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Article A Proportional Hazards Mixture Cure Model for Subgroup Analysis: Inferential Method and an Application to Colon Cancer Data(MDPI, 2025) Liu, Kai; Balakrishnan, Narayanaswamy; Peng, YingweiWhen determining subgroups with heterogeneous treatment effects in cancer clinical trials, the threshold of a variable that defines subgroups is often pre-determined by physicians based on their experience, and the optimality of the threshold is not well studied, particularly when the mixture cure rate model is considered. We propose a mixture cure model that allows optimal subgroups to be estimated for both the time to event for uncured subjects and the cure status. We develop a smoothed maximum likelihood method for the estimation of model parameters. An extensive simulation study shows that the proposed smoothed maximum likelihood method provides accurate estimates. Finally, the proposed mixture cure model is applied to a colon cancer study to evaluate the potential differences in the treatment effect of levamisole plus fluorouracil therapy versus levamisole alone therapy between younger and older patients. The model suggests that the difference in the treatment effect on the time to cancer recurrence for uncured patients is significant between patients younger than 67 and patients older than 67, and the younger patient group benefits more from the combined therapy than the older patient group.Article On the Notion of Discrete-Time Signature and Some Associated Properties and Results(Cambridge University Press, 2026) Balakrishnan, Narayanaswamy; Yi, He; Goroncy, AgnieszkaIn this work, by considering coherent systems comprising independent components with discrete lifetimes, we introduce the notion of discrete-time signature and then discuss some of its properties. With the use of the introduced signature, a stochastic ordering result is also established. We then introduce transformation formulas for the discrete-time signature to facilitate the comparison of systems of different sizes. Some examples are also presented to illustrate all the results developed here.Article Circular One/Two/Three-Dimensional Consecutive k-Type Systems(Springer, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangIn this paper, several circular one/two/three-dimensional consecutive k-type systems are studied, including circular consecutive-k-out-of-n: F systems, circular l-consecutivek-out-of-n: F systems without/with overlapping, circular connected-(k(1), k(2))-outof-(n(1), n(2)): F systems, circular l-connected-(k(1), k(2))-out-of-(n(1), n(2)): F systems without/ with overlapping, circular connected-(k(1), k(2), k(3))-out-of-(n(1), n(2), n(3)): F systems, and circular l-connected-(k(1), k(2), k(3))-out-of-(n(1), n(2), n(3)): F systems without/with overlapping. Reliability functions of these systems are studied using finite Markov chain imbedding approach (FMCIA). Some illustrative examples are provided, and possible applications and generalizations of the established results are also mentioned.Article Generalized Chi-Squared Based Goodness-of-Fit Tests under Progressive Type-II Censoring for Exponential and Weibull Distributions(Taylor & Francis Inc, 2026) Schmiedt, Anja B.; Balakrishnan, Narayanaswamy; Cramer, ErhardWe propose new goodness-of-fit tests for exponentiality based on progressively Type-II censored data. These tests utilize scale-invariant statistics obtained from the Mahalanobis norm of normalized order statistics, leading to three test statistics, corresponding to L-2(-), L-1(-), and L-infinity-norms of centered uniform spacings. Exact and asymptotic distributions of these statistics are presented. A power study evaluates the proposed tests against existing benchmarks across various alternative distributions and censoring plans, demonstrating superior performance in cases with small and moderate sample sizes. Furthermore, we extend the methodology to approximate goodness-of-fit tests for Weibull distributions via power transformation, ensuring robustness w.r.t. the approximated significance level under unknown shape parameters. An illustrative data example confirms the practical applicability of our tests. Our findings highlight the potential for further extending goodness-of-fit tests under progressive Type-II censoring to other null distributions.Article Acceptance Sampling Plan under Step Stress Accelerated Life Test for One-Shot Devices(Elsevier Sci Ltd, 2026) Lin, Chien-Tai; Balakrishnan, Narayanaswamy; Ling, Man-HoAcceptance sampling plans are essential in many manufacturing industries, serving as a statistical method to decide whether to accept or reject a batch of products based on the quality of a sample. The demand for reliable assessment methodologies has grown increasingly important as devices become more reliable under normal operating conditions. This trend poses challenges for reliability assessments, especially when there is limited failure data from tests. To address this, accelerated life testing (ALT) is widely employed to induce rapid failures for reliability analysis. This paper presents a comprehensive study of acceptance sampling plans under step-stress ALT (SSALT) for one-shot devices. We propose a tailored acceptance sampling plan that incorporates SSALT principles to enhance decision-making regarding product acceptance. Our methodology enables the determination of acceptable quality thresholds while evaluating the associated producer's and consumer's risks. Moreover, it can determine the minimum sample size required to ensure that both risks are adequately addressed, as well as to identify the optimal percentage of test items to examine at various testing stages. Two illustrative examples