Assume that phi(& centerdot;) is a non-negative, continuously differentiable weight function and phi '(& centerdot;) is nondecreasing on [0,infinity), and let 0= 0) of dimension alpha >= 1 and starting at zero such that E[tau(mu)]
We examine the optimality properties of the Gini index estimator under complex survey design involving stratification, clustering, and sub-stratification. While Darku et al. (Econometrics, 8, 26, 2020) considered only stratification and clustering and did not provide theoretical guarantees, this study addresses these limitations by proposing two procedures, that is, a purely sequential method and a two-stage method. Under suitable regularity conditions, we establish uniform continuity in probability for the proposed estimator, thereby contributing to the development of random central limit theorems under sequential sampling frameworks. Furthermore, we show that the resulting procedures satisfy both asymptotic first-order efficiency and asymptotic consistency. Simulation results demonstrate that the proposed procedures achieve the desired optimality properties across diverse settings. The practical utility of the methodology is further illustrated through an empirical application using data collected by the National Sample Survey agency of India.
In this paper, we consider the prediction problem of the future failure times based on observed Type-II right-censored data from Lehmann family of distribution under simple step stress life testing (SSLT) scenario. We explore various point predictors including best unbiased predictor (BUP), maximum likelihood predictor (MLP), and conditional median predictor (CMP) of the censored items. Furthermore, we extend our analysis to include the derivation of prediction intervals (PIs) using pivotal quantity method and highest conditional density (HCD) method. We conduct an extensive simulation study to evaluate the effectiveness of the point and interval predictors. The criteria employed for comparative evaluation include prediction bias and root mean square prediction error for point predictors, alongside average length and coverage probability for prediction intervals. Finally, we have analyzed two real life data sets for illustrative purposes. The first data set contains the failure times of 35 solar lighting devices from an SSLT experiment, while the second data set contains the lifetimes of nanocrystalline embedded high-k device data.
This article develops and explores the sequential detection and quickest detection problems for marked Poisson processes. In particular, mixed-differential-difference equations that describe the error probabilities and average sample numbers of the sequential probability ratio test (SPRT), and the average time between false alarms and the average delay until detection of the cumulative sum (CUSUM) test are derived. A numerical technique for solving these equations is described, and this method is used to analyze several examples, revealing some basic properties of these tests.
Federated monitoring of large-scale data streams offers a computationally efficient framework by aggregating local detection statistics into a global monitoring rule. While fusion policies for local detection statistics on the log-likelihood-ratio (LLR) scale, e.g., notably the Cumulative Sum (CUSUM) statistics, are well documented, the properties of fusion on the likelihood-ratio (LR) scale, such as Shiryaev-Roberts-Pollak (SRP) statistics, remain underexplored. In this paper, we conduct a comprehensive numerical study evaluating thirteen fusion policies across both LLR and LR scales. We demonstrate that these two scales yield fundamentally distinct detection patterns: while LLR-scale fusion exhibits a characteristic imbalance in detection delays between sparse and dense changes, the simple sum of local LR-scale detection statistics inherently possesses a "self-shrinkage" property. Crucially, we observe that applying additional explicit shrinkage to local LR-scale detection statistics provides negligible benefit, suggesting that the LR-scale fusion is naturally robust to high-dimensional sparsity. Leveraging these insights, we propose a Rank-Weighted-Sum (RWS) rule for LR-scale fusion, as well as a discussion on general alpha-scale fusion for future research.
Breast cancer remains a leading malignancy among women globally, necessitating precise monitoring of therapeutic outcomes to improve survival and quality of life. Conventional statistical quality control tools often face limitations due to patient heterogeneity. This study proposes a novel log-logistic risk-adjusted cumulative sum (RACUSUM) control chart for monitoring breast cancer treatments, incorporating an accelerated failure time (AFT) model to adjust for individual patient risk factors. This risk adjustment enhances the chart's sensitivity and specificity in detecting deviations in treatment effectiveness. A significant advancement is the multi-objective optimization of the RACUSUM design, balancing rapid detection of process shifts with cost-efficiency. Multi-objective particle swarm optimization (MOPSO) generates optimal solution sets, which are prioritized using the data envelopment analysis (DEA) and weighted aggregated sum product assessment (WASPAS) methods to support clinical decision-making. Validation through a case study at a breast cancer center demonstrated superior performance compared to traditional models, achieving improved monitoring accuracy with minimal incremental cost. This integrated approach offers clinicians and healthcare administrators a robust tool for enhancing breast cancer care quality, ultimately aiming to reduce adverse events and improve patient outcomes.
