Epistemic_Statements  

CORD-19:07815f7f4d5711d2c737bfc2562bd07bf4644189 JSONTXT 9 Projects

Annnotations TAB TSV DIC JSON TextAE Lectin_function

Id Subject Object Predicate Lexical cue
T1 275-329 Epistemic_statement denotes As a consequence, there will be no observations on the
T2 330-371 Epistemic_statement denotes Contents lists available at ScienceDirect
T3 373-533 Epistemic_statement 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.
T4 534-838 Epistemic_statement 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.
T5 1765-1965 Epistemic_statement 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.
T6 2280-2465 Epistemic_statement 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.
T7 2466-2599 Epistemic_statement denotes However, a careful reflection and analysis shows them to be quite different ideas; see Jammalamadaka and Mangalam (2003) for details.
T8 3000-3298 Epistemic_statement 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.
T9 3473-3626 Epistemic_statement denotes In different types of censoring, the relationship between nonparametric maximum likelihood estimator (NPMLE) and SCE has been studied by various authors.
T10 4159-4274 Epistemic_statement denotes They also pointed out that an SCE provides only a local maximum of the likelihood equation and may not be an NPMLE.
T11 5314-5460 Epistemic_statement denotes In this note, we aim to establish connections between the middle-censoring and interval censoring by investigating the self-consistency algorithm.
T12 5461-5731 Epistemic_statement 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.
T13 6049-6104 Epistemic_statement denotes Here, we consider the case when there is no truncation.
T14 6258-6433 Epistemic_statement 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.
T15 6434-6505 Epistemic_statement denotes Based on the notations defined above, the likelihood is proportional to
T16 6506-6590 Epistemic_statement denotes where P F 0 ðA i Þ denotes the probability that is assigned to the interval by F 0 .
T17 7375-7678 Epistemic_statement denotes Thus, it suffices to maximize O ¼ s 2 R J : 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 .
T18 7801-7889 Epistemic_statement denotes Next, we shall show that the estimatorF M satisfies self-consistent equation (1.3), i.e.
T19 8103-8141 Epistemic_statement denotes Furthermore, d j ðŝÞ can be written as
T20 8142-8224 Epistemic_statement denotes Consider an initial estimatorF ð0Þ , which puts mass only on ðq j ,p j Þ ðj ¼ 1, .
T21 8234-8275 Epistemic_statement denotes LetF ð1Þ denote the first step estimator.
T22 8276-8436 Epistemic_statement 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 .
T23 8615-8677 Epistemic_statement denotes Next, we consider the following two cases: Case 1: q j = p j .
T24 9041-9109 Epistemic_statement denotes However, our proof is based on the EM algorithm of Turnbull (1976) .
T25 9255-9452 Epistemic_statement 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.
T26 9460-9595 Epistemic_statement 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 .
T27 9596-9735 Epistemic_statement 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).
T28 9992-10273 Epistemic_statement 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.
T29 10274-10377 Epistemic_statement denotes If such a patient should relapse, then the time to clinical relapse can often be accurately determined.
T30 10378-10575 Epistemic_statement 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.
T31 10576-10986 Epistemic_statement 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) ).
T32 11165-11253 Epistemic_statement denotes 1)) considered a mixture interval censorship model to characterize MIC data as follows:
T33 11254-11376 Epistemic_statement 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:
T34 11377-11667 Epistemic_statement 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.
T35 11791-11855 Epistemic_statement denotes (2.8) can be written as where Q n is the empirical version of Q.
T36 12339-12340 Epistemic_statement denotes &
T37 12341-12426 Epistemic_statement denotes We have demonstrated how middle-censored data relate to mixed interval-censored data.
T38 12427-12694 Epistemic_statement 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.