are provided to demonstrate the developed optimal acceptance sampling plans. This research highlights the significant impact of the choice of reliability index and stress pattern on inspection allocation.Article A General Type of Linear Consecutive-K Systems(Springer, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangIn this paper, some well-known consecutive k-type systems, including linear consecutive-k-out-of-n: F systems and linear l-consecutive-k-out-of-n: F systems without/with overlapping, are generalized by using more general failure patterns. Finite Markov chain imbedding approach (FMCIA) is applied in a new way for evaluating reliabilities of these generalized new systems. Some illustrative examples are provided for demonstrating the theoretical results established here and also for showing the efficiency of the computational process. Finally, some possible applications and generalizations are mentioned.Article Nonparametric Tests for Comparing Reliabilities of Coherent Systems at Specific Mission Time(IEEE-Inst Electrical Electronics Engineers Inc, 2026) Xu, Xuan; Zhu, Xiaojun; Balakrishnan, Narayanaswamy; Ng, Hon Keung TonyReliability analysis of coherent systems is critical for evaluating the performance of systems whose functionality depends on the reliability of their components. Traditional parametric methods for comparing reliabilities of coherent systems assume a specific probability distribution for component lifetimes, which may result in inaccurate results when these model assumptions are violated. This article introduces nonparametric procedures using system-level data with known signatures to compare the reliabilities of systems. The proposed methodology avoids parametric distributional assumptions for component lifetimes while relying on the standard assumption in signature-based reliability analysis. Specifically, a two-sample likelihood ratio test procedure is proposed to demonstrate a component or system with superior reliability. Monte Carlo simulations are performed to evaluate the performance of the proposed methods. Furthermore, we examine the effect of system structure on test power and determine favourable structures to enhance the power performance of the test. Practical examples are used to illustrate the proposed test procedures.Article Multi-State Linear Three-Dimensional Consecutive k-Type Systems(Cambridge Univ Press, 2026) Yi, He; Balakrishnan, Narayanaswamy; Li, XiangConsecutive $k$-type systems have become important in both reliability theory and applications; in spite of a large literature existing on them, three-dimensional consecutive $k$-type systems have not yet been studied for multi-state case. In this paper, we introduce several different types of multi-state linear three-dimensional consecutive $k$-type systems for the first time, with due consideration to possible overlapping of failure blocks. The finite Markov chain imbedding approach is then used for the derivation of their reliability functions with state spaces and transition matrices provided in a novel way, and the involved computational process is illustrated through several numerical examples. Finally, some possible applications of the work and potential extensions are pointed out.Article Robust Divergence-Based Tests of Hypotheses for Simple Step-Stress Accelerated Life-Testing Under Gamma Lifetime Distributions(Elsevier, 2026) Balakrishnan, Narayanaswamy; Jaenada, Maria; Pardo, LeandroMany modern devices are highly reliable, with long lifetimes before their failure. Conducting reliability tests under actual use conditions may require therefore impractically long experimental times to gather sufficient data for developing accurate inference. To address this, Accelerated Life Tests (ALTs) are often used in industrial experiments to induce product degradation and eventual failure more quickly by increasing certain environmental stress factors. Data collected under such increased stress conditions are analyzed, and results are then extrapolated to normal operating conditions. These tests typically involve a small number of devices and so pose significant challenges, such as interval-censoring. As a result, the outcomes are particularly sensitive to outliers in the data. Additionally, a comprehensive analysis requires more than just point estimation; inferential methods such as confidence intervals and hypothesis testing are essential to fully assess the reliability behaviour of the product. This paper presents robust statistical methods based on minimum divergence estimators for analyzing ALT data of highly reliable devices under step-stress conditions and Gamma lifetime distributions. Robust test statistics generalizing the Rao test and divergence-based tests for testing linear null hypothesis are then developed. These hypotheses include in particular tests for the significance of the identified stress factors and for the validity of the assumption of exponential lifetimes.Article On Optimal Allocation of Redundancies in Random Weighted Coherent Systems(Wiley, 2026) Sahoo, Tanmay; Hazra, Nil Kamal; Balakrishnan, NarayanaswamyUnlike the ordinary coherent systems, a random weighted coherent system captures components' disparity. Its performance is measured by some appropriate measure of contributions made by all of its components. In this paper, we consider random weighted coherent systems, defined based on the survival capacity, with three different survival mechanisms, namely, Types-I, II, and III. Then, we study the optimal active redundancy allocation policy as well as the optimal assembly method of random weights in a random weighted coherent system. We provide several numerical examples to illustrate all the results established here.
