Approximate Confidence Intervals Research Articles

Rainfall data modeling using improved adaptive type-II progressively censored Weibull-exponential samples

To get enough data from experiments that last for a long time, a recently unique improved adaptive Type-II progressive censoring technique has been suggested. This study, taking this scheme into consideration, concentrates on some conventional and Bayesian estimation tasks for parameter and reliability indicators, where the underlying distribution is the Weibull-exponential. From a traditional point of view, the likelihood methodology is explored for gaining point and approximate confidence interval estimates. Apart from the standard method, the Bayesian methodology is investigated to obtain the Bayesian point and credible intervals by taking advantage of the Markov chain Monte Carlo technique and the squared error loss function. To differentiate between the traditional and Bayesian estimates, a simulation analysis proceeds under various conditions. In order to put the suggested strategies into application, a pair of rainfall data sets are evaluated and numerous precision criteria are employed to pick the best progressive censoring plan.

Inference of record failure times from unit generalized Rayleigh distribution

ABSTRACT In this paper, problems of estimation and prediction for record values are discussed for a two-parameter unit generalized Rayleigh distribution. Maximum likelihood estimators of model parameters along with their existence and uniqueness are established, and corresponding approximate confidence intervals are constructed based on asymptotic theory. In addition, alternative generalized inferential approach is provided for comparison based on the proposed pivotal quantities. Further, prediction intervals of remaining useful life are also presented under classical and generalized inferential methods when the unit generalized Rayleigh record values are available. Extensive numerical studies and two real-life examples are conducted to show the performance of the proposed methods. The results indicate that both classical and generalized estimation methods work satisfactorily and the proposed generalized inferential approach performs superior to traditional estimates in general.

A Note on Confidence Intervals for a Binomial p: Andersson–Nerman vs. Wilson

ABSTRACTNumerous approximate confidence intervals in closed form have been suggested over the years for the parameter in the binomial distribution. One of the oldest and still most advocated is the Wilson (score) interval, which in this article will be compared with the less well‐known Andersson–Nerman (henceforth AN) interval. These intervals are quite closely related, but it will be shown analytically and illustrated by examples that the coverage probability of the AN interval always equals or exceeds that of the Wilson interval, while also having uniformly larger expected length. Asymptotic expressions for the coverage probability and expected length of the AN interval are provided, using Edgeworth and Taylor expansions, respectively. The well‐behaved Wilson and AN pivots are furthermore contrasted with the problematic Wald pivot. The latter gives rise to the Wald interval, which is probably one of the best known and most used procedures altogether in the history of statistical inference. Unfortunately, this interval performs poorly, but even to this day it is to be found in many current textbooks. Therefore, it is still of relevance to search for attractive alternatives based on sound statistical principles.

Optimal Estimation of Reliability Parameters for Modified Frechet-Exponential Distribution Using Progressive Type-II Censored Samples with Mechanical and Medical Data

The aim of this research is to estimate the parameters of the modified Frechet-exponential (MFE) distribution using different methods when applied to progressive type-II censored samples. These methods include using the maximum likelihood technique and the Bayesian approach, which were used to determine the values of parameters in addition to calculating the reliability and failure functions at time t. The approximate confidence intervals (ACIs) and credible intervals (CRIs) are derived for these parameters. Two bootstrap techniques of parametric type are provided to compute the bootstrap confidence intervals. Both symmetric loss functions such as the squared error loss (SEL) and asymmetric loss functions such as the linear-exponential (LINEX) loss are used in the Bayesian method to obtain the estimates. The Markov Chain Monte Carlo (MCMC) technique is utilized in the Metropolis–Hasting sampler approach to obtain the unknown parameters using the Bayes approach. Two actual datasets are utilized to examine the various progressive schemes and different estimation methods considered in this paper. Additionally, a simulation study is performed to compare the schemes and estimation techniques.

Analysis of dependent complementary competing risks data from a generalized inverted family of lifetime distributions under a maximum ranked set sampling procedure with unequal samples

This paper explores analysis of a dependent complementary competing risks model when the failure causes are distributed by the proposed generalized inverted family of lifetime distributions. Under maximum ranked set sampling with unequal samples (MRSSU), statistical inference of model parameters and reliability indices is discussed under classical frequentist and Bayesian approaches, respectively. Maximum likelihood estimators along with their existence and uniqueness are obtained for model parameters, and associated approximate confidence intervals are constructed in consequence. Bayesian estimation is also performed with respect to general flexible priors, and the Markov Chain Monte Carlo (MCMC) algorithm is proposed for complex posterior computation. The study further examines classical and Bayesian estimations with order restriction of parameters when additional historical information is available in the MRSSU scenario. Finally, the performance of different results is evaluated through numerical simulations and a real data example is presented for demonstrating the application of our methods.

