Adaptive Progressive Hybrid Censoring (APHC)

• Adaptive Progressive Hybrid Censoring (APHC) is a scheme that integrates progressive, hybrid, and adaptive censoring to ensure exactly m failures while managing test duration and survivor count.
• It employs a predetermined removal plan and a censoring threshold to optimize statistical information and control costs in accelerated life testing experiments.
• MLE techniques, including Newton–Raphson, are used to estimate Weibull parameters with proven asymptotic properties, enhancing model reliability compared to traditional methods.

Adaptive Progressive Hybrid Censoring (APHC) encompasses a family of censoring schemes in life-testing experiments that combine features of progressive, hybrid, and adaptive censoring in order to balance statistical information with practical constraints such as cost, time, and sample size. In APHC, units are removed from test adaptively depending on observed failure times and a predefined censoring time, permitting the experiment to capture a fixed number of failures while retaining control over the survivor count and test duration. The scheme is designed to mitigate the risk of incomplete information, often observed in classical censoring approaches, and is especially relevant in accelerated life testing (ALT) frameworks under Weibull or Marshall-Olkin bivariate Weibull lifetime models (Konar et al., 11 Jan 2026, &&&1&&&).

1. Definition and Structure of the APHC Scheme

Adaptive Progressive Hybrid Censoring is defined by the following components:

• $n$: Total number of identical units placed on test.
• $m$: Desired number of observed failures ($1 \leq m \leq n$).
• $T$: Pre-specified censoring time ($T > 0$).
• $R_1, \ldots, R_m$: Deterministic plan for progressive removal (nonnegative integers with $\sum_{i=1}^m R_i \leq n - m$).

The chronological mechanism is as follows:

1. Start with $n$ test units.
2. Observe failures $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(m)}$.
3. For each failure $i$ ($1 \leq i \leq j$), if $X_{(i)} < T$, remove $R_i$ survivors at $X_{(i)}$, where $j = \max\{i: X_{(i)} < T\}$.
4. After time $T$, no further removals at intermediate failures; at the $m$th failure ($X_{(m)}$), remove all remaining survivors and terminate the test.
5. The test is adaptive in the sense that removal actions depend on the comparison between observed failure times and $T$.

This procedure ensures exactly $m$ failures are always observed, regardless of the distribution of failures relative to $T$, providing robust information for inference.

2. Likelihood Formulation and Parameter Estimation

Suppose the lifetimes are i.i.d. Weibull with density $$f(x; \alpha, \lambda) = (\alpha/\lambda) (x/\lambda)^{\alpha - 1} \exp[-(x/\lambda)^\alpha]$$. The likelihood under APHC conditions on the number of failures before $T$ and incorporates both observed failures and censored survivors. The joint likelihood is given by:

• Combinatorial constant:

$$D_j = \prod_{i=1}^m \left[n - i + 1 - \sum_{k=1}^{\max(i-1, j)} R_k \right].$$

• Log-likelihood:

$$\ell(\alpha, \lambda) = \log D_j + m \log(\alpha/\lambda) + \sum_{i=1}^m \left[ (\alpha-1)\log(X_{(i)}/\lambda) - (X_{(i)}/\lambda)^\alpha \right] - \sum_{i=1}^j R_i (X_{(i)}/\lambda)^\alpha - \left(n-m-\sum_{i=1}^jR_i\right) (X_{(m)}/\lambda)^\alpha.$$

Maximum likelihood estimation requires solving the following equations numerically for the shape parameter $\alpha$ and scale $\lambda$:

• Scale:

$$\lambda^\alpha = \frac{1}{m}\left\{ \sum_{i=1}^m X_{(i)}^\alpha + \sum_{i=1}^j R_i X_{(i)}^\alpha + (n-m-\sum_{i=1}^j R_i) X_{(m)}^\alpha \right\}$$

• Shape:

$$0 = \frac{m}{\alpha} + \sum_{i=1}^m \log X_{(i)} - m \frac{ \sum_{i=1}^m X_{(i)}^\alpha \log X_{(i)} + \sum_{i=1}^j R_i X_{(i)}^\alpha \log X_{(i)} + (n-m-\sum_{i=1}^j R_i) X_{(m)}^\alpha \log X_{(m)} }{ \sum_{i=1}^m X_{(i)}^\alpha + \sum_{i=1}^j R_i X_{(i)}^\alpha + (n-m-\sum_{i=1}^j R_i) X_{(m)}^\alpha }$$

Convergence is ensured by strict monotonicity of the score function in $\alpha$ (Konar et al., 11 Jan 2026).

