CORD-19-Sentences  

CORD-19:07815f7f4d5711d2c737bfc2562bd07bf4644189 JSONTXT 9 Projects

Annnotations TAB TSV DIC JSON TextAE

Id Subject Object Predicate Lexical cue
TextSentencer_T1 0-71 Sentence denotes The nonparametric maximum likelihood estimator for middle-censored data
TextSentencer_T1 0-71 Sentence denotes The nonparametric maximum likelihood estimator for middle-censored data
TextSentencer_T2 73-81 Sentence denotes Abstract
TextSentencer_T2 73-81 Sentence denotes Abstract
TextSentencer_T3 82-139 Sentence denotes on any interval [a,b], a r b for which F 0 ðbÞ 4F 0 ðaÀÞ.
TextSentencer_T3 82-139 Sentence denotes on any interval [a,b], a r b for which F 0 ðbÞ 4F 0 ðaÀÞ.
TextSentencer_T4 140-274 Sentence denotes Observe that if A 0 (x)=1 on any interval where F 0 has a positive mass, then censoring occurs with probability 1 on such an interval.
TextSentencer_T4 140-274 Sentence denotes Observe that if A 0 (x)=1 on any interval where F 0 has a positive mass, then censoring occurs with probability 1 on such an interval.
TextSentencer_T5 275-329 Sentence denotes As a consequence, there will be no observations on the
TextSentencer_T5 275-329 Sentence denotes As a consequence, there will be no observations on the
TextSentencer_T6 330-371 Sentence denotes Contents lists available at ScienceDirect
TextSentencer_T6 330-371 Sentence denotes Contents lists available at ScienceDirect
TextSentencer_T7 373-533 Sentence denotes In this note, we consider data subjected to middle censoring where the variable of interest becomes unobservable when it falls within an interval of censorship.
TextSentencer_T7 373-533 Sentence denotes In this note, we consider data subjected to middle censoring where the variable of interest becomes unobservable when it falls within an interval of censorship.
TextSentencer_T8 534-838 Sentence denotes We demonstrate that the nonparametric maximum likelihood estimator (NPMLE) of distribution function can be obtained by using Turnbull's (1976) EM algorithm or self-consistent estimating equation (Jammalamadaka and Mangalam, 2003) with an initial estimator which puts mass only on the innermost intervals.
TextSentencer_T8 534-838 Sentence denotes We demonstrate that the nonparametric maximum likelihood estimator (NPMLE) of distribution function can be obtained by using Turnbull's (1976) EM algorithm or self-consistent estimating equation (Jammalamadaka and Mangalam, 2003) with an initial estimator which puts mass only on the innermost intervals.
TextSentencer_T9 839-1029 Sentence denotes The consistency of the NPMLE can be established based on the asymptotic properties of self-consistent estimators (SCE) with mixed interval-censored data (Yu et al., 2000 (Yu et al., , 2001 .
TextSentencer_T9 839-1029 Sentence denotes The consistency of the NPMLE can be established based on the asymptotic properties of self-consistent estimators (SCE) with mixed interval-censored data (Yu et al., 2000 (Yu et al., , 2001 .
TextSentencer_T10 1030-1050 Sentence denotes & 2011 Elsevier B.V.
TextSentencer_T10 1030-1050 Sentence denotes & 2011 Elsevier B.V.
TextSentencer_T11 1051-1071 Sentence denotes All rights reserved.
TextSentencer_T11 1051-1071 Sentence denotes All rights reserved.
TextSentencer_T12 1072-1187 Sentence denotes Middle censoring occurs when a data point falls inside a random censoring interval whereby it becomes unobservable.
TextSentencer_T12 1072-1187 Sentence denotes Middle censoring occurs when a data point falls inside a random censoring interval whereby it becomes unobservable.
TextSentencer_T13 1188-1312 Sentence denotes For some individuals the exact values are available while for others the corresponding intervals of censorship are observed.
TextSentencer_T13 1188-1312 Sentence denotes For some individuals the exact values are available while for others the corresponding intervals of censorship are observed.
TextSentencer_T14 1313-1764 Sentence denotes We mention two situations where middle censoring occurs. (i) In a follow-up study, if the childhood learning center where the observations are being taken, is closed for a period, due to an external emergency such as the outbreak of severe acute respiratory syndrome (SARS). (ii) In a clinical trial, where the clinic where the observations are being taken, is closed for a period, due to an external emergency such as the outbreak of war or a strike.