Classical and Bayesian inference of Cpc$\mathcal {C}_{pc}$ for Wilson–Hilferty distribution under progressively first‐failure type‐II censoring samples

AbstractThis study uses two frequentist approaches and the Bayesian method of estimation using progressively first‐failure type‐II censored data to estimate process capability index (PCI), , for the Wilson–Hilferty (WH) distribution. A competitive maximum product of spacing (MPS) method for estimation of is proposed in the frequentist method as an alternative to conventional likelihood (LK)‐based estimation. We have also taken into account the PS function in the Bayesian setup as an alternative to the conventional LK function, and both LK and PS‐based Bayesian estimates are given for . Further, the Bayes estimates are evaluated by applying Tierney–Kadane (TK) approximation method and Markov chain Monte Carlo (MCMC) method with the help of both LK and PS functions and Jeffrey's prior. Additionally, the approximate confidence intervals based on the suggested frequentist techniques as well as the Bayes highest posterior density (HPD) credible intervals are built for the index . In addition, two bootstrap confidence intervals are obtained. In the simulation exercise, the performance of the Bayes and traditional estimates of is assessed in terms of their mean squared errors, and the average width and coverage probabilities of the CIs and HPD intervals are compared. An actual data set from the electronic industries is reanalyzed in order to show the efficacy of the proposed index and estimation methodology.

Reliability inference for dual constant-stress accelerated life test with exponential distribution and progressively Type-II censoring

Accelerated life test provides a feasible and effective way to rapidly derive lifetime information by exposing products to higher-than-normal operating conditions. However, most of the previous research on accelerated life test has focused on the application of a single stress factor and a traditional censoring scheme. This article considers the reliability inference for a dual constant-stress accelerated life test model with exponential distribution and progressively Type-II censoring. Point estimates for model parameters are provided using maximum likelihood estimation and the weighted least squares method based on random variable transformation. In addition, we construct asymptotic confidence intervals, approximate confidence intervals, and bootstrap confidence intervals for the parameters of interest. Finally, extensive simulation studies and an illustrative example are presented to investigate the performance of the proposed methods.

Inference for depending competing risks from Marshall–Olikin bivariate Kies distribution under generalized progressive hybrid censoring

This paper explores inferences for a competing risk model with dependent causes of failure. When the lifetimes of competing risks are modelled by a Marshall–Olikin bivariate Kies distribution, classical and Bayesian estimations are studied under generalized progressive hybrid censoring. The existence and uniqueness results for maximum likelihood estimators of unknown parameters are established, whereas approximate confidence intervals are constructed using the observed Fisher information matrix. In addition, Bayes estimates are explored based on a flexible Gamma-Dirichlet prior information. Furthermore, when there is a priori order information on competing risk parameters being available, traditional classical likelihood and Bayesian estimates are also established under restricted parameter case. The behavior of the proposed estimators is evaluated through extensive simulation studies, and a real data study is presented for illustrative purposes.

Likelihood and pivotal-based reliability inference for inverse exponentiated Rayleigh distribution under block progressive censoring

ABSTRACT In this paper, focus is on the estimation of survival characteristics, specifically the reliability function, hazard rate function, median time to failure, and differences in different test facilities, using block progressive censored data. Estimations are carried out through both maximum likelihood and pivotal methods, assuming the lifetime distribution of test units follows an inverse exponentiated Rayleigh distribution. The paper derives maximum likelihood estimators for unknown parameters, exploring their existence and uniqueness properties. Approximate confidence intervals for survival characteristics are constructed using the delta method and likelihood theory. Moreover, point and generalized confidence interval estimators are developed through a pivotal quantity-based method. A simulation study is conducted to compare the performance of the proposed approaches, revealing that the pivotal quantity-based approach yields superior estimation results. Finally, the proposed estimates are applied to analyze two real datasets, illustrating their practical applicability.

