We studied the consistency of the semi-parametric maximum likelihood estimator (SMLE) under the Cox model with right-censored (RC) data.

Let Y be a random survival time, X a p-dimensional random covariate. Conditional on X = x, Y satisfies the Cox model if its hazard function satisfies

(1.1)

where h_o is the baseline hazard function, i.e., h_o (y) = f_o (y) /S_o (y-), f_o is a density function, S_o (y) = S(y|0) P (Y > y |X = 0), F_o = 1 - S_o, τ_Y = sup{t:S_Y(t) > 0}, h(y|x) = , S(·|·) f(·|·) orF(·|·)) is the conditional survival function (density function (df) or cumulative distribution function (cdf)) of Y given X = x. The restriction y<τ_Y is not in the original definition of the PH model, but is necessary if S_o is discontinuous at τ_Y (see Remark 1 [1])

In this paper, we shall make use of the assumptions as follows:

AS1. Suppose that C is a random variable with the df f_C (t) and the survival function S_C (t), X takes at least p +1 values, say 0 , x₁, ..., x_p, where x₁, ..., x_p are linearly independent, (Y,X) and C are independent. Let (Y₁,X₁,C₁), ..., (Y_n,X_n,C_n) be i.i.d. random vectors from (Y,X,C). M = min(Y,C) and δ = 1(Y ≤ C), where 1(A) is the indicator function of the event A. Let (M₁, δ₁X₁), ..., (M_n, δ_n, X_n) be i.i.d. RC observations from (M, δ, X) with the df are as follows:

(1.2)

and S(t|x) is a function of (S_o, β) (see Eq. (1.1)), but not f_x and f_C (the df’s of X and C).

Due to (AS1) and Eq. (1.2), the generalized likelihood function can be written as:

(1.3)

which coincides with the standard form of the generalized likelihood [2]. Eq. (1.3) is identical to the next expression:

(1.4)

where η_n = min{|M_i-M_j|: M_i ≠ M_j, i, j {1,2, ..., n}}. This form allows S_o to be arbitrary (discrete or continuous, or others), thus is more convenient in the later proofs. If Y is continuous then S(t|x) = (S(t|0))^exp(X'β) = (S_o (t))^exp(X'β), but

(1.5)

If Y is discrete then S(t|x) = ∏_s≤t(1 - h(s|x)) = ∏_s≤t(1 - h₀ (s)e^X'β) If Y has a mixture distribution, then S(t|x)= p (S₀₁(t))^exp(X'β) + (1 - p) ∏_s≤t(1 - h₀₂(s)e^X'β where p (0,1), h₀₁ and h₀₂ are two hazard functions. h₀ (t) = ph₀₁ + (1 - p)h₀₂ and S₀ (t) = pS₀₁ + (1 - p)S₀₂

The SMLE of (S_o, β) maximizes L (S, b) overall possible survival function S and bR^p, denoted by (). The SMLE of S(t|x) is denoted by (t|x), which is a function of (). The computation issue of the SMLE under the Cox model has been studied, but its consistency has not been established under the model [3]. Their simulation results suggest that the SMLE is more efficient than the partial likelihood estimator under the Cox model.

The partial likelihood estimator is a common estimator under the Cox model, which maximizes the partial likelihood: , where D is the collection of indices of the exact observations and R_i is the risk set {j: M_j ≥ Y_j}. The asymptotic properties of the estimator are well understood [4].

The consistency of the SMLE under the continuous Cox model with interval-censored (IC) data has been established, making use of the following result [5]:

The Shannon-Kolmogorov (S-K) inequality. Let f_o and f be two densities with respect to (w.r.t.) a measure μ and ∫ f₀ (t)ln f₀ (t)dμ(t) is finite. Then, ∫ f₀ (t)ln f₀ (t)dμ(t) ≥ ∫ f₀ (t)ln f (t)dμ(t), with equality iff f = f_o a.e. w.r.t. μ.

Under the Cox model with IC data, the S-K inequality becomes E (lnL(S_o, β)) ≥E (lnL(S, b)) (S, b), where L(∙, ∙) is the likelihood function of the Cox model with IC data, which is different from L ( ∙, ∙) in Eq. (1.3) and S is a baseline survival function and bR^p. Their approach cannot be extended to the Cox model with RC data as the key assumption (in the S-K inequality) [3].