TextSentencer_T14 1313-1764 Sentence denotes We mention two situations where middle censoring occurs. (i) In a follow-up study, if the childhood learning center where the observations are being taken, is closed for a period, due to an external emergency such as the outbreak of severe acute respiratory syndrome (SARS). (ii) In a clinical trial, where the clinic where the observations are being taken, is closed for a period, due to an external emergency such as the outbreak of war or a strike.
TextSentencer_T15 1765-1965 Sentence denotes For situations (i) and (ii) where during a fixed time interval (this fixed interval is indeed, a random interval (denoted by (U,V)) relative to individual's lifetime) the observation was not possible.
TextSentencer_T15 1765-1965 Sentence denotes For situations (i) and (ii) where during a fixed time interval (this fixed interval is indeed, a random interval (denoted by (U,V)) relative to individual's lifetime) the observation was not possible.
TextSentencer_T16 1966-2279 Sentence denotes If some children (or patients) develop a skill (or disease) of interest during this time, we are not able to observe the exact age T of these children (or patients) at the time of skill (or disease) development, rather only the information that the event of interest occurred during a certain time interval (U,V).
TextSentencer_T16 1966-2279 Sentence denotes If some children (or patients) develop a skill (or disease) of interest during this time, we are not able to observe the exact age T of these children (or patients) at the time of skill (or disease) development, rather only the information that the event of interest occurred during a certain time interval (U,V).
TextSentencer_T17 2280-2465 Sentence denotes At first glance, middle censoring, where a random middle part is missing, appears as complementary to the idea of double censoring in which the middle part is what is actually observed.
TextSentencer_T17 2280-2465 Sentence denotes At first glance, middle censoring, where a random middle part is missing, appears as complementary to the idea of double censoring in which the middle part is what is actually observed.
TextSentencer_T18 2466-2599 Sentence denotes However, a careful reflection and analysis shows them to be quite different ideas; see Jammalamadaka and Mangalam (2003) for details.
TextSentencer_T18 2466-2599 Sentence denotes However, a careful reflection and analysis shows them to be quite different ideas; see Jammalamadaka and Mangalam (2003) for details.
TextSentencer_T19 2600-2692 Sentence denotes Let T i , i=1,y,n, be a sequence of i.i.d. random variables with distribution function F 0 .
TextSentencer_T19 2600-2692 Sentence denotes Let T i , i=1,y,n, be a sequence of i.i.d. random variables with distribution function F 0 .
TextSentencer_T20 2693-2863 Sentence denotes Independent of T i 's, let (U i ,V i ), i=1,y,n, be i.i.d. extended real-valued random variables with joint distribution function K 0 ðx,yÞ ¼ PðU i r x,V i r yÞ such that
TextSentencer_T20 2693-2863 Sentence denotes Independent of T i 's, let (U i ,V i ), i=1,y,n, be i.i.d. extended real-valued random variables with joint distribution function K 0 ðx,yÞ ¼ PðU i r x,V i r yÞ such that
TextSentencer_T21 2864-2999 Sentence denotes interval and that prevents us from distinguishing any two distributions which are identical outside [a,b] but differing only on [a,b] .
TextSentencer_T21 2864-2999 Sentence denotes interval and that prevents us from distinguishing any two distributions which are identical outside [a,b] but differing only on [a,b] .
TextSentencer_T22 3000-3298 Sentence denotes In many censoring situations, if we were to try to estimate the distribution function via the EM algorithm the resulting equation takes the form F S ðtÞ ¼ EF S ½E n jX, ð1:2Þ as described by Tsai and Crowley (1985) , where E n is the empirical distribution function and X denotes the observed data.
TextSentencer_T22 3000-3298 Sentence denotes In many censoring situations, if we were to try to estimate the distribution function via the EM algorithm the resulting equation takes the form F S ðtÞ ¼ EF S ½E n jX, ð1:2Þ as described by Tsai and Crowley (1985) , where E n is the empirical distribution function and X denotes the observed data.
TextSentencer_T23 3299-3396 Sentence denotes This equation was first introduced and referred to as self-consistency equation by Efron (1967) .
TextSentencer_T23 3299-3396 Sentence denotes This equation was first introduced and referred to as self-consistency equation by Efron (1967) .
TextSentencer_T24 3397-3472 Sentence denotes A solutionF S of (1.2) is called a self-consistent estimator (SCE) of F 0 .
TextSentencer_T24 3397-3472 Sentence denotes A solutionF S of (1.2) is called a self-consistent estimator (SCE) of F 0 .
TextSentencer_T25 3473-3626 Sentence denotes In different types of censoring, the relationship between nonparametric maximum likelihood estimator (NPMLE) and SCE has been studied by various authors.
TextSentencer_T25 3473-3626 Sentence denotes In different types of censoring, the relationship between nonparametric maximum likelihood estimator (NPMLE) and SCE has been studied by various authors.
