MABLE Maximum Approximate Bernstein Likelihood Estimation in Proportional Hazards Model for Interval-Censored Data
Mathematical Methods AIMech:MABLEMaximumApproximateBernsteinLikelihoodProportionalHazardsIntervalCensoring DRAFT
Aliases: MABLE Proportional Hazards Estimator, Maximum Approximate Bernstein Likelihood Estimator for Interval-Censored Data, Bernstein Polynomial Cox Proportional Hazards Estimator
A semiparametric survival regression method for Cox's proportional hazards model under interval-censored data that parameterizes the baseline density on [0, tau_n] as a degree-m Bernstein polynomial mixture of Beta densities f_m(t|x_0; p) = tau_n^{-1} sum_{i=0}^m p_i beta_{mi}(t/tau_n) and maximizes the resulting full approximate log-likelihood ell_m(gamma, p) over regression coefficients gamma and simplex weights p in S_{m*} using an alternating Newton-Raphson and multiplicative fixed-point iteration anchored at the empirical baseline covariate x_hat_0(gamma) = argmin_{x_i} gamma^T x_i.
Identity Notes & Concept Scope
No AIO class represents Maximum Approximate Bernstein Likelihood Estimation (MABLE) for interval-censored proportional hazards models; RegressionAnalysis (AIO:RegressionAnalysis) is the closest broad match covering semiparametric regression estimation.
Claims & Grounded Evidence
4 Claims
Mechanism
#empirical-baseline-concave-bernstein-loglikelihood-and-multiplicative-fixed-point
Choosing the baseline covariate x_0 to satisfy gamma^T x_0 <= min_{1<=i<=n} gamma^T x_i ensures eta_i = exp(gamma^T (x_i - x_0)) >= 1 for all subjects, making all entries of the Hessian H(gamma, p) = d^2 ell_m(gamma, p) / (dp dp^T) nonpositive and H(gamma, p) negative semi-definite on the simplex S_{m*} (Lemma 1), so the multiplicative fixed-point iteration p_j^{[s+1]} = p_j^{[s]} Psi_bar_j(gamma, p^{[s]}) with lambda_n(gamma) = sum_{i=1}^n exp(gamma^T x_tilde_i) remains nonnegative and converges globally to the maximizer p_tilde(gamma) (Equations 10-13, Theorems 1-2).
Mechanism
#changepoint-degree-selection-and-asymptotic-normality
MABLE selects the optimal Bernstein polynomial degree m_hat by maximizing the change-point log-likelihood increment ratio statistic R(m_i) over candidate degrees {m_0, ..., m_k} (Section 2.3) and achieves almost-sure consistency ||gamma_hat - gamma_0||^2 <= C n^{-1+epsilon} and D_i^2(p_hat; x_hat_0) <= C n^{-1+epsilon} (Theorem 4) alongside root-n asymptotic normality sqrt(n)(gamma_tilde - gamma_0) -> N(0, I^{-1}) when m >= C n^{1/rho} (Theorem 5).
Application
#smooth-survival-and-hazard-estimation-under-interval-censoring-application
MABLE is applied to estimate smooth baseline survival curves, hazard rates, and covariate hazard ratios from interval-censored and right-censored clinical survival data (such as the Stanford Heart Transplant and Ovarian Cancer studies), avoiding the jagged step-function artifacts of Turnbull's NPMLE.
Limitation
#right-tail-unidentifiability-and-high-degree-simplex-nonuniqueness-limitation
When the largest observed time tau_n is right-censored with F(tau_n|x_0) < 1, the baseline density on (tau_n, infinity) is nonparametrically unidentifiable and requires an external tail specification alpha(t - tau_n) (Section 2.1-2.2, Equations 4-5); moreover, if the Bernstein degree m exceeds the number of distinct Turnbull inspection endpoints, the maximizer p_hat on the simplex may not be unique even when the fitted survival curve at the endpoints is invariant (Theorem 1, Section 5.1).
⚠️ Curated Limitations & Negative Results
- When the largest observed time tau_n is right-censored with F(tau_n|x_0) < 1, the baseline density on (tau_n, infinity) is nonparametrically unidentifiable and requires an external tail specification alpha(t - tau_n) (Section 2.1-2.2, Equations 4-5); moreover, if the Bernstein degree m exceeds the number of distinct Turnbull inspection endpoints, the maximizer p_hat on the simplex may not be unique even when the fitted survival curve at the endpoints is invariant (Theorem 1, Section 5.1).
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:RegressionAnalysis
Regression Analysis
|
broad_match | MABLE estimates regression coefficients and smooth baseline survival/density functions in Cox proportional hazards regression models under interval censoring. |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
r834-primary-pdf
|
Zhong Guan
|
primary paper
full_text
|
ID: https://arxiv.org/abs/1906.08882
sha256:6ec93d42d9c6…
|
r834-arxiv-abs
|
Zhong Guan
|
primary paper
abstract_only
|
ID: https://arxiv.org/abs/1906.08882
sha256:d0c8dd675d2e…
|
r834-metadata-cr
|
Zhong Guan
|
primary paper
abstract_only
|
ID: https://arxiv.org/abs/1906.08882
sha256:85744c284115…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini
Created: 2026-10-19T15:17:00Z
Updated: 2026-10-19T15:17:00Z
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