions: Primary: 62G05, 62G30; Secondary: 62G99 1 INTRODUCTION Esimaion of he unknown disribuion of a random variable is of fundamenal imporance in saisics. In areas such as reliabiliy, biomery, general medical follow-up sudies and clinical rials, he disribuion funcion of he underlying lifeime or more specifically. he survival funcion is of paramoun ineres. In hese siuaions, he random variable of ineres is he lifeime and he observaions refer o imes of occurrence of an even such as deah due o a cerain cause under sudy, or imes for equipmen failure. When complee daa are available, he Empirical Disribuion Funcion (EDF) is used and i has many desirable properies. However, in many pracical siuaions, i is quie common o have incomplee daa, making he sandard empirical disribuion funcion (EDF) unavailable. Ofen, such incomplee observaion of he daa resuls from a random censoring mechanism. When observaions are censored o he righ, he produc limi esimaor due o Kaplan and Meier (1958) is used in place of he EDF and similar esimaors exis for he lef-censored case. Gehan (1965) and Turnbull (1974) and ohers considered doubly-censored daa (where boh lef and righ censoring occur simulaneously) and esimaors for he disribuion funcion have been developed. Groeneboom and Wellner * Corresponding auhor. ISSN prin; ISSN online # 2003 Taylor & Francis Ld DOI: =

2 254 S. R. JAMMALAMADAKA AND V. MANGALAM (1992) and Geskus and Groeneboom (1996) sudied he case of inerval-censored daa where one can only observe a censoring even and wheher he ime of he even of ineres, say deah, occurred before or afer he occurrence of he censoring even. Nonparameric Maximum Likelihood Esimaors (NPMLE) for he disribuion of ineres have been sudied by various auhors for all hese cases. A Self Consisen Esimaor (SCE) is usually obained by solving a se of equaions called he self-consisency equaions (see Efron, 1967; Tarpey and Flury, 1996), and under some condiions his coincides wih he NPMLE. Tsai and Crowley (1985) have shown ha many of hese cases can be unified by applying a generalized maximum likelihood principle. They also poin ou ha solving he self-consisency equaion is essenially equivalen o applying he EM algorihm for he corresponding missing daa problem. See Dempser and Laird (1977) and McLachlan and Krishnan (1997) on he EM algorihm. In his paper we consider an imporan variaion and generalizaion of censoring where a daa poin becomes unobservable if i fell inside a random inerval. When ha happens we observe a censorship indicaor and he inerval of censorship. We will refer o his as middle-censoring. Lef censoring, righ censoring and double censoring are special cases of his middle censorship by suiable choice of his censoring inerval, which can be infinie. Middle-censoring where a random middle par is missing appears a firs glance, as complemenary o he idea of double-censoring where he middle is wha is acually observed. However, if one considers hese wo schemes carefully along wih he resuling daa ses (see nex secion), hey urn ou o be quie disinc ideas. Before discussing he esimaor, we consider some siuaions where his ype of censoring may arise. In general, in any lifeime sudy, if he subjec is emporarily wihdrawn from he sudy we will have an inerval of censorship. I can be equipmen failure ha could occur during a period where observaion is no possible or is no being made. In biomedical sudies, he paien under observaion may be absen from sudy for a shor period during which ime he even of ineres may occur. As an example of double censoring, Turnbull (1974) refers o a sudy of African infan precociy by Leiderman e al. (1973), where esablishing he for infan developmen for a communiy in Kenya was he purpose. A sample of 65 children are considered and each child was esed monhly o see if (s)he had learned o accomplish cerain asks. The ime from birh o he learning ime was he variable of ineres. In heir analysis, double-censoring occurred due o lae arrivals (he child had already learned he skills before enering he sudy) and losses (he child had no acquired he skill by he end of ime sudy). We envisage a scenario where here are no lae enries or losses as such, bu during a fixed ime inerval (his fixed inerval is indeed, a random inerval relaive o he individual s lifeime) he observaion was no possible, such as he emporary closure of he clinic due o an oubreak of say, war. If some children, of varying ages, developed he skill during his ime, we do no observe he exac age of hese children a he ime of skill developmen, raher only he informaion ha he even of ineres occurred during a cerain ime inerval. These ideas can, of course, be exended o more general random ses of censorship such as union of inervals or even more complicaed ses bu we have no explored his in deail. In Secion 2, we derive he self-consisency equaion for he middle censored case and show ha he NPMLE indeed saisfies he self-consisency equaion. A simple example which shows how one compues he NPMLE is also given. In Secion 3 we explore condiions under which he self-consisen esimaor (SCE) is consisen and prove he consisency in an imporan special case. Secion 4 illusraes he SCE for middle-censored case for a simulaed daa se as well as for a real daa se on Melanoma survival, from Andersen e al. (1993). A compuer program which allows he compuaion of he SCE is available by wriing o he auhors.