TextSentencer_T26 3627-3781 Sentence denotes In the case of right censoring the product-limit estimator (Kaplan and Meier, 1958) is the NPMLE, and Efron (1967) showed that it is also self-consistent.
TextSentencer_T26 3627-3781 Sentence denotes In the case of right censoring the product-limit estimator (Kaplan and Meier, 1958) is the NPMLE, and Efron (1967) showed that it is also self-consistent.
TextSentencer_T27 3782-3907 Sentence denotes In the double-censoring case, Mykland and Ren (1996) provided a necessary and sufficient condition for an SCE to be an NPMLE.
TextSentencer_T27 3782-3907 Sentence denotes In the double-censoring case, Mykland and Ren (1996) provided a necessary and sufficient condition for an SCE to be an NPMLE.
TextSentencer_T28 3908-4049 Sentence denotes In the middle censored cases the self-consistent estimator (SCE) (see Jammalamadaka and Mangalam, 2003 )F S satisfies the following equation:
TextSentencer_T28 3908-4049 Sentence denotes In the middle censored cases the self-consistent estimator (SCE) (see Jammalamadaka and Mangalam, 2003 )F S satisfies the following equation:
TextSentencer_T29 4050-4158 Sentence denotes ð1:3Þ Jammalamadaka and Mangalam (2003) showed that the NPMLE satisfies the self-consistency equation (1.3).
TextSentencer_T29 4050-4158 Sentence denotes ð1:3Þ Jammalamadaka and Mangalam (2003) showed that the NPMLE satisfies the self-consistency equation (1.3).
TextSentencer_T30 4159-4274 Sentence denotes They also pointed out that an SCE provides only a local maximum of the likelihood equation and may not be an NPMLE.
TextSentencer_T30 4159-4274 Sentence denotes They also pointed out that an SCE provides only a local maximum of the likelihood equation and may not be an NPMLE.
TextSentencer_T31 4275-4455 Sentence denotes Furthermore, they showed that the NPMLE will have all its mass on the uncensored observations except when it so happens that a censored interval contains no uncensored observation.
TextSentencer_T31 4275-4455 Sentence denotes Furthermore, they showed that the NPMLE will have all its mass on the uncensored observations except when it so happens that a censored interval contains no uncensored observation.
TextSentencer_T32 4456-4597 Sentence denotes The consistency of the SCEF S was established by Jammalamadaka and Mangalam (2003) for the special case when either U i or V i is degenerate.
TextSentencer_T32 4456-4597 Sentence denotes The consistency of the SCEF S was established by Jammalamadaka and Mangalam (2003) for the special case when either U i or V i is degenerate.
TextSentencer_T33 4598-4772 Sentence denotes Jammalamadaka and Iyer (2004) proposed an approximation to the distribution function F 0 (denoted by F 0 0 ) for which a modified self-consistent estimatorF 0 S was obtained.
TextSentencer_T33 4598-4772 Sentence denotes Jammalamadaka and Iyer (2004) proposed an approximation to the distribution function F 0 (denoted by F 0 0 ) for which a modified self-consistent estimatorF 0 S was obtained.
TextSentencer_T34 4773-4893 Sentence denotes They established the asymptotic properties ofF 0 S and provided an upper bound of the difference between F 0 and F 0 0 .
TextSentencer_T34 4773-4893 Sentence denotes They established the asymptotic properties ofF 0 S and provided an upper bound of the difference between F 0 and F 0 0 .
TextSentencer_T35 4894-5194 Sentence denotes Mangalam et al. (2008) showed that under condition (A) every censoring interval contains at least one uncensored observation, i.e. d i ¼ 0 implies that there exists j such that d j ¼ 1 and X j 2 ðU i ,V i Þ, the solution of Eq. (1.3) will be unique, and as a consequenceF n will be equal to an NPMLE.
TextSentencer_T35 4894-5194 Sentence denotes Mangalam et al. (2008) showed that under condition (A) every censoring interval contains at least one uncensored observation, i.e. d i ¼ 0 implies that there exists j such that d j ¼ 1 and X j 2 ðU i ,V i Þ, the solution of Eq. (1.3) will be unique, and as a consequenceF n will be equal to an NPMLE.
TextSentencer_T36 5195-5313 Sentence denotes Mangalam et al. (2008) proposed a technique for obtaining the NPMLE by dividing the original problem into subproblems.
TextSentencer_T36 5195-5313 Sentence denotes Mangalam et al. (2008) proposed a technique for obtaining the NPMLE by dividing the original problem into subproblems.
