Smooth confidence intervals for the survival function under random right censoring

A goodness of fit test for the survival function under random right censoring

“…S 0 (x) denotes the assumed true parametric model and is completely known. By Corollary 1 in [2], Theorem 1 and under H 0 , the proposed test statistic, T n , is obtained by replacingŜ(x) byŜ n (x) and S(x) by S 0 (x) in the expression of I n . By lemma 3, the scaled and centered T n , say T n, * , can be written as…”

“…where the second term on the RHS of (4) results from applying the bias expression ofŜ n (x) (Theorem 1, [2]) on (2) and…”

“…Validation of the theorem's applicability is done through two conditional Lindeberg and variance conditions given in Corollary 3.1 of [10]. Proceeding with the term in (b), Lemma 2 uses direct calculation to show that the expression there equals σ 2 2 plus a negligible (under optimal bandwidth) term. Lemma 3 shows that the expected ISE value equals the term in (c) plus the result of Lemma 2.…”

“…Lemma 3 shows that the expected ISE value equals the term in (c) plus the result of Lemma 2. Thus, the result of Lemma 3 is used to shape the LHS of Theorem 1 as well as to cancel out the σ 2 2 term resulting from Lemma 2. Finally, in Lemma 4 the term in (d) is written as a sum of independent and identically distributed random variables and so, by a central limit theorem, verification of the theorem's conditions and straightforward calculations, is proved that the term is asymptotically normally distributed.…”

“…The smooth estimate is taken to be the estimate of [15], given in Section 2 below. The absence in the literature of a goodness of fit test based on the estimate employed here, together with the advantages this estimate offers, discussed in detail in [2], necessitate the development of such a test and extend its scope of application.…”