Effective process monitoring is essential in engineering and industrial systems to ensure product reliability and prevent costly failures of critical components. In such contexts, censoring techniques and control charts are indispensable tools for detecting deviations from desired operational standards. This study offers a new methodology to develop a novel Cumulative Sum (CUSUM) control chArt for monitoRing the Progressive Type-II (PT-II) CensOred data's meaN from Burr-X dIstribution (MARCONI will be used hereafter for this monitoring frameworks). Named in honor of Guglielmo Marconi for his groundbreaking work in signal detection, the MARCONI control chart extends similar concepts to the domain of statistical process control. Unlike classical CUSUM or Shewhart charts, which assume simple distributional forms and cannot be directly applied under censoring without bias, MARCONI integrates three innovations: (i) censoring-adjusted mean statistics, (ii) distribution-specific formulations leveraging the Burr-X model, and (iii) adaptive, two-phase control limits that preserve nominal in-control performance under censoring. Through comprehensive simulation studies, we evaluate MARCONI's performance in detecting small to moderate process shifts, achieving significantly lower average run lengths (ARLs) under varying censoring conditions. Performance metrics including ARL, standard deviation of run length (SDRL), and detection probability are analyzed across various shift magnitudes and censoring levels. The chart's effectiveness is validated through application to stress-rupture life data of Kevlar 49/epoxy strands, demonstrating its superior performance, via lower out-of-control (OOC) ARLs compared to a Shewhart-type chart. To the best of our knowledge, this is the first SPC framework that unifies PT-II censoring, heavy-tailed Burr-X modeling, and CUSUM principles into a single tool, offering engineers a statistically rigorous and practically relevant solution for monitoring reliability in high-risk environments.
Tze Leung Lai made seminal contributions to sequential analysis, particularly in sequential hypothesis testing, changepoint detection and nonlinear renewal theory. His work established fundamental optimality results for the sequential probability ratio test and its extensions, and provided a general framework for testing composite hypotheses. In changepoint detection, he introduced new optimality criteria and computationally efficient procedures that remain influential. He applied these and related tools to problems in biostatistics. In this article, we review these key results in the broader context of sequential analysis.
Recent advancements in designing optimal sampling frameworks, particularly under cost or budget constraints, have gained significant attention. Several studies aim to maximize statistical power for hypothesis tests within these constraints. However, existing methods lack the flexibility to accommodate interventions while maximizing power for four specific tests under budgetary limitations. To address this gap, we propose a two-stage procedure that estimates the optimal sample sizes for two groups and simultaneously tests four hypotheses with maximum power under budget constraints. Through extensive simulation analyses, we demonstrate the performance of the proposed procedure. To illustrate the application of our procedure, we implemented a PERMA (Positive Emotions, Engagement, Relationships, Meaning, and Accomplishment) intervention experiment. This experiment involved two groups, viz. experimental group and control group. This was conducted in a natural setting to enhance the well-being of school students in India. The experimental group participated in a five-day PERMA intervention, while the control group did not receive any intervention. Participants in both groups completed pre- and post-tests using the WHO-5 well-being measure. Given the limited budget for this experiment, optimizing sample size to achieve maximum statistical power was critical. This study provides a practical framework for optimizing sample sizes in budget-constrained experiments, particularly in psychological research.
Capture-recapture is a well-established sampling technique used to estimate population size and related demographic parameters such as birth, mortality, immigration, and emigration rates. Traditional methods rely on a fixed-size recapture sample. In this work, we propose a sequential capture-recapture method where individuals are sampled one by one until a fixed number of previously marked individuals is observed. We derive an unbiased estimator of the population size and establish its variance analytically. A stability-based stopping rule is also introduced to optimize the sampling effort while preserving estimation accuracy. Simulation results demonstrate that the proposed estimator outperforms classical estimators in terms of efficiency and stability. Approximate confidence intervals are also derived and empirically shown to achieve near-nominal coverage probabilities.