TextSentencer_T37 5314-5460 Sentence denotes In this note, we aim to establish connections between the middle-censoring and interval censoring by investigating the self-consistency algorithm.
TextSentencer_T37 5314-5460 Sentence denotes In this note, we aim to establish connections between the middle-censoring and interval censoring by investigating the self-consistency algorithm.
TextSentencer_T38 5461-5731 Sentence denotes In Section 2, we shall demonstrate that the NPMLE of F 0 can be obtained by using the EM algorithm of Turnbull (1976) or the self-consistent estimating equation (Jammalamadaka and Mangalam, 2003) with an initial estimator which puts mass only on the innermost intervals.
TextSentencer_T38 5461-5731 Sentence denotes In Section 2, we shall demonstrate that the NPMLE of F 0 can be obtained by using the EM algorithm of Turnbull (1976) or the self-consistent estimating equation (Jammalamadaka and Mangalam, 2003) with an initial estimator which puts mass only on the innermost intervals.
TextSentencer_T39 5732-5872 Sentence denotes Furthermore, we establish the consistency and asymptotic normality of the NPMLE by using the results of Yu et al. (2000 Yu et al. ( , 2001 .
TextSentencer_T39 5732-5872 Sentence denotes Furthermore, we establish the consistency and asymptotic normality of the NPMLE by using the results of Yu et al. (2000 Yu et al. ( , 2001 .
TextSentencer_T40 5873-5877 Sentence denotes 2.1.
TextSentencer_T40 5873-5877 Sentence denotes 2.1.
TextSentencer_T41 5878-5988 Sentence denotes Self-consistency Turnbull (1976) characterized the NPMLE in the presence of interval censoring and truncation.
TextSentencer_T41 5878-5988 Sentence denotes Self-consistency Turnbull (1976) characterized the NPMLE in the presence of interval censoring and truncation.
TextSentencer_T42 5989-6048 Sentence denotes Frydman (1994) later corrected Turnbull's characterization.
TextSentencer_T42 5989-6048 Sentence denotes Frydman (1994) later corrected Turnbull's characterization.
TextSentencer_T43 6049-6104 Sentence denotes Here, we consider the case when there is no truncation.
TextSentencer_T43 6049-6104 Sentence denotes Here, we consider the case when there is no truncation.
TextSentencer_T44 6105-6257 Sentence denotes Turnbull (1976) , Frydman (1994) and Alioum and Commenges (1996) , we consider nonparametric estimation of F 0 using the independent observation A i 's.
TextSentencer_T44 6105-6257 Sentence denotes Turnbull (1976) , Frydman (1994) and Alioum and Commenges (1996) , we consider nonparametric estimation of F 0 using the independent observation A i 's.
TextSentencer_T45 6258-6433 Sentence denotes Since Turnbull's EM algorithm can be used to tackle the case when A i is a single point set, the connection between middle censoring and interval censoring can be established.
TextSentencer_T45 6258-6433 Sentence denotes Since Turnbull's EM algorithm can be used to tackle the case when A i is a single point set, the connection between middle censoring and interval censoring can be established.
TextSentencer_T46 6434-6505 Sentence denotes Based on the notations defined above, the likelihood is proportional to
TextSentencer_T46 6434-6505 Sentence denotes Based on the notations defined above, the likelihood is proportional to
TextSentencer_T47 6506-6590 Sentence denotes where P F 0 ðA i Þ denotes the probability that is assigned to the interval by F 0 .
TextSentencer_T47 6506-6590 Sentence denotes where P F 0 ðA i Þ denotes the probability that is assigned to the interval by F 0 .
TextSentencer_T48 6591-6613 Sentence denotes We define an NPMLE asF
TextSentencer_T48 6591-6613 Sentence denotes We define an NPMLE asF
TextSentencer_T49 6614-6729 Sentence denotes For censored data, using graph theory, Gentleman and Vandal (2001) presented methods for finding the NPMLE of F 0 .
TextSentencer_T49 6614-6729 Sentence denotes For censored data, using graph theory, Gentleman and Vandal (2001) presented methods for finding the NPMLE of F 0 .
TextSentencer_T50 6730-6865 Sentence denotes For censored and truncated data, Hudgens (2005) employed a graph theoretical approach to describe the support set of the NPMLE of F 0 .
TextSentencer_T50 6730-6865 Sentence denotes For censored and truncated data, Hudgens (2005) employed a graph theoretical approach to describe the support set of the NPMLE of F 0 .
TextSentencer_T51 6866-7112 Sentence denotes Define innermost intervals H j , j=1,y,J, induced by A 1 ,y,A n to be all the disjoint intervals which are non-empty intersections of these A i 's (e.g. A k ¼ A k \ A k is an intersection of A i 's) such that A i \ H j ¼ | or H j for all i and j.