That is, finite E (lnL (S_o, β)), may not hold. Indeed, if Y has a df and β = 0, then L

A related inequality is as follows.

The Kullback-Leibler (K-L) information inequality. Let f_o and f be two densities w.r.t. a measure μ. Then ∫ f₀ (t)ln (f₀ /f)(t)dμ(t) ≥ 0, with equality iff f = f_oa.e. w.r.t. μ.

The K-L inequality says that ∫ f₀ (t)ln (f₀ /f)(t)dμ(t) exists, though it maybe ∞. The two inequalities are not equivalent. In fact,

In this note, we show that the SMLE under the Cox model is consistent, making use of the Kullback-Leibler information inequality [6]

2. The Main Results. Notice that under the assumption that h_o exists, S_o, f_o, F_o and h_o are equivalent, in the sense that given one of them, the other 3 functions can be derived. Thus, the Cox model is applicable only to the distributions that the density functions exist, that is, Y is either continuous, or discrete, or the mixture of the previous two. Since the expression of S(t|x) varies in these three cases, for simplicity, we only prove the consistency of the SMLE under the Cox model in the first two cases.

Theorem 1. Under the Cox model with RC data, if Y is either continuous or discrete, and ifS_o (τ_M) <1, then the SMLE () is consistent t D (see Eq. (1.2)).

The proof of Theorem 1 makes use of a modified K-L inequality. K-L inequality requires that f₀ and f are both densities w.r.t. the measure μ. That is ∫ f(t)dμ(t = 1. However, in our case, we encounter the case that ∫ f(t)dμ(t) [0,1].

Lemma 1 (the modified K-L inequality). If f_i ≥ 0, μ₁ is a measure, ∫ f₁(t)dμ₁(t = 1 and ∫ f₂(t)dμ₁(t ≤ 1, then ∫ f₁(t)ln dμ₁(t) ≥ 0, with equality iff f₁ = f₂ a.e. w.r.t. μ₁.

Proof. In view of the K-L inequality, it suffices to prove the inequality ∫ f₁(t)ln dμ₁(t) ≥ 0 under the additional assumptions that ∫ f₂(t)dμ₁(t < 1, ∫ f₁(t)dμ₂(t = 0 and ∫ f₂(t)dμ(t < 1, where μ₂ is a measure and μ = μ₁ + μ₂ Since ∫ f₂(t)dμ(t) = 1, f₁ and f₂ are df's w.r.t. μ.

Proof of Theorem 1. Let Ω₀ be the subset of the sample space Ω such that the empirical distribution function (edf) , (t, s, x) based on (M_i, δ_i, X_i) converges to F(t,s,x), the cdf of (M, δ, X). It is well-known that P(Ω₀,) =1. Notice that the SMLE () is a function of (ω, n), say (_o,n (t)(ω), _o,n (t_n)(ω) , where ω Ω and n is the sample size. Hereafter, fix an ω Ω₀ , since (=_n(ω)) is a sequence of vectors in R^p, there is a convergent subsequence with the limit β*, where the components of β_* can be ±∞. Moreover, S_o (= S_o,n (∙)(ω)) is a sequence of bounded non-increasing functions, Helly’s selection theorem ensures that given any subsequence of _o, there exists a further subsequence which is convergent. Without loss of generality (WLOG), we assume that _o → S_* and → β_*. Of course, (β_*, S_*) depends on ω( Ω₀ ). We prove in Theorem 2 for the discrete case and in Theorem 3 for the continuous case that:

(2.1)

Since ω can be arbitrary in Ω₀ and P(Ω₀ ) = 1, the SMLE is consistent.

Before we prove Theorems 2 and 3, we present a preliminary result.

Lemma 2 (Proposition 17 in Royden (1968), page 231). Suppose thatμ_n is a sequence of measures on the measurable space (J, ) such that μ_n(B) μ(B),B, g_n and f_n are non-negative measurable functions, and (f_n, g_n)(x) = (f, g)(x) Then,

Corollary 1. Suppose that μ_n is a sequence of measures on the measurable space (J , B) such thatμ_n (B) → μ (B), B, f andf_n (n ≥ 1) are integrable functions that are bounded below andf(x)_n→∞ = lim f_n(x). Then ∫ f dμ ≤ lim_n→∞ ∫ f_n dμ_n.