This study aims to propose a reliability acceptance sampling plan (RASP) that evaluates the acceptability of products using their lifetimes under a hybrid censoring scheme. In RASP, lifetime characteristics are used to assess product quality, ensuring that the plan accurately captures the reliability performance of the product. To design this plan, we have determined the optimal sample size using a compound optimal design (COD) strategy, which balances two objectives simultaneously. We have demonstrated the methodology using the Gumbel distribution; however, it can be extended to any location-scale family of distributions, making it versatile for various practical scenarios. As a part of our approach, we derive the Fisher information matrix for hybrid-censored data and employ it to define the imprecision-based optimality criteria. Further, we have used two criteria simultaneously to obtain the COD and address the complexities of hybrid censoring. The COD has been applied by introducing a graphical solution technique, simplifying the computation process and result interpretability. A detailed sensitivity analysis is performed to evaluate the robustness of the proposed strategy against misspecified model parameters. To demonstrate the proposed methodology in a real-life scenario, we illustrate the procedure using a real dataset.
In fields such as industry and finance, large amounts of data with heterogeneous characteristics are often generated. Traditional statistical models struggle to address the modeling problems associated with such data. The joint mean and variance model provides a new approach to handle these data. Since its introduction, various challenges related to the model, such as parameter estimation, variable selection, empirical likelihood inference, and statistical diagnostics, have been extensively studied. However, in practical applications, many heterogeneous datasets undergo structural changes at certain points due to various factors. Ignoring such changes during data analysis might lead to errors in the results, and more severely, it may affect decision-making, causing significant losses. To address this issue, this paper focuses on the joint mean and variance model under a normal distribution and proposes a change point test method based on the Modified Information Criterion (MIC) to tackle the change point problem in heteroscedastic data. Simulation results reveal that the MIC-based test method significantly outperforms the classical likelihood ratio test. Finally, the method is applied to the analysis of the China Securities 2000 Index return dataset, successfully identifying the location of the change points within the data.
This paper presents the design procedure of a generalized multiple dependent state sampling plan for strategically reducing the average sample number while ensuring the transmuted Weibull distributed mean life based on time-truncated life tests and optimizing the total inspection cost. Lifetime has been an essential quality characteristic of every product, especially electronic devices, due to its applications in human's day-to-day life. In general, the higher the economic cost and usage of the products, the greater the expectation on the lifetime of the products. Hence, this study considers the economical design of the proposed sampling plan using two points on the operating characteristic curve to ensure the products' mean life with optimal cost. Optimal parameters determination, application of the proposed plan, and advantages of the proposed plan are discussed in this study. The results show that the proposed plan can provide quality assurance with minimum time and at a lower cost rather than the other existing sampling plans.
We are concerned in this paper with sequentially planned hypothesis tests for continuous-time processes with stationary and independent increments. The problem arises when any observation, to be admitted to the analysis, should be paid for. In this situation, the continuous observation is impossible, and any feasible testing procedure can only be based on a finite number of observations. Respectively, the classical sequential analysis, typically based on optimal stopping, does not help with the optimization, because the problem arises how to assign the waiting times to the next observation, before the stopping occurs. We propose in this paper a construction of strategies that optimize both the decision rules and also the timing of individual measurements, for a rather general class of continuous-time processes with stationary and independent increments. We also propose an algorithmic scheme for its computational implementation. For a particular case of a Wiener process with a linear drift, the proposed algorithms are implemented in the R programming language, and the program code are made available in a public GitHub repository on the Internet. The proposed method is numerically compared with the classical sequential probability ratio test for the Wiener process and with the optimal one-stage test, provided that all the three have the same error probabilities. Many other numerical examples with graphical illustrations are provided.
We look at a sequence of Binomial random variables where the success rate increases from theta 1 to theta 2. We assume that both the success rates before and after the change are known. The number of trials at each time point is fixed, but may not be the same at each time point. We calculate the probability that the change has occurred. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop late.