TextSentencer_T51 6866-7112 Sentence denotes Define innermost intervals H j , j=1,y,J, induced by A 1 ,y,A n to be all the disjoint intervals which are non-empty intersections of these A i 's (e.g. A k ¼ A k \ A k is an intersection of A i 's) such that A i \ H j ¼ | or H j for all i and j.
TextSentencer_T52 7113-7236 Sentence denotes Let the endpoints of the innermost intervals be q j and p j , j=1,y,J, where 0 r q 1 rp 1 r q 2 rp 2 r Á Á Á rq J rp J r 1:
TextSentencer_T52 7113-7236 Sentence denotes Let the endpoints of the innermost intervals be q j and p j , j=1,y,J, where 0 r q 1 rp 1 r q 2 rp 2 r Á Á Á rq J rp J r 1:
TextSentencer_T53 7237-7374 Sentence denotes Peto (1973) showed that the NPMLE of F 0 assigns weight, say s 1 , . . . ,s J , to the corresponding innermost intervals H 1 ,y,H J only.
TextSentencer_T53 7237-7374 Sentence denotes Peto (1973) showed that the NPMLE of F 0 assigns weight, say s 1 , . . . ,s J , to the corresponding innermost intervals H 1 ,y,H J only.
TextSentencer_T54 7375-7418 Sentence denotes Thus, it suffices to maximize O ¼ s 2 R J :
TextSentencer_T54 7375-7418 Sentence denotes Thus, it suffices to maximize O ¼ s 2 R J :
TextSentencer_T55 7419-7678 Sentence denotes Based on the estimatorsŝ j 's, an estimatorF M ðtÞ of F 0 (t) can be uniquely defined for t 2 ½p j ,q j þ 1 Þ byF M ðp j Þ 1 F M ðq j þ 1 ÀÞ ¼ŝ 1 þ Á Á Á þŝ j , but is not uniquely defined for t being in an open innermost interval (q j ,p j ) with q j o p j .
TextSentencer_T55 7419-7678 Sentence denotes Based on the estimatorsŝ j 's, an estimatorF M ðtÞ of F 0 (t) can be uniquely defined for t 2 ½p j ,q j þ 1 Þ byF M ðp j Þ 1 F M ðq j þ 1 ÀÞ ¼ŝ 1 þ Á Á Á þŝ j , but is not uniquely defined for t being in an open innermost interval (q j ,p j ) with q j o p j .
TextSentencer_T56 7679-7800 Sentence denotes To avoid ambiguity we defineF M ðtÞ ¼ŝ 1 þ Á Á Á þŝ jÀ1 þ s j ðtÀq j Þ=ðp j Àq j Þ if t 2 ðq j ,p j and 0 o q j o p j o1.
TextSentencer_T56 7679-7800 Sentence denotes To avoid ambiguity we defineF M ðtÞ ¼ŝ 1 þ Á Á Á þŝ jÀ1 þ s j ðtÀq j Þ=ðp j Àq j Þ if t 2 ðq j ,p j and 0 o q j o p j o1.
TextSentencer_T57 7801-7889 Sentence denotes Next, we shall show that the estimatorF M satisfies self-consistent equation (1.3), i.e.
TextSentencer_T57 7801-7889 Sentence denotes Next, we shall show that the estimatorF M satisfies self-consistent equation (1.3), i.e.
TextSentencer_T58 7890-7895 Sentence denotes ð2:8Þ
TextSentencer_T58 7890-7895 Sentence denotes ð2:8Þ
TextSentencer_T59 7896-7906 Sentence denotes Theorem 1.
TextSentencer_T59 7896-7906 Sentence denotes Theorem 1.
TextSentencer_T60 7907-7940 Sentence denotes The NPMLEF M satisfies Eq. (2.8).
TextSentencer_T60 7907-7940 Sentence denotes The NPMLEF M satisfies Eq. (2.8).
TextSentencer_T61 7941-7947 Sentence denotes Proof.
TextSentencer_T61 7941-7947 Sentence denotes Proof.
TextSentencer_T62 7948-8102 Sentence denotes First, notice that for each ðq j ,p j Þ ðj ¼ 1, . . . ,JÞ either q j =p j or q j o p j and there is no uncensored observation in (q j ,p j ) if q j op j .
TextSentencer_T62 7948-8102 Sentence denotes First, notice that for each ðq j ,p j Þ ðj ¼ 1, . . . ,JÞ either q j =p j or q j o p j and there is no uncensored observation in (q j ,p j ) if q j op j .