Proof. Let k = inf_n inf_xf_n(x). If k ≥ 0 then the corollary follows from Lemma 2. Otherwise, let f_n^-(x) = 0 Λ f_n(x), f_n⁺(x) = 0 v f_n(x), f^-(x) = 0 Λ f(x) and f ⁺(x) = 0 v f(x). Then, f_n⁺ → f ⁺ and f_n^- → f ^- point wisely, as, f_n → f

lim_n→∞ ∫ f_n dμ_n = lim_n→∞ ∫ (f_n⁺ + f_n^-)dμ_n = lim_n→∞ [∫ f_n⁺ dμ_n + f_n^- dμ_n] ≥ ∫ lim_n→∞f_n⁺ dμ + ∫ lim_n→∞f_n^- dμ (by Lemma2, as f_n⁺ (x) is nonnegative and |f^- (x)| ≤ k) = ∫ f ⁺ dμ + ∫ f^- dμ = ∫ (f ⁺ + f^-)dμ = ∫ f dμ.

Theorem 2. Under the discrete Cox model with RC data, Eq. (2.1) holds.

Proof. For the given ω Ω₀ and (S_*, β_*) in the proof of Theorem 1, as assumed, () (ω) → (S_*, β_*). Defining h_*(t) = and h_*(t|x) = h_*(t)^eβ_*'x (for S_*(t -) > 0) yeilds S_*(t|x) and f_*(t|x), which are continuous functions of S_* and β_*. Consequently, (·|·) → S_*(·|·).

Let G_n(S₀ , β) = lnL(S₀ , β)/n (see Eq.(1.3)). Then, the SMLE () satisfies

(2.2)

.

where B is a measurable set in R^p+1. To apply Lemma 2,

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

and v_n converges set wisely to a finite measure v (see (2.9)), by a similar argument as in (2.4), (2.6), (2.7) and (2.8), we have:

(2.10)

Thus, ∫ lndF(t, 0, x) + ∫ lndF(t, 1, x). Hence, (S₀ (t),β) = (S_*(t),β)tD by the 2nd statement of the K-L inequality.

Theorem 3.Under the Cox model with RC data, if Y is continuous then Eq. (2.1) holds.

Proof. For the given ωΩ₀ and (S_*,β_*) in the proof of Theorem 1, as well as (ω) and (t|x)(ω), we have S_*(t|x) = (S_*(t))^exp(β_*'x). By a similar argument as in proving Eq. (2.8), we can show:

(2.11)

In view of Eq. (1.4) due to Y is continuous, we denote:

(2.12)

(2.13)

as S_* is a monotone function, S_*^' exists a.e., and so do S_*^'(t|x) and F_*^'(t|x). We have

(2.14)

The reason is as follows. For each (t, x) such that F^'(t|x) > 0 and Eq. (2.13) holds,

F_*^'(t|x) /F^'(t|x) (=f_*(t|x) /f (t|x)) is finite. Then, there exists n_o such that G(t, x, n) < 1 + F_*^'(t|x) /F'(t|x) for n ≥ n_o . On the other hand, G(t, x, n) is finite for n =1, ..., n_o . Thus, G(t, x, n) < k for some k. Since Eq. (2.1) holds a.e. and ∫ 1dF(t, s, x) = 1, Eq. (2.14) holds.

We shall prove in Lemma 3 that

(2.15)

.

(2.16)

.

The last inequality further implies that ∫lnd F(t,0,x) + ∫lnd F(t,1,x) = 0. Thus, (S₀ (t),β) = (S_*(t),β_*) t D by the 2nd statement of the K-L inequality and by the assumption ASI.

Lemma 3. Inequality (2.15) holds.

Proof. Let k ≥ 1 and , where B is a measurable set and

.

Since H((S (t-η_n|x) - (S (t|x))/(( (t-η_n|x) - ( (t|x))) ≥ - 1/e and v_n converges set wisely to a finite measure v by a similar argument as in (2.4), (2.6), (2.7) and (2.8), we can show that:

.