A finite-horizon variant of the quickest change detection (QCD) problem that is of relevance to learning in non-stationary environments is studied. The metric characterizing false alarms is the probability of a false alarm occurring before the horizon ends. The metric that characterizes the delay is latency, which is the smallest value such that the probability that detection delay exceeds this value is upper bounded to a predetermined latency level. The objective is to minimize the latency (at a given latency level), while maintaining a low false alarm probability. Under the pre-specified latency and false alarm levels, a universal lower bound on the latency, which any change detection procedure needs to satisfy, is derived. Change detectors are then developed, which are order-optimal in terms of the horizon. The case where the pre- and post-change distributions are known is considered first, and then the results are generalized to the non-parametric case when they are unknown except that they are sub-Gaussian with different means. Simulations are provided to validate the theoretical results.
To enhance the effectiveness of sequential detection methods, we introduce a kernel smoothing moving average (KSMA) chart that places greater emphasis on recent observations. This new approach is benchmarked against established methods, including the finite moving average (MA), exponentially weighted moving average (EWMA), and CUSUM charts, evaluated in terms of the conditional average detection delay time (ADDT) for a given in-control average run length (ARL0) under persistent change. A tailored kernel function is proposed to maximize asymptotic efficiency across the signal strength spectrum while maintaining consistently high performance. The comparative study considers both fixed-mean and random-mean change models in multivariate settings. Our results indicate that for moderate signal strengths with fixed mean changes, the KSMA procedure outperforms the alternatives, whereas the EWMA procedure is superior when the signal strength is small. Overall, both the EWMA and KSMA procedures perform competitively and consistently surpass the other methods across fixed and random mean change scenarios. Applications to Dow Jones stock prices and EEG monitoring further illustrate the practical value of the proposed approach. Additional dimension reduction techniques for sparse signal detection are also briefly discussed.
Response-adaptive designs in clinical trials use ongoing patient outcome data to adjust the randomization process, aiming to assign more patients to treatments that are showing better performance during the trial. One approach to implement such adaptive randomization is through urn models, which provide a probabilistic framework for updating treatment allocations. In this article, we study the performance of the hybrid selection and testing procedure with curtailment proposed earlier by Buzaianu and Razaila for comparing binomial treatments with a standard, when implemented using the generalized P & oacute;lya's urn design (GPUD) and the cyclic play-the-winner (PWC) design as sampling plans. These adaptive designs meet the same probability requirements in terms of power and size as the curtailment procedure using the vector-at-a-time (VT) sampling studied by Buzaianu and Razaila. Simulations show that integrating curtailment with GPUD and PWC designs significantly lowers the expected sample size compared to the VT design. Moreover, these adaptive designs allocate fewer subjects to inferior treatments relative to both their non-curtailed two-stage counterparts and traditional fixed-sample-size procedures.
This study focuses on a monitoring procedure aimed at sequentially detecting parameter changes in bivariate time series models of counts. For this task, we utilize a bivariate integer-valued generalized autoregressive conditional heteroscedastic (BINGARCH) model to capture the characteristics of the time series of counts. The residuals obtained from fitting the BINGARCH model are employed in constructing the monitoring processes. Firstly, we introduce the BINGARCH model along with its inferential procedure. Next, we present two types of detectors for monitoring in BINGARCH models, and the control limits for these detectors are determined based on the corresponding limit theorems. To evaluate the performance of the monitoring methods, we conduct Monte Carlo simulations. Additionally, we provide a real data analysis for illustration. The results from these empirical studies confirm the validity of the proposed monitoring procedures.
For the classical problem of sequential binary hypothesis testing (SHT) studied by Wald for independent and identically distributed (i.i.d.) data, the optimal test is the sequential probability ratio test (SPRT). When the data is no longer i.i.d., the SPRT's exact or strong optimality property is lost. However, the SPRT is asymptotically optimal for general non-i.i.d. data under mild conditions, as the error probabilities go to zero. In this paper, the problem of SHT is studied for a special class of non-i.i.d. processes, and an exact optimal solution is obtained. This special class is the class of statistically periodic processes encountered in many applications in science and engineering. The objective for SHT chosen in the paper is an average risk criterion containing time-varying penalties for collecting samples and making decision errors. It is shown that the optimal solution is the SPRT with a time-varying sequence of thresholds. It is further shown that a constant threshold variant is asymptotically optimal. The constant threshold test is then applied to electrocardiogram (ECG) data to perform energy-efficient detection of heart arrhythmia.