Different estimation techniques and data analysis for constant-partially accelerated life tests for power half-logistic distribution

Partial accelerated life tests (PALTs) are employed when the results of accelerated life testing cannot be extended to usage circumstances. This work discusses the challenge of different estimating strategies in constant PALT with complete data. The lifetime distribution of the test item is assumed to follow the power half-logistic distribution. Several classical and Bayesian estimation techniques are presented to estimate the distribution parameters and the acceleration factor of the power half-logistic distribution. These techniques include Anderson–Darling, maximum likelihood, Cramér von-Mises, ordinary least squares, weighted least squares, maximum product of spacing and Bayesian. Additionally, the Bayesian credible intervals and approximate confidence intervals are constructed. A simulation study is provided to compare the outcomes of various estimation methods that have been provided based on mean squared error, absolute average bias, length of intervals, and coverage probabilities. This study shows that the maximum product of spacing estimation is the most effective strategy among the options in most circumstances when adopting the minimum values for MSE and average bias. In the majority of situations, Bayesian method outperforms other methods when taking into account both MSE and average bias values. When comparing approximation confidence intervals to Bayesian credible intervals, the latter have a higher coverage probability and smaller average length. Two authentic data sets are examined for illustrative purposes. Examining the two real data sets shows that the value methods are workable and applicable to certain engineering-related problems.

A Measure of Departure from Marginal Homogeneity using Continuation Odds for Square Contingency Tables with Ordered Categories

For the analysis of square contingency tables with ordinal classifications, the marginal homogeneity (MH) model is the one of important models. Some measures for analyzing the degree of departure from the MH model have been proposed. This study proposes a new measure using the continuation odds. Continuation odds may be considered as discrete time hazard. The proposed measure is expressed in the form of Cressie-Read’s power-divergence (including the Kullback-Leibler divergence) or Patil and Taillie’s diversity index (including Shannon entropy). This study derives a plug-in estimator of the proposed measure and an approximate confidence interval for the proposed measure. Through numerical examples, we evaluate the performances of them. Additionally. the usefulness of the proposed measure is demonstrated by applying it to real data that the row and column variables are the discrete survival time variables.

Tests for one and two mean vectors and simultaneous confidence intervals with monotone incomplete data

In this study, we consider the problems of testing for a mean vector and testing the equality of two mean vectors with monotone missing data. We propose new test statistics similar to the simplified Hotelling’s T 2-type test statistic for one-sample and two-sample problems under general-step monotone missing data. Approximate upper percentiles of this new statistics are provided by asymptotic expansion along with transformed test statistics based on Bartlett adjustment. Approximate simultaneous confidence intervals for pairwise comparisons among mean vectors are also presented. Furthermore, we investigated the asymptotic behavior of the proposed statistics using a Monte Carlo simulation, and the approximate accuracy of the proposed approximations are provided and discussed. Finally, the numerical power for several statistics is given.

Different Statistical Inference Algorithms for the New Pareto Distribution Based on Type-II Progressively Censored Competing Risk Data with Applications

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

An index for measuring degree of departure from symmetry for ordinal square contingency tables

For the analysis of square contingency tables with the same row and column ordinal classifications, this study proposes an index for measuring the degree of departure from the symmetry model using new cumulative probabilities. The proposed index is constructed based on the Cressie and Read’s power divergence, or the weighted average of the Patil and Taillie’s diversity index. This study derives a plug-in estimator of the proposed index and an approximate confidence interval for the proposed index. The estimator of the proposed index is expected to reduce the bias more than the estimator of the existing index, even when the sample size is not large. The proposed index is identical to the existing index under the conditional symmetry model. Therefore, assuming the probability structure in which the conditional symmetry model holds, the performances of plug-in estimators of the proposed and existing indexes can be simply compared. Through numerical examples and real data analysis, the usefulness of the proposed index compared to the existing index is demonstrated.

Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model

Accelerated life tests play a very critical role in reliability analysis because highly reliable products are being produced with recent advanced technologies to sustain the market demand and competition. A progressive-stress accelerated life test is one of the kinds of accelerated life tests that allows applied stress to change continuously. Considering the importance of progressive-stress accelerated life tests, this paper deals with progressive-stress accelerated life tests when the lifetimes of units follow the log-logistic distribution and are progressively type-II censored where the associated scale parameter conforms to the inverse power law. Different estimators of model parameters are derived using maximum likelihood and Bayesian methods. Interval estimation is also considered. In the sequel, approximate confidence, bootstrap, and Bayes credible intervals are constructed. Bayes estimators are obtained under squared error, LINEX, and entropy loss functions using proper and improper prior distributions. A simulation study is conducted based on various censoring schemes. The coverage percentages and interval widths are computed via Monte Carlo simulations. Bayes predictive estimates and intervals are also obtained. Finally, two different accelerated life test data are analyzed for illustration purposes.