3. Asymptotic Properties and Inferential Theory

Consistency and asymptotic normality of the maximum likelihood estimators under APHC are established. If the sample size grows ($m_n \to \infty$), the unique solution $\hat{\alpha}_n$ for the shape parameter is consistent: $\hat{\alpha}_n \xrightarrow{p} \alpha_0$. Under additional conditions (such as $m/n \to \rho \in (0,1]$, homogeneous $R_i = R$, finite moments), the estimator is asymptotically normal:

$$\sqrt{n}(\hat{\alpha}_n - \alpha_0) \xrightarrow{d} N(0, \rho I_\alpha^{-1}),$$

where $I_\alpha$ is an appropriate Fisher-information-type quantity (Konar et al., 11 Jan 2026).

By the delta method, the scale estimator $\hat{\lambda}_n$ achieves similar properties. The existence and uniqueness of these MLEs under the adaptive scheme have been rigorously proven (Dutta et al., 2023).

4. Comparison with Related Censoring Schemes

APHC is distinguished from Progressive Hybrid Censoring (PHC):

• In PHC, units are removed at specified failure times, but if the $m$th failure time exceeds $T$, the experiment stops—possibly with fewer than $m$ failures and with survivors right-censored at $T$.
• APHC adapts removal actions after $T$ to ensure the experiment proceeds until exactly $m$ failures are observed, with deferred removals if failures are delayed. This increases the information content while preserving experimental constraints.

Compared to standard right-censoring (Type I or II), APHC provides intermediate control between cost (linked to survivor removals) and information (number and timing of failures), offering a more flexible and robust middle ground (Konar et al., 11 Jan 2026).

Adaptive Type-II Progressive Hybrid Censoring (AT-II PHC) extends these ideas to competing risks and Marshall-Olkin bivariate Weibull models, ensuring exactly $m$ observable failures under general dependence and cause-of-failure ambiguity (Dutta et al., 2023).

5. Implementation Algorithms and Numerical Considerations

Implementation under APHC for Weibull lifetimes proceeds as follows:

1. Collect ordered failure times and determine $j = \max\{i: X_{(i)} < T\}$.
2. Compute:
   • $$A(\alpha) = \sum_{i=1}^m X_{(i)}^\alpha + \sum_{i=1}^j R_i X_{(i)}^\alpha + (n-m-\sum_{i=1}^jR_i) X_{(m)}^\alpha$$
   • $$B(\alpha) = \sum_{i=1}^m X_{(i)}^\alpha \log X_{(i)} + \sum_{i=1}^j R_i X_{(i)}^\alpha \log X_{(i)} + (n-m-\sum_{i=1}^jR_i) X_{(m)}^\alpha \log X_{(m)}$$
3. Define the shape function:

$$S(\alpha) = \frac{m}{\alpha} + \sum_{i=1}^m \log X_{(i)} - m \cdot [B(\alpha) / A(\alpha)]$$

1. Solve $S(\alpha) = 0$ via Newton–Raphson, updating:

$$\alpha_{\text{new}} = \alpha_{\text{old}} - S(\alpha_{\text{old}}) / S'(\alpha_{\text{old}})$$

2. Update $$\hat{\lambda} = [A(\hat{\alpha}) / m ]^{1/\hat{\alpha}}$$.

Fast convergence is observed due to the strict monotonicity and unimodality of the involved score function (Konar et al., 11 Jan 2026).

6. Regression Modeling in the ALT Context

In accelerated life testing (ALT), APHC-captured data supports regression of Weibull parameters on external stress variables (e.g., temperature $T$, voltage $V$):

• Log-linear models are posited:

$$\lambda(T, V) = a_0 + a_1 (1/T) + a_2 \log V, \ \alpha(T, V) = c_0 + c_1 (1/T) + c_2 \log V.$$

• After separate estimation of $(\hat{\lambda}_i, \hat{\alpha}_i)$ at each stress combination $(T_i, V_i)$, these estimates are regressed via OLS with appropriate variance corrections (e.g., Murphy–Topel) to account for plug-in variability (Konar et al., 11 Jan 2026).

This two-step approach enables quantification of the stress–lifetime relationship under complex censoring.

7. Extensions and Optimization for Competing Risks

For dependent competing risks (e.g., Marshall–Olkin bivariate Weibull), APHC is extended to AT-II PHC. The observed likelihood accommodates unknown and simultaneous failure causes, and MLEs are computed via Newton–Raphson or fixed-point approaches. Bayesian inference (using gamma-Dirichlet priors and MCMC) provides Bayes estimates and credible intervals, with convergence diagnostics such as the Gelman–Rubin $G$-statistic (Dutta et al., 2023).

Optimality of the censoring plan $(T, R)$ can be pursued with respect to:

• A-optimality: Minimize trace of asymptotic covariance.
• D-optimality: Minimize determinant of asymptotic covariance.
• F-optimality: Maximize trace of the information matrix.

Simulation studies and real data analyses confirm the practical efficiency and accuracy of APHC-based estimation, especially when plans are chosen under optimality criteria (Dutta et al., 2023).