3 2 SELF-CONSISTENCY AND THE NPMLE Le X i, i ¼ 1,..., n, be a sequence of independen idenically disribued (i.i.d.) random variables wih unknown disribuion F 0. Le Y i ¼ (L i, R i ) be a sequence of i.i.d. random vecors, independen of X s, wih unknown bivariae disribuion G such ha P(L i < R i ) ¼ 1. While X denoes he variable of ineres, Y represens he censoring mechanism. Using he noaion we observe d i ¼ I[X i =2 (L i, R i )], Z i ¼ X i when d i ¼ 1 i:e:, if X i =2 (L i, R i ) ¼ (L i, R i ) when d i ¼ 0 i:e:, if X i 2 (L i, R i ) Tha is, we eiher observe he original value X i, if here is no censoring or he inerval of censoring (L i, R i ) when here is censoring. In many censoring siuaions, if we were o ry o esimae he disribuion funcion via he EM algorihm he resuling equaion akes he form ^F() ¼ E ^F [E n()jz] as described by Tsai and Crowley (1985), where E n is he empirical disribuion funcion. This equaion was firs inroduced and referred o as self-consisency equaion by Efron (1967). In he middle censored case he SCE F n, saisfies he equaion F n () ¼ 1 X n d i I(X i ) þ d n i I(R i ) þ d F n () F n (L i ) i I[ 2 (L i,r i )] (1) F n (R i ) F n (L i ) i¼1 where d i ¼ 1 d i. (For he res of he paper we will follow he convenion ha x, for any variable or funcion x, indicaes 1 x). As in he case of doubly-censored daa, here is no explici closed form soluion o he equaion and has o be compued by he ieraive formula ^F (mþ1) () ¼ E ^F (m) [E n ()jz]: The convergence of he algorihm is assured by Theorem 2.1 of Tsai and Crowley (1985) provided ha he iniial esimaor gives posiive mass o all observed poins. See Remark 2.1 below regarding he choice of he iniial esimaor. For a general discussion on selfconsisency and is relaion o EM algorihm, see Tarpey and Flury (1996). Le F denoe he se of all disribuion funcions on he line. For F 2F he likelihood of he sample is given by L(F) ¼ Yn i¼1 [F(X i ) F(X i )] d i [F(R i ) F(L i )] 1 d i : Denoing by DF(x) ¼ F(x) F(x ), f(f) (1=n) log L(F) is given by X n ESTIMATION FOR MIDDLE-CENSORED DATA 255 f(f) ¼ 1 [d i log(df(x i )) þ d n i log [F(R i ) F(L i )]] i¼1 ¼ {I[x =2 (l, r)] log DF(x) þ I[x 2 (l, r)] log[f(r ) F(l)]} dp n (x, l, r)

4 256 S. R. JAMMALAMADAKA AND V. MANGALAM where P n is he empirical measure of {(X i, L i, R i ): 1 i n}. The maximizer of f is clearly he NPMLE. In he nex heorem, we show ha he NPMLE for middle censored daa saisfies he self-consisency equaion. Bu before ha we need he following lemma. LEMMA 1 Define F( ^ x) A (x) ¼ F(x) F() if F() > 0 ¼ 0 oherwise (2) where ^ x ¼ min(, x). Then K(x) ¼ F(x) þ ha (x) defines a class of disribuion funcions for h sufficienly close o zero. Proof Noe ha we need o show his only for F() > 0 since A 0 when F() ¼ 0. F( ^ x) K(x) ¼ (1 h)f(x) þ h F() is a convex combinaion of wo cdf s and hence is a cdf for 0 h < 1. For negaive h, wrie K ¼ F ha wih h > 0 so ha K(x) ¼ (1 þ h)f(x) h F( ^ x) : F() Clearly K( 1) ¼ 0 and K(1) ¼ 1. I is also righ-coninuous so i remains o show ha i is monoone. We check his separaely on ( 1, ] and [, 1). For x in ( 1, ], K(x) ¼ (1 þ h)f(x) h F(x) F() (1 F()) ¼ F(x) 1 h : F() This is clearly bounded by 1 and is non-negaive if h (F()=1 F()) and in his case, K is monoone non-decreasing. Similarly on [, 1), K(x) ¼ (1 þ h)f(x) h ¼ F(x) h(1 F(x)): Again his is bounded by 1 and is non-negaive so long as h (F(x)=1 F(x)) which is assured if h (F()=1 F()) since x. Monooniciy of K is clear. j THEOREM 1 The NPMLE saisfies he equaion F() ¼ 1 n X n i¼1 d i I(X i ) þ d i I(R i ) þ d F() F(L i ) i I[ 2 (L i, R i )] : (3) F(R i ) F(L i )