Parameter estimation of Chen distribution under improved adaptive type-II progressive censoring

ABSTRACT This study focuses on the estimation of the two unknown parameters of Chen distribution, characterized by a bathtub-shaped hazard rate function, within an improved adaptive type-II progressive censored data framework. Maximum likelihood estimation is proposed for the two parameters, and the establishment of approximate confidence intervals is based on asymptotic normality. Bayesian estimation is also conducted under both symmetric and asymmetric loss functions, utilizing the proposed importance sampling and Metropolis–Hastings algorithm. Lastly, the performance of various estimation methods is evaluated through Monte Carlo simulation experiments, and the proposed estimation approach is illustrated using a real dataset.

A new extended Pham distribution for modelling cancer data

In this study, a novel extended Pham model is offered and examined. This model can fit positive, negative, and approximately symmetric data. In addition, it features four alternative hazard rate shapes, namely increasing, decreasing, bathtub, and upside-down bathtub shapes. The new model's main properties, such as quantiles, moments, and mean deviations, are derived. To evaluate the estimation of distribution parameters, two approaches are studied. From a classical standpoint, the maximum likelihood technique is used to determine the point and approximate confidence intervals of all unknown parameters. The Bayesian estimation method, on the other hand, is considered via the Markov chain Monte Carlo technique for obtaining Bayes point estimates alongside Bayes credible intervals. We present various Monte Carlo simulation studies to investigate the efficacy of maximum likelihood and Bayes estimation findings for the model's parameters. Numerical investigations showed that the offered Bayes estimates outperformed those derived using the conventional method. We also analyzed two sets of real data taken from the medical sector; the first represents survival rates for cancer patients who received a combination of radiotherapy and chemotherapy, and the second consists of remission times for bladder cancer patients. These clinical scenarios reveal that the new model outperforms other important competing models based on specific statistical criteria. Ultimately, for both clinical data applications, the proposed extended Pham distribution is the best choice compared to the traditional Pham and the other well-known eight lifetime models.

Assessing high‐quality process performance using the quality‐yield index: An innovative methodology

AbstractManufacturers must meet high‐quality standards and exceed customer expectations to stay competitive due to significant technological advancements in recent decades. While implementing the yield measure is useful for achieving process performance by focusing on products that fall within specified limits, it does not accommodate specific customer requirements, particularly when a product's quality characteristic deviates from target value. To address this need, the quality‐yield index (Q‐yield) has been proposed, which combines the process‐yield index and loss‐based capability index, providing a more advanced performance measure. However, the Q‐yield index's confidence interval is challenging to derive due to the complicated sampling distribution involved. Several existing methods have attempted to construct an approximate confidence interval but none have performed well. Therefore, this article proposes an innovative approach, called the generalized confidence intervals (GCIs), that utilizes the idea of generalized pivotal quantities to establish the confidence interval for the Q‐yield index. The proposed approach is evaluated through simulations and compared to existing methods. The results reveal that the proposed approach provides the most accurate results for constructing the lower confidence bound of the Q‐yield index. This approach is recommended to evaluate process performance using the Q‐yield index for high‐quality customer requirements.

Approximate confidence intervals for the Population Proportion based on linear model

Approximate confidence intervals for the Population Proportion based on linear model

Statistical inference of unified hybrid censoring scheme for generalized inverted exponential distribution with application to COVID-19 data

Using a unified hybrid censoring scheme, this study explores statistical inferences for products with lifetimes following the generalized inverted exponential distribution. The model’s parameters are estimated using the maximum likelihood approach. In addition, likelihood functions and asymptotic theories are employed to generate approximate confidence intervals. Moreover, Bayesian estimates based on classical likelihood functions are investigated, considering both asymmetric and symmetric loss functions with prior information. It is recommended to approximate the Bayes estimates using Gibbs sampling, which utilizes the Markov chain Monte Carlo technique to establish credible intervals for the parameters. A numerical example is provided to demonstrate the effectiveness of the proposed methods. Furthermore, a simulation study is conducted to illustrate the confidence intervals and the statistical characteristics of the parameters.