Splitting the sample at the largest uncensored observation

Maller, Ross; Resnick, Sidney I.; Shemehsavar, Soudabeh

doi:10.3150/21-bej1417

Cited by 3 publications

(10 citation statements)

References 14 publications

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“…where 𝑎 𝑛 ∼ (𝑛𝑎 𝐺 ) 1∕𝛾 and 𝑏 𝑛 ∼ (𝑛𝑎 𝐺 ) 1∕(1+𝛾) . Then according to Case 3 of Theorem 3.1 of Maller, Resnick & Shemehsavar (2022) (with the slowly varying functions specified there taken as 1), (𝑈 𝑛 , 𝑉 𝑛 ) ⇒ (𝑈, 𝑉 ) as 𝑛 → ∞, where 𝑈 and 𝑉 are independently distributed with…”

Section: Discussionmentioning

confidence: 99%

“…The censoring distribution 𝐺 is required to have a finite right extreme but this is inherently satisfied in the important Case 1 and will usually be the case in practice. Equation ( 14 14) and ( 16) are special cases of those imposed in Theorem 3.1 of Maller, Resnick & Shemehsavar (2022) (where an extra regularly varying function is allowed as a factor in ( 14) and ( 16), but we omit this). The conditions are such that 𝑀 𝑢 (𝑛) and 𝑀(𝑛) are in extreme domains of attraction; see (A10), (A11), and (A20) of the Appendix.…”

Section: Asymptotic Distribution Of the Test Statisticmentioning

confidence: 99%

“…This is the “random censorship” model of Gill (1980), p. 22, commonly assumed in theoretical analyses of survival data, and, as Gill discusses, is very reasonable in many practical situations. Adopting the notation in Maller & Resnick (2021) and Maller, Resnick & Shemehsavar (2022), the mixture cure model is formulated as follows: A sample of size

n

consists of observations on the sequence of i.i.d. 2‐vectors

(T_{i} = T_{i}^{*} \land U_{i}, C_{i} = 1 (T_{i}^{*} \leq U_{i}); i \in ℕ_{n} = {1, 2, \dots, n})

.…”

Section: Notation and Assumptions: The Mixture Cure Modelmentioning

confidence: 99%

“…From this we are able to derive its asymptotic distribution in various scenarios. Our analysis draws on the formulae in Maller & Resnick (2021) and Maller, Resnick & Shemehsavar (2022) for the joint distribution of the largest uncensored and the largest survival time in the sample.…”

Section: Introductionmentioning

confidence: 99%

“…Rather than (A23) we now have Now, 𝑃 𝐴 𝑛 (𝑢, v; 𝑧) has limit Pr(𝑁(0, 1) ≤ 𝑧) as 𝑛 → ∞, so again we have convergence to normality and arrive at (17) with 𝜈 = 𝜈 𝐴 . Case 3: This proof proceeds as for Cases 2a and 2b except that Case 3 of Theorem 3.1 of Maller, Resnick & Shemehsavar (2022) is used. ◼…”

mentioning

confidence: 99%

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Finite sample and asymptotic distributions of a statistic for sufficient follow‐up in cure models

2023

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The existence of immune or cured individuals in a population and whether there is sufficient follow‐up in a sample of censored observations on their lifetimes to be confident of their presence are questions of major importance in medical survival analysis. Here we give a detailed analysis of a statistic designed to test for sufficient follow‐up in a sample. Assuming an i.i.d. censoring model, we obtain exact finite‐sample and asymptotic distributions for the statistic, and use these to calculate the power of a test based on it. A particularly useful finding is that the asymptotic distribution of the test statistic is parameter‐free in the null case when follow‐up is insufficient. The methods are illustrated with application to a glioma cancer dataset.

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Section: Discussionmentioning

confidence: 99%

Section: Asymptotic Distribution Of the Test Statisticmentioning

confidence: 99%

n

consists of observations on the sequence of i.i.d. 2‐vectors

(T_{i} = T_{i}^{*} \land U_{i}, C_{i} = 1 (T_{i}^{*} \leq U_{i}); i \in ℕ_{n} = {1, 2, \dots, n})

.…”

Section: Notation and Assumptions: The Mixture Cure Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Finite sample and asymptotic distributions of a statistic for sufficient follow‐up in cure models

2023

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Testing for Sufficient Follow‐Up in Censored Survival Data by Using Extremes

Xie,

Escobar‐Bach,

Van Keilegom

2024

Biometrical J

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In survival analysis, it often happens that some individuals, referred to as cured individuals, never experience the event of interest. When analyzing time‐to‐event data with a cure fraction, it is crucial to check the assumption of “sufficient follow‐up,” which means that the right extreme of the censoring time distribution is larger than that of the survival time distribution for the noncured individuals. However, the available methods to test this assumption are limited in the literature. In this article, we study the problem of testing whether follow‐up is sufficient for light‐tailed distributions and develop a simple novel test. The proposed test statistic compares an estimator of the noncure proportion under sufficient follow‐up to one without the assumption of sufficient follow‐up. A bootstrap procedure is employed to approximate the critical values of the test. We also carry out extensive simulations to evaluate the finite sample performance of the test and illustrate the practical use with applications to leukemia and breast cancer data sets.

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