TextSentencer_T63 8103-8141 Sentence denotes Furthermore, d j ðŝÞ can be written as
TextSentencer_T63 8103-8141 Sentence denotes Furthermore, d j ðŝÞ can be written as
TextSentencer_T64 8142-8233 Sentence denotes Consider an initial estimatorF ð0Þ , which puts mass only on ðq j ,p j Þ ðj ¼ 1, . . . ,JÞ.
TextSentencer_T64 8142-8233 Sentence denotes Consider an initial estimatorF ð0Þ , which puts mass only on ðq j ,p j Þ ðj ¼ 1, . . . ,JÞ.
TextSentencer_T65 8234-8275 Sentence denotes LetF ð1Þ denote the first step estimator.
TextSentencer_T65 8234-8275 Sentence denotes LetF ð1Þ denote the first step estimator.
TextSentencer_T66 8276-8436 Sentence denotes Without changing the innermost intervals and likelihood function, we can transform data by moving all right censored points between p j À 1 and q j to p j À 1 .
TextSentencer_T66 8276-8436 Sentence denotes Without changing the innermost intervals and likelihood function, we can transform data by moving all right censored points between p j À 1 and q j to p j À 1 .
TextSentencer_T67 8437-8532 Sentence denotes Similarly, move all left censored points between p j À 1 and q j to q j (see Li et al., 1997) .
TextSentencer_T67 8437-8532 Sentence denotes Similarly, move all left censored points between p j À 1 and q j to q j (see Li et al., 1997) .
TextSentencer_T68 8533-8614 Sentence denotes Hence, we have Hence, F (1) also puts mass only on ðq j ,p j Þ ðj ¼ 1, . . . ,JÞ.
TextSentencer_T68 8533-8614 Sentence denotes Hence, we have Hence, F (1) also puts mass only on ðq j ,p j Þ ðj ¼ 1, . . . ,JÞ.
TextSentencer_T69 8615-8657 Sentence denotes Next, we consider the following two cases:
TextSentencer_T69 8615-8657 Sentence denotes Next, we consider the following two cases:
TextSentencer_T70 8658-8677 Sentence denotes Case 1: q j = p j .
TextSentencer_T70 8658-8677 Sentence denotes Case 1: q j = p j .
TextSentencer_T71 8678-8720 Sentence denotes When q j =p j , we havê Case 2: q j op j .
TextSentencer_T71 8678-8720 Sentence denotes When q j =p j , we havê Case 2: q j op j .
TextSentencer_T72 8721-8855 Sentence denotes When q j op j , since there is no uncensored observations in (q j ,p j ), we havê First, we have P n i ¼ 1 d i I ½q j o T i o p j ¼ 0.
TextSentencer_T72 8721-8855 Sentence denotes When q j op j , since there is no uncensored observations in (q j ,p j ), we havê First, we have P n i ¼ 1 d i I ½q j o T i o p j ¼ 0.
TextSentencer_T73 8856-9040 Sentence denotes Next, note that given an interval (L i ,R i ) and d i ¼ 0, we either have ðq j ,p j Þ DðL i ,R i Þ or This conclusion is the same as in Theorem 1 of Jammalamadaka and Mangalam (2003) .
TextSentencer_T73 8856-9040 Sentence denotes Next, note that given an interval (L i ,R i ) and d i ¼ 0, we either have ðq j ,p j Þ DðL i ,R i Þ or This conclusion is the same as in Theorem 1 of Jammalamadaka and Mangalam (2003) .
TextSentencer_T74 9041-9109 Sentence denotes However, our proof is based on the EM algorithm of Turnbull (1976) .
TextSentencer_T74 9041-9109 Sentence denotes However, our proof is based on the EM algorithm of Turnbull (1976) .
TextSentencer_T75 9110-9254 Sentence denotes It is obvious that if condition (A) holds then q j =p j for all j, and the solutionŝ j 's will have all its mass on the uncensored observations.
TextSentencer_T75 9110-9254 Sentence denotes It is obvious that if condition (A) holds then q j =p j for all j, and the solutionŝ j 's will have all its mass on the uncensored observations.
TextSentencer_T76 9255-9459 Sentence denotes Furthermore, if we start with an initial estimator which puts weight 1/J on q j = p j for uncensored observations and on (q j +p j )/2 for censored observations, we can obtain an NPMLE by using Eq. (2.8).
TextSentencer_T76 9255-9459 Sentence denotes Furthermore, if we start with an initial estimator which puts weight 1/J on q j = p j for uncensored observations and on (q j +p j )/2 for censored observations, we can obtain an NPMLE by using Eq. (2.8).
