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Asymptotic normality of the Cox partial-likelihood estimator

Statement

Assume the hypotheses of Cox partial-likelihood score expansion and estimator linearization and AC. Then $\sqrt{n} ({\hat{β}}_{n} - β_{0}) \Rightarrow N_{p} (0, I^{- 1}) .$ This terminal normal limit follows from the separately proved influence expansion; it is not used to prove that expansion.

Facts & Assumptions

Given: The measurable consistent score root and the expansion from Cox partial-likelihood score expansion and estimator linearization, the IID subject influence variables from Cox score martingale representation and random-risk-mean replacement, and AC.

[F1]

Cox score martingale representation and random-risk-mean replacement: $ξ_{i}$ are IID centered square-integrable vectors with covariance $I$ .

[F2]

Cox partial-likelihood score expansion and estimator linearization: $\sqrt{n} ({\hat{β}}_{n} - β_{0}) = I^{- 1} n^{- 1 / 2} \sum_{i} ξ_{i} + o_{P} (1)$ .

[F3]

IID terminal influence limit from a separately established expansion: under AC, an IID measurable estimator with centered finite-second-moment influence function and a separately established $o_{P} (n^{- 1 / 2})$ expansion is asymptotically normal with influence-function covariance.

[F4]

Covariance matrix of a finite-dimensional random vector: covariance is $E [(X - E X) (X - E X)^{T}]$ for a square-integrable finite-dimensional random vector.

[A1]

AC is required by the terminal influence-limit theorem. The Axiom of Choice

Proof

technique · Identify the one-subject influence function and apply the terminal IID influence theorem only after the estimator expansion is established

1.1A1F1F4givenalgebra

Define $ϕ (O_{i}) = I^{- 1} ξ_{i}$ . By [F1], $E ϕ = 0$ and $ϕ$ is square-integrable. Since $I$ is symmetric positive definite, [F4] gives $Cov (ϕ) = I^{- 1} Cov (ξ_{i}) (I^{- 1})^{T} = I^{- 1} I I^{- 1} = I^{- 1}$ . The map is a measurable function of one subject record because $M_{i}$ , $Z_{i}$ , and the compensator $A_{i}$ are subject-specific and $\overset{ˉ}{z}$ is deterministic.

2.1A1F2F3step 1.1∎

By [F2], the expansion is ${\hat{β}}_{n} - β_{0} = n^{- 1} \sum_{i = 1}^{n} ϕ (O_{i}) + r_{n}$ with $r_{n} = o_{P} (n^{- 1 / 2})$ . The sample records are IID, ${\hat{β}}_{n}$ is measurable, $ϕ$ is centered with finite second moment, and AC holds. These are exactly the hypotheses of [F3]; applying it with covariance from step 1.1 proves the stated limit.

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Sources

Lu Tian, STAT331 Revisiting Cox's PH Model: Asymptotic Theory, PDF pp. 6–10 (standard reference, not scraped)
Qiqing Yu, Survival Analysis (Lecture Notes), §5.2, PDF pp. 58–63 (standard reference, not scraped)