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- W2012175659 abstract "Let $X_1, cdots, X_n$ be i.i.d. $P(X > u) = F(u)$ and $Y_1, cdots, Y_n$ be i.i.d. $P(Y > u) = G(u)$, where both $F$ and $G$ are unknown continuous distributions. For $i = 1, cdots, n$ set $delta_i = 1$ if $X_i leq Y_i$ and 0 if $X_i > Y_i$ and $Z_i = min {X_i, Y_i}$. One way to estimate $F$ from the observations $(Z_i, delta_i) i = 1, cdots, n$ is by means of the product limit (PL) estimator, $F^ast_n$ (Kaplan-Meier, [8]). In this paper it is shown that $F^ast_n$ is uniformly almost sure consistent with rate $O(sqrt{log n} / sqrt n)$, that is $P(sup_{0 leq u leq T}|F^ast_n(u) - F(u)| = O(sqrt{log n/n})) = 1.$ Assuming that $F$ is distributed according to a Dirichlet process (Ferguson, [3]) with parameter $alpha$, Susarla and Van Ryzin ([11]) obtained the Bayes estimator $F^alpha_n$ of $F$. In the present paper a similar result is established for the Bayes estimator, namely: $P(sup_{0 leq u leq T}| F^alpha_n(u) - F(u)| = O(sqrt{(log n)^{1 + gamma}} / sqrt n)) = 1 quad (gamma > 0).$" @default.
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- W2012175659 date "1981-01-01" @default.
- W2012175659 modified "2023-09-29" @default.
- W2012175659 title "Strong Uniform Consistency for Nonparametric Survival Curve Estimators from Randomly Censored Data" @default.
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- W2012175659 doi "https://doi.org/10.1214/aos/1176345337" @default.
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