29
is with probability 1 as n → ∞, Fn(⅛) → F(t) and Gn(t) → G(t) for all t ∈ [o, ∞). ”
Theorem 3. “The copula-graphic estimator is a maximum likelihood estimator. ”
Theorem 4. “For the independence copula C(x,y)=xy, when t ≤ tn, the largest
observed time, the copula-graphic estimates of marginal survival functions are exactly
the Kaplan-Meier estimates. ”
Note: Zheng and Klein (1995) was written before Zheng and Klein (1994), al-
though it was published later than Zheng and Klein (1994).
Zheng and Klein (1994)
Zheng and Klein (1994) apply the copula method to construct an estimator of the
marginal survival function based on dependent competing risk data.
In a competing risks framework, we define X, Y, T and δ same as in Zheng and
Klein (1995). Zheng and Klein (1994) show that the marginal survival function can
be estimated:
⅜)
⅜)
n
* 5√≥ *] +∑(1 -tχ>t∖χ>t^γ = ti∖ *
2=1 ti<t
(2.27)
n
< E/ [f≈ ≥ t∖ + [y > t∖γ
i=l ti<t
> ti,X = ti] .
(2.28)
Here, both S(t) and R(f) are self-consistent estimators. When X and Y are de-
pendent with a known copula C(u, υ), we have:
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