5 Proof If F maximizes f, hen he direcional derivaive of f owards A should be zero a F i.e., saisfies he equaion f(f þ ha ) f(f) 0 ¼ lim h!0 h log[df(x) þ hda (x)] log DF(x) ¼ I[x =2 (l, r)] lim h!0 h þi[x 2 (l, r)] lim h!0 log[f(r ) F(l) þ h(a (r ) A (l))] log [F(r ) F(l)] h dp n (x, l, r) ESTIMATION FOR MIDDLE-CENSORED DATA 257 as he inegral involved is really a finie sum and hence inerchanging of limi and inegraion is valid. When F() > 0, he firs of he wo limis inside he inegral is log[df(x) þ e] log DF(x) DA (x) lim ¼ DA (x) e!0 e DF(x) The second limi is similarly equal o A (r ) A (l) F(r ) F(l) F() ^ F(r ) F( ^ I) ¼ 1 F()[F(r ) F(l)] I(x ) ¼ 1: F() where F() ^ F(r ) sands for F() if < r and F(r ) oherwise. Thus we ge or 1 ¼ I[x =2 (l, r)] I(x ) þ I[x 2 (l, r)] F() F() ^ F(r ) F( ^ I) F()[F(r ) F(l)] dp n (x, l, r) F() ¼ I[x =2 (l, r)]i(x ) þ I(l < x < r ) þ I[x, 2 (l, r)]: F() F(l) dp n (x, l, r): F(r ) F(l) RHS of he above is same as RHS of (3). j I is a quesion of considerable ineres o ask if NPMLE will have all is mass on he uncensored observaions. The answer is yes, provided all censored inervals conain a leas one uncensored observaion. When here is a censoring inerval empy of uncensored observaions, clearly some mass has o be aached o ha inerval or he likelihood would be zero. Tha he weighs are concenraed on he uncensored observaions when all censoring inervals are non-empy is a consequence of he following proposiion. PROPOSITION 1 If each observed censored inerval (L i, R i ) conains a leas one uncensored observaion X j, j 6¼ i, hen any disribuion funcion ha saisfies (3) aaches all is mass on he uncensored observaions.