TextSentencer_T77 9460-9595 Sentence denotes However, similar to interval-censored data, the self-consistent NPMLE of F 0 is not uniquely defined for x 2 ðq j ,p j Þ if q j o p j .
TextSentencer_T77 9460-9595 Sentence denotes However, similar to interval-censored data, the self-consistent NPMLE of F 0 is not uniquely defined for x 2 ðq j ,p j Þ if q j o p j .
TextSentencer_T78 9596-9735 Sentence denotes An SCE with an initial estimator which puts weight on intervals other than (q j , p j ) can lead to a less efficient estimator (not NPMLE).
TextSentencer_T78 9596-9735 Sentence denotes An SCE with an initial estimator which puts weight on intervals other than (q j , p j ) can lead to a less efficient estimator (not NPMLE).
TextSentencer_T79 9736-9805 Sentence denotes In this section, we shall investigate large sample properties ofF M .
TextSentencer_T79 9736-9805 Sentence denotes In this section, we shall investigate large sample properties ofF M .
TextSentencer_T80 9806-9861 Sentence denotes First, we introduce mixed interval censored (MIC) data.
TextSentencer_T80 9806-9861 Sentence denotes First, we introduce mixed interval censored (MIC) data.
TextSentencer_T81 9862-9991 Sentence denotes A data set is called a MIC data when it consists of both exact observations and case 2 interval censoring data (i.e. L i o R i ).
TextSentencer_T81 9862-9991 Sentence denotes A data set is called a MIC data when it consists of both exact observations and case 2 interval censoring data (i.e. L i o R i ).
TextSentencer_T82 9992-10273 Sentence denotes Mixed IC data arises in clinical follow-up studies where a tumor maker (e.g., Ca 125 in ovarian cancer) is available, a patient whose marker value is consistently on the high (or low) end of normal range in repeated testing is usually under close surveillance for possible relapse.
TextSentencer_T82 9992-10273 Sentence denotes Mixed IC data arises in clinical follow-up studies where a tumor maker (e.g., Ca 125 in ovarian cancer) is available, a patient whose marker value is consistently on the high (or low) end of normal range in repeated testing is usually under close surveillance for possible relapse.
TextSentencer_T83 10274-10377 Sentence denotes If such a patient should relapse, then the time to clinical relapse can often be accurately determined.
TextSentencer_T83 10274-10377 Sentence denotes If such a patient should relapse, then the time to clinical relapse can often be accurately determined.
TextSentencer_T84 10378-10575 Sentence denotes However, if a patient is not under close surveillance, and would seek assistance only after some tangible symptoms have appeared, then time to relapse would be subject to case 2 interval censoring.
TextSentencer_T84 10378-10575 Sentence denotes However, if a patient is not under close surveillance, and would seek assistance only after some tangible symptoms have appeared, then time to relapse would be subject to case 2 interval censoring.
TextSentencer_T85 10576-10986 Sentence denotes For MIC data, several models have been proposed, and the asymptotic properties of the NPMLE have been investigated under the assumption that either the censoring variables take on finite many values (see Huang, 1999; Yu et al., 1998 Yu et al., , 2000 , or the censoring and survival distributions are strictly increasing and continuous and they have ''positive separation'' (see Huang, 1999, Assumption (A3) ).
TextSentencer_T85 10576-10986 Sentence denotes For MIC data, several models have been proposed, and the asymptotic properties of the NPMLE have been investigated under the assumption that either the censoring variables take on finite many values (see Huang, 1999; Yu et al., 1998 Yu et al., , 2000 , or the censoring and survival distributions are strictly increasing and continuous and they have ''positive separation'' (see Huang, 1999, Assumption (A3) ).
TextSentencer_T86 10987-11139 Sentence denotes For MIC data, define (Y i ,Z i ) as a pair of extended random censoring times (1 allowed) with PðY i o Z i Þ ¼ 1, and T i is independent of (Y i ,Z i ).
TextSentencer_T86 10987-11139 Sentence denotes For MIC data, define (Y i ,Z i ) as a pair of extended random censoring times (1 allowed) with PðY i o Z i Þ ¼ 1, and T i is independent of (Y i ,Z i ).
TextSentencer_T87 11140-11164 Sentence denotes Yu et al. (2000, see (2.
TextSentencer_T87 11140-11164 Sentence denotes Yu et al. (2000, see (2.
