The name is absent

28

can be extended to the competing risk framework.

Zheng and Klein (1995)

Zheng and Klein (1995) show that when the copula function is known, the compet-
ing risks data is sufficient to identify the marginal survival functions, and construct a
suitable and consistent estimator. When the event and censoring times are indepen-
dent, the proposed estimator reduces to the Kaplan-Meier estimator.

Specifically, in a competing risks framework, let X be the time until an event of
interest occurs, Y be the time until a competing risk occurs. That is, X can not be
observed if a competing risk happens. Assume the copula of X and Y is known and a
competing risk sample is denoted as (T, J), where T = min(X, Y) and δ = I(X ≤ Y).
I(A) is the indicator function of the set A.

Zheng and Klein (1995) prove several important theorems, which enable us to
further utilize their proposed method.

Theorem 1. “Suppose the marginal distribution functions of (X,Y) are continuous
and strictly increasing in (0,∞). Suppose the copula, C, of (X,Y), is known, and
μ_c{E~) > 0 for any open set E in [0,l]×[0,l]. Then F and G, the marginal distribution
functions of X and Y, are uniquely determined by {k(t'),p1(t),p2(tft > 0}.”

Theorem 2. “Suppose that two marginal distribution functions F, G, are continu-
ous and strictly increasing on (0, ∞), and the assumed copula has density function
u(x,y) > 0 on [0,l]× [0,1]. Then F_n and G_n are strongly consistent for F and G. That

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