6 258 S. R. JAMMALAMADAKA AND V. MANGALAM Proof Le F be a disribuion saisfying (3) and le x 1, x 2,..., x m be he uncensored observaions. For any x le DF(x) ¼ F(x) F(x ) be he weigh F associaes o x. We need o show ha P m j¼1 DF(x j) ¼ 1. From (3) i follows ha DF(x j ) ¼ 1 n þ 1 n Summing (4) over all uncensored observaions, we ge X m X n i¼1 DF(x j ) (1 d i )I[x j 2 (L i, R i )] : (4) F(R i ) F(L i ) X n DF(x j ) ¼ m n þ 1 (1 d i ) P j I[x j 2 (L i, R i )]DF(x j ) (5) n F(R j¼1 i¼1 i ) F(L i ) For each censoring inerval (L i, R i ), le a i be he slack beween he mass associaed o he inerval and he sum of weighs of all uncensored observaions in he inerval. Then a i ¼ F(R i ) F(L i ) Xm I[x j 2 (L i, R i )]DF(x j ) (6) j¼1 and a i s are all non-negaive. From (5) and (6), i follows ha 1 X a i ¼ 1 1 X a i n F(R i ) F(L i ) or X ai ¼ 1 X a i (7) n F(R i ) F(L i ) where he sum is over all censored observaions. As every inerval conains a leas one uncensored observaion, i follows from (4) ha F(R i ) F(L i ) (1=n) and hence (7) implies ha a i a i ¼ (8) n(f(r i ) F(L i )) for each i. Now if here exiss i such ha a i > 0, (F(R i ) F(L i ))(1=n) þ a i > (1=n) conradicing (8). j We have now proved ha he NPMLE will have all is mass on he uncensored observaions excep when i so happens ha a censored inerval conains no uncensored observaion. If his happens, we are in a siuaion similar o ha of righ censored daa where he larges observaion is censored. While in he righ censored case he exra mass is usually lef unassigned, for middle-censored daa here is a naural way of handling his. When a censored inerval conains no uncensored poins, we le he mass ha corresponds o ha inerval be assigned o is midpoin. Thus our iniial esimaor may give equal mass o all uncensored observaions and o he midpoins of hose finie censored inervals ha conain no uncensored observaions. If an infinie censoring inerval happens o be empy of uncensored observaions, one can hen assign he mass o any arbirary poin inside his inerval for he esimaor o have a maximum. Consider he following example where n ¼ 5 and z 1 ¼ 2, z 2 ¼ 4, z 3 ¼ 6, z 4 ¼ (1, 5) and z 5 ¼ (3, 7). Le p 1, p 2, p 3 be he masses o be assigned o z 1, z 2, z 3 respecively. The likelihood funcion is given by p 1 p 2 p 3 ( p 1 þ p 2 )( p 2 þ p 3 )

7 ESTIMATION FOR MIDDLE-CENSORED DATA 259 and, as p i s add up o 1 and he roles of p 1 and p 3 are inerchangeable, we can simplify he problem o ha of maximizing (x 2 )(1 p 2x)(1 x) 2 wih p 1 ¼ p 3 ¼ x and p 2 ¼ 1 2x. The soluion, hen, is given by x ¼ (5 ffiffiffi p 5 )=10 so ha p1 ¼ p 3 ¼ (5 ffiffi p 5 )=10 and p2 ¼ 1= ffiffi 5 is he soluion o he NPMLE. In his example he ieraions of he self-consisency equaion rapidly converged o he NPMLE. The SCE, being a resul of convergence of he EM algorihm, provides a local maximum of he likelihood equaion [see, for example, Mykland and Ren, 1996] and may no coincide wih he NPMLE. Examples of cases when an SCE is no he NPMLE can be consruced by considering siuaions where wo empy censoring inervals overlap. For insance, if we have 1, 2, (3, 6), (4, 7) as he daa, we could assign 0.25 mass o 1, 2, 4.5 and 5.5 o ge an SCE. The NPMLE will assign 0.25 each on 1 and 2, bu assign 0.5 on some poin, say 5, on he overlap area (4, 6). Boh esimaors are self-consisen, bu he laer has higher likelihood. This happens whenever here are empy, overlapping inervals. In he nex secion we shall show he srong consisency of self-consisen esimaors for cerain cases. This will demonsrae ha SCE and NPMLE are, a leas for hese special cases, asympoically equivalen and hence will be approximaely he same for large samples. 3 CONSISTENCY OF SELF-CONSISTENT ESTIMATORS Define P and Q, sub-disribuion funcions on R and R 2 respecively, by and heir empirical versions P n and Q n by P() ¼ P(X, d ¼ 1) Q(l, r) ¼ P(L l, R r, d ¼ 0) P n () ¼ 1 n Q n (l, r) ¼ 1 n X n i¼1 X n i¼1 I(X i, d i ¼ 1) I(L i l, R i r, d i ¼ 0): Then by Glivenko-Canelli Lemma, i follows ha P n and Q n converge almos surely o P and Q respecively and he convergence in each case is uniform on he respecive domain. Also, (1) can be wrien in erms of P n and Q n as follows: Fn () ^ F n (r ) F n ( ^ l) F n () ¼ P n () þ dq n (l, r) (9) F n (r ) F n (l) By Helly s Theorem, 9 a subsequence n k and a non-decreasing funcion F bounded by 0 and 1 such ha on a se of probabiliy 1, F nk ()!F() for each. PROPOSITION 2 If {j n } is a sequence of funcions on R 2 which converge uniformly o bounded coninuous funcion j, hen j n (l, r)dq n (l, r)! j(l, r) dq(l, r):