TextSentencer_T88 11165-11252 Sentence denotes 1)) considered a mixture interval censorship model to characterize MIC data as follows:
TextSentencer_T88 11165-11252 Sentence denotes 1)) considered a mixture interval censorship model to characterize MIC data as follows:
TextSentencer_T89 11254-11376 Sentence denotes Replacing (Y i ,Z i ) and (Y i ,Z i ] in (2.14) with (U i ,V i ), we obtain the model for middle-censored data as follows:
TextSentencer_T89 11254-11376 Sentence denotes Replacing (Y i ,Z i ) and (Y i ,Z i ] in (2.14) with (U i ,V i ), we obtain the model for middle-censored data as follows:
TextSentencer_T90 11377-11667 Sentence denotes ( Hence, although the sampling scheme of MIC data seems to be quite different in character from that of middle-censored data described in Section 1, the resulting observations (L i , R i ) would reduce to the observations from middle-censoring data when there is no left or right censoring.
TextSentencer_T90 11377-11667 Sentence denotes ( Hence, although the sampling scheme of MIC data seems to be quite different in character from that of middle-censored data described in Section 1, the resulting observations (L i , R i ) would reduce to the observations from middle-censoring data when there is no left or right censoring.
TextSentencer_T91 11668-11674 Sentence denotes Proof.
TextSentencer_T91 11668-11674 Sentence denotes Proof.
TextSentencer_T92 11675-11770 Sentence denotes Let Q denote the empirical version of the joint distribution function of (L i ,R i ) (i=1,y,n).
TextSentencer_T92 11675-11770 Sentence denotes Let Q denote the empirical version of the joint distribution function of (L i ,R i ) (i=1,y,n).
TextSentencer_T93 11771-11855 Sentence denotes It follows that Eq. (2.8) can be written as where Q n is the empirical version of Q.
TextSentencer_T93 11771-11855 Sentence denotes It follows that Eq. (2.8) can be written as where Q n is the empirical version of Q.
TextSentencer_T94 11856-11928 Sentence denotes In (2.15), if F(t)=F(r À )=F(l), then we encounter 0 0 in the integrand.
TextSentencer_T94 11856-11928 Sentence denotes In (2.15), if F(t)=F(r À )=F(l), then we encounter 0 0 in the integrand.
TextSentencer_T95 11929-11961 Sentence denotes In this case, we define 0 0 ¼ 1.
TextSentencer_T95 11929-11961 Sentence denotes In this case, we define 0 0 ¼ 1.
TextSentencer_T96 11962-12209 Sentence denotes Notice that Eq. (2.15) is exactly the same as Eq. (2.3) of Yu et al. (2000) and Eq. (2.2) of Yu et al. (2001) , which is a self-consistent equation of F 0 for the model in Yu et al. (2000 Yu et al. ( , 2001 with mixed interval-censored (MIC) data.
TextSentencer_T96 11962-12209 Sentence denotes Notice that Eq. (2.15) is exactly the same as Eq. (2.3) of Yu et al. (2000) and Eq. (2.2) of Yu et al. (2001) , which is a self-consistent equation of F 0 for the model in Yu et al. (2000 Yu et al. ( , 2001 with mixed interval-censored (MIC) data.
TextSentencer_T97 12210-12340 Sentence denotes By Theorems 2.1, 2.2 and 3.1 of Yu et al. (2001) , the strong consistency and asymptotic normality ofF M andF S are established. &
TextSentencer_T97 12210-12340 Sentence denotes By Theorems 2.1, 2.2 and 3.1 of Yu et al. (2001) , the strong consistency and asymptotic normality ofF M andF S are established. &
TextSentencer_T98 12341-12426 Sentence denotes We have demonstrated how middle-censored data relate to mixed interval-censored data.
TextSentencer_T98 12341-12426 Sentence denotes We have demonstrated how middle-censored data relate to mixed interval-censored data.
TextSentencer_T99 12427-12694 Sentence denotes With some modification of the definition for intervals (q j , p j )'s, we can obtain the NPMLE of distribution function by using EM algorithm of Turnbull (1976) or self-consistent estimating equation (Jammalamadaka and Mangalam, 2003) with a proper initial estimator.
TextSentencer_T99 12427-12694 Sentence denotes With some modification of the definition for intervals (q j , p j )'s, we can obtain the NPMLE of distribution function by using EM algorithm of Turnbull (1976) or self-consistent estimating equation (Jammalamadaka and Mangalam, 2003) with a proper initial estimator.
TextSentencer_T100 12695-12909 Sentence denotes The consistency and asymptotic normality of the NPMLE can be established based on the asymptotic properties of selfconsistent estimators (SCE) with mixed interval censored data (Yu et al., 2000 (Yu et al., , 2001 .
TextSentencer_T100 12695-12909 Sentence denotes The consistency and asymptotic normality of the NPMLE can be established based on the asymptotic properties of selfconsistent estimators (SCE) with mixed interval censored data (Yu et al., 2000 (Yu et al., , 2001 .