8 260 S. R. JAMMALAMADAKA AND V. MANGALAM Proof Noe ha, j n (l, r)dq n (l, r) j n (l, r)dq(l, r) (j n (l, r) j(l, r)) dq n (l, r) þ j(l, r)dq n (l, r) j(l, r) dq(l, r) kj n jk dq n þ j(l, r)dq n (l, r) j(l, r) dq(l, r) where kk represens he supremum norm. Now he firs erm on he RHS of he inequaliy goes o zero since Ð dq n ¼ 1 while he second erm goes o zero because he sequence of empirical measures Q n converge o Q weakly and j is a bounded coninuous funcion. j LEMMA 2 Any subsequenial limi F of F n will saisfy he equaion F() ^ F(r ) F( ^ l) F() ¼ P() þ dq(l, r) (10) F(r ) F(l) Proof For a fixed, aking limis in (9) hrough he subsequence n k as k!1and using Proposiion 2 wih j n (l, r) ¼ F n() ^ F n (r ) F n ( ^ l), F n (r ) F n (l) he resul follows. j When P and Q are wrien in erms of F 0 and G, (10) is equivalen o F() F(l) F() F 0 () ¼ F(r ) F(l) (F 0(r) F 0 (l)) þ F 0 (l) F 0 () dg(l, r): (11) l<<r From (11), i follows ha F(1) ¼ 1 and F( 1) ¼ 0. Noe ha if F ¼ F 0, (11) is auomaically saisfied. If we were able o show ha (11) has a unique soluion, hen i follows ha F 0 is he only limi poin of {F n }. Then we will have ha on a se of probabiliy 1, F n (x)! F 0 (x) for each x and by coninuiy of F 0 uniformiy of he almos sure convergence follows. A necessary condiion for consisency is wha we call idenifiabiliy. Le A() ¼ P(L < < R). The condiion is ha A be no idenically 1 on any inerval [a, b], a b for which F 0 (b) > F 0 (a ). Observe ha if A 1 on any inerval where F 0 has a posiive mass, hen censoring occurs wih probabiliy 1 on such an inerval. As a consequence, here will be no observaions on his inerval and ha prevens us from disinguishing any wo disribuions which are idenical ouside [a, b] bu differing only on [a, b]. This condiion will be referred o as he idenifiabiliy condiion and is a requiremen for consisen esimaion of F 0. LEMMA 3 Le h ¼ (F F 0 ) and g() ¼ c(l, r)i(l < < r)dg(l, r) R 2

9 ESTIMATION FOR MIDDLE-CENSORED DATA 261 where c(l, r) ¼ (h(r) h(l)=f(r ) F(l)). Then Adh ¼ gdf (12) Proof From (11) we ge h() ¼ ¼ l<<r l<<r h(r) h(l) (F() F(l)) þ h(l) h() dg(l, r) F(r ) F(l) [(F() F(l))c(l, r) þ h(l)]dg(l, r) þ h()a(): So, where h() A() ¼ l<<r [(F() F(l))c(l, r) þ h(l)]dg(l, r) ¼ F()g() þ C() (13) C() ¼ l<<r Differeniaing boh sides of (13) w.r.., we ge [h(l) F(l)c(l, r)]dg(l, r): h()da() A()dh() ¼ g()df() þ F()dg() þ dc(): To show ha (12) holds, clearly i is sufficien o show ha F()dg() þ dc() h()da() ¼ 0 (14) If B() ¼ Ð l<<r H(l, r)dg(l, r) for some funcion H, hen i can be shown ha db() ¼ ( Ð 1 H(, r)df RjL (rj))df L () ( Ð 1 H(l, )df LjR(lj))dF R () where F RjL ( j) is he condiional disribuion funcion of R given L ¼. Hence, applying his o g and C, 1 LHS of(14) ¼ F() c(, r)df RjL (rj) df L () F() c(l, )df LjR (lj) df R () 1 1 þ (h() F()c(, r))df RjL (rj) df L () (h(l) F(l)c(l, ))df LjR (lj) df R () h()da() 1 ¼ df R () 1 h()df LjR (lj) þ df L () 1 h()df RjL (rj) h()da() ¼ h()(df L () df R ()) h()da() ¼ 0 because Ð 1 df LjR(lj) ¼ Ð 1 df RjL (rj) ¼ 1 and A() ¼ F L () F R (): j Thus, if he only funcion h saisfying (12) is he zero funcion, hen we would have proved he srong consisency of he SCE. We have no ye been able o show ha his (12) has a unique soluion in he general case, bu we give below a proof for he special case when one of he end poins of he censoring inerval is degenerae. Alhough he resul is saed for he case when L is degenerae (for insance, censoring if i occurs, sars on a cerain birhday of he individual), he proof works equally well when R is degenerae.

10 262 S. R. JAMMALAMADAKA AND V. MANGALAM THEOREM 2 Assume ha F 0 and F R are coninuous and L l 0. Assume he idenifiabiliy condiion is saisfied. Then he only funcion F ha saisfies 11Þ is F 0 and hence he SCE is uniformly srongly consisen. Proof In his special case (13) becomes h() A() ¼ I( > l 0 ) 1 [(F() F(l 0 ))c(l 0, r) þ h(l 0 )]df R (r) (15) As A() ¼ 1 P[ 2 (l 0, R)] ¼ 1 I( > l 0 ) F R (), A() ¼ 1 for all l 0 and A() ¼ F R () for all > l 0 ; hence from (15), h() ¼ 0 for all l 0. In paricular, h(l 0 ) ¼ 0. Thus (15) becomes 1 F() F(l 0 ) h() A() ¼ I( > l 0 ) F(r ) F(l 0 ) h(r)df R(r): Similarly, (12) holds wih 1 h(r) g() ¼ I( > l 0 ) F(r ) F(l 0 ) df R(r): (16) Noe ha from he assumpions of he heorem i follows ha F, h and g are coninuous on (l 0, 1). We aim o show ha h 0on(l 0, 1). Assuming 9 0 > l 0 such ha h( 0 ) > 0, we will arrive a a conradicion. The proof is similar if h( 0 ) < 0. j As h(l 0 ) ¼ h(1) ¼ 0, 9 1 < 2 such ha 1 l 0, 2 1, h( 1 ) ¼ h( 2 ) ¼ 0 and h() > 0on ( 1, 2 ). From (16), on (l 0, 1), g() ¼ Ð 1 c(r)df R (r) where c(r) ¼ (h(r)=f(r ) F(l 0 )). CLAIM 1 g( 1 ) 0. Suppose no. Then 9 such ha g > 0on( 1, ). We shall now show ha A() > 0on ( 1, ). If 9 2 ( 1, ) such ha A() ¼ 0, hen A() ¼ 0 for all 2 (l 0,) so ha by he idenifiabiliy condiion, df 0 () ¼ 0 for all 2 (l 0, ). From (12) df() ¼ 0on( 1, ); so dh() ¼ df() df 0 () ¼ 0on( 1, ) which implies h 0 here, conrary o our assumpion. From (12), dh() ¼ ( g()=1 A())dF(), so Ð 1 ( g()=1 A())dF() ¼ h( ) h( 1 ) ¼ h( ). Now, LHS 0 conradicing he fac ha h( ) > 0. This proves Claim 1. CLAIM 2 g( 2 ) 0. Suppose no. Then 9 2 ( 0, 2 ) such ha g < 0on(, 2 ). Similar o he previous siuaion, we have A() > 0on(, 2 ). As earlier, h( 2 ) h( ) ¼ Ð 2 ( g()=1 A())dF() 0, implying h( ) 0 which is conradicion. Thus Claim 2 is proved. On ( 1, 2 ), dg() ¼ c()df R () ¼ ( h()df R ()=F( ) F(l 0 )) 0, so g is decreasing here. Thus from Claim 1 and Claim 2 i follows ha g 0on( 1, 2 ). (Noe ha he argumen Ð goes hrough even if 2 ¼1). From (12), Adh 0 on ( 1, 2 ). As g() ¼ 1 c(r)df R (r), F R is a consan ( 1, 2 ) and hence A is a consan on (1, 2 ). If c > 0, h 0on( 1, 2 ) which is a conradicion. If c ¼ 0, A 0on( 1, 2 ) and hence by he idenifiabiliy condiion F 0 is consan on ( 1, 2 ). As h( 1 ) ¼ h( 2 ) ¼ 0, F( 1 ) ¼ F( 2 ), so F is a consan on ( 1, 2 ). So h is a consan on ( 1, 2 ), which means h 0on( 1, 2 ). This is a conradicion. 4 ILLUSTRATIVE EXAMPLES A simulaion sudy was performed o measure he performance of self-consisen esimaors where an exponenial (mean ¼ 10) random variable was middle-censored by random inervals

11 ESTIMATION FOR MIDDLE-CENSORED DATA 263 FIGURE 1 The EDF and SCE for he simulaed exponenial daa. wih lef end poins being exponenial (mean ¼ 5) and inerval widhs being an independen exponenial (mean ¼ 5). A sample of size 100 was aken and his resuled in 22 of hem being censored. Figure 1 shows he SCE along wih he original exponenial disribuion funcion F 0. The maximum disance kf n F 0 k is which is very good compared o he Kolmogorov Smirnov disance, namely he maximum disance of he EDF. E n of he uncensored daa from he rue disribuion, which is ke n F 0 k¼0:0715. The auhors also ried ou various oher survival disribuions such as gamma and Weibull ha were censored by inervals whose lef ends were disribued as exponenial, gamma, Weibull or uniform and inerval widh was a posiive random variable or a consan. In all hese cases, he resuling esimaors for middle censoring were in very close agreemen wih he EDF of he original uncensored daa. I is clear ha he amoun of censoring in any of hese cases, is approximaely P(L X R). Finally we applied our echniques o an acual daa se on melanoma survival colleced a Odense Universiy Hospial, Denmark [see Andersen e al., 1993]. The sample conains 205 daa poins, ranging from 10 o The daa were censored by a random inerval whose lef end was an exponenial random variable wih mean 2000 and widh was exponenial wih mean Over 23% of daa were censored resuling in 157 uncensored observaions. The SCE F n is given in Figure 2 while he EDF E n of he survival daa is in Figure 3. They are shown super-imposed in Figure 4, o see how close hey are. Indeed, he maximum disance kf n E n k beween hem is while he maximum relaive disance k((f n E n )=E n )k urns ou o be sill a small FIGURE 2 SCE for he censored melanoma survival daa.

12 264 S. R. JAMMALAMADAKA AND V. MANGALAM FIGURE 3 EDF for he uncensored melanoma survival daa. FIGURE 4 EDF and SCE superimposed. Acknowledgemens We would like o hank an anonymous referee whose persisence led o a much more readable paper. References Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N. (1993). Saisical Models Based on Couning Processes. Springer-Verlag, New York. Dempser, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplee daa via he EM algorihm. J. Roy. Sais. Soc. Ser., B39, 1 38 (wih discussion). Efron, B. (1967). The wo-sample problem wih censored daa. Proc. Fifh Berkeley Symp. Mah. Sa. Probab., Vol. 4. Universiy of California Press, Berkeley, pp Gehan, E. A. (1965). A generalized wo-sample Wilcoxon es for doubly censored daa. Biomerika, 52, Geskus, R. B. and Groeneboom, P. (1996). Asympoically opimal esimaion of smooh funcionals for inerval censoring. J. Sais. Neerlandica, 50, Groeneboom, P. and Wellner, J. A. (1992). Informaion Bounds and Non-parameric Maximum Likelihood Esimaion. DMV Seminar, Vol. 19. Birkhäuser Verlag, Basel. Gu, M. G. and Zhang, C.-H. (1993). Asympoic properies of self-consisen esimaors based on doubly censored daa. Ann. Sais., 21, Kaplan, E. L. and Meier, P. (1958). Nonparameric esimaion from incomplee observaions. J. Amer. Sais. Assoc., 53, Leiderman, P. H., Babu, D., Kagia, J., Kraemer, H. C. and Leiderman, G. F. (1973). African infan precociy and some social influences during he firs year. Naure, 242, McLachlan, G. J. and Krishnan, T. (1997). The EM Algorihm and Exensions. John Wiley and Sons, Inc., New York.

13 ESTIMATION FOR MIDDLE-CENSORED DATA 265 Mykland, P. A. and Ren, J. (1996). Algorihms for compuing self-consisen and maximum likelihood esimaors wih doubly censored daa. Ann. Sais., 24, Tarpey, T. and Flury, B. (1996). Self-consisency: A fundamenal concep in saisics. Saisical Science, 11, Tsai, W. Y. and Crowley, J. (1985). A large sample sudy of generalized maximum likelihood esimaors from incomplee daa via self-consisency. Ann. Sais., 13, Turnbull, B. W. (1974). Nonparameric esimaion of a survivorship funcion wih doubly censored daa. J. Amer. Sais. Assoc., 69,

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