Asymptotic Confidence Intervals Research Articles

Generalized fiducial inference for the generalized logistic distribution: Censored and uncensored cases

. This article primarily considers the statistical inference of the scale parameter, shape parameter, and reliability of the generalized logistic distribution in both censored and uncensored cases. For the progressively Type-II censored and complete samples, the frequentist method is utilized to construct asymptotic confidence intervals, and the Bayesian inference method is employed for constructing the point estimators as well as posterior credible intervals. Then the generalized fiducial method is applied to construct the fiducial point estimators and the fiducial confidence intervals. Furthermore, the non parametric generalized fiducial method is introduced to estimate the survival function. Simulation results demonstrate that the generalized fiducial method consistently outperforms other methods in terms of mean square error, average length, and empirical coverage. Finally, two real datasets are used to illustrate the proposed methods.

New and Efficient Estimators of Reliability Characteristics for a Family of Lifetime Distributions under Progressive Censoring

Estimation of reliability and stress–strength parameters is important in the manufacturing industry. In this paper, we develop shrinkage-type estimators for the reliability and stress–strength parameters based on progressively censored data from a rich class of distributions. These new estimators improve the performance of the commonly used Maximum Likelihood Estimators (MLEs) by reducing their mean squared errors. We provide analytical asymptotic and bootstrap confidence intervals for the targeted parameters. Through a detailed simulation study, we demonstrate that the new estimators have better performance than the MLEs. Finally, we illustrate the application of the new methods to two industrial data sets, showcasing their practical relevance and effectiveness.

Importance sampling method for Feldman-Cousins confidence intervals

In various high-energy physics contexts, such as neutrino-oscillation experiments, several assumptions underlying the typical asymptotic confidence interval construction are violated, such that one has to resort to computationally expensive methods like the Feldman-Cousins method for obtaining confidence intervals with proper statistical coverage. By construction, the computation of intervals at high confidence levels requires fitting millions or billions of pseudoexperiments, while wasting most of the computational cost on overly precise intervals at low confidence levels. In this work, a simple importance sampling method is introduced that reuses pseudoexperiments produced for all tested parameter values in a single mixture distribution. This results in a significant error reduction on the estimated critical values, especially at high confidence levels, and simultaneously yields a correct interpolation of these critical values between the parameter values at which the pseudoexperiments were produced. The theoretically calculated performance is demonstrated numerically using a simple example from the analysis of neutrino oscillations. The relationship to similar techniques applied in statistical mechanics and p-value computations is discussed. Published by the American Physical Society 2024

Reliability estimation in multicomponent stress-strength model using weighted exponential-Lindley distribution

This research investigates the reliability estimation in multicomponent stress-strength model when both the stress and strengths are drawn from Weighted Exponential-Lindley distribution. Reliability assessment is carried out using classical and Bayesian approaches. The Bayes estimates of the reliability in the multicomponent stress-strength model are derived under a squared error loss function using informative and non-informative priors for the parameters. Further, Lindley's approximation and Gibbs sampling method are used to develop Bayes estimators for the system's reliability due to the lack of explicit forms. Additionally, an asymptotic confidence interval and the highest probability density credible interval are constructed to gauge system performance. A simulation study is conducted to assess the performance of reliability estimators. Finally, a real data set is analysed for illustrative purposes.

Statistical inference of reliability for a K-out-of-N: G system with switching failure under Poisson shocks

Based on the stochastic uncertainty of the system's operating environment, this research presents statistical inferences on the mean time to failure (MTTF) of a K-out-of-N: G non-repairable system model with switching failure under Poisson shocks. The standby component is switched to the operating component when an operating component fails, with a switching failure probability of p. The MTTF of the system is derived by using the Markov process theory and the Laplace transform for two cases where the shock threshold is a constant value or a random variable. The maximum likelihood estimator (MLE) of the MTTF is obtained, and based on this estimator, asymptotic confidence interval estimation and hypothesis testing are performed. Based on the setting of the basic parameter values, the MTTF under two different cases of the shock threshold is compared. The effect of each parameter on the MTTF is analyzed in numerical simulation. The effectiveness of the above statistical inference methods is also verified by numerical simulation.

Analysis of Differentially Private Synthetic Data: A Measurement Error Approach

Differentially private (DP) synthetic datasets have been receiving significant attention from academia, industry, and government. However, little is known about how to perform statistical inference using DP synthetic datasets. Naive approaches that do not take into account the induced uncertainty due to the DP mechanism will result in biased estimators and invalid inferences. In this paper, we present a class of maximum likelihood estimator (MLE)-based easy-to-implement bias-corrected DP estimators with valid asymptotic confidence intervals (CI) for parameters in regression settings, by establishing the connection between additive DP mechanisms and measurement error models. Our simulation shows that our estimator has comparable performance to the widely used sufficient statistic perturbation (SSP) algorithm in some scenarios but with the advantage of releasing a synthetic dataset and obtaining statistically valid asymptotic CIs, which can achieve better coverage when compared to the naive CIs obtained by ignoring the DP mechanism.

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

Statistical inference for a two-parameter Rayleigh distribution under generalized progressive hybrid censoring scheme

The two-parameter Rayleigh distribution, as an extended distribution of the Rayleigh distribution, has been widely applied in reliability analysis. With the introduction of the location parameter, two-parameter Rayleigh distribution becomes more flexible in fitting real-data. In this paper, based on generalized progressively hybrid censored(GPHC) sample from the two-parameter Rayleigh distribution, Classical and Bayesian inferences are discussed. The Newton-Raphson(NR) and Expectation-Maximization(EM) algorithms are used to compute the maximum likelihood estimates(MLEs). As well as the asymptotic confidence interval(ACI) estimation is obtained through the asymptotic distribution theory of maximum likelihood estimation(MLE) and computation of the observed Fisher information matrix. In Bayesian frame, the estimation of unknown parameters and prediction of future observable are taken into consideration. Due to Bayesian estimation is challenging to compute precisely and for the purpose of comparison, the Lindley's approximation, the Tierney-Kadane(TK) approximation and Markov chain Monte Carlo(MCMC) method are employed to obtain Bayesian estimates. Then, combining the MCMC algorithm mentioned in the article, the one- and two- samples Bayesian prediction are obtained. Finally, the simulation results are provided and a real-life data set is used for illustration purpose.

Estimation of stress-strength reliability in s-out-of-k system for new flexible exponential distribution under progressive type-II censoring

This article addresses stress-strength reliability estimation in s-out-of-k systems under progressive type-II censoring. The system consists of components facing common stress, where stress and strength variable follows independent new flexible exponential (NFE) distributions, sharing a common parameter. The estimation procedure is carried out using MLE, Bayesian estimation and uniformly minimum variance unbiased estimation (UMVUE) techniques. MLE is derived for known and unknown common parameters. Bayesian estimates are obtained using Lindley’s approximation and Markov chain Monte Carlo techniques due to the complexity of obtaining the exact Bayesian estimates, when the common parameter is unknown. Moreover, UMVUE is derived when the common parameter is known. The 95% asymptotic confidence interval, highest posterior density credible intervals, and Bayesian credible intervals, are constructed. An extensive simulation study is carried out to compare the performance of various censoring schemes and estimation methods. Finally, two progressively censored real datasets are analyzed for illustration.

A new over-dispersed count model based on Poisson-Geometric convolution

A new two-parameter discrete distribution, namely the PoiG distribution, is derived by the convolution of a Poisson variate and an independently distributed geometric random variable. This distribution generalizes both the Poisson and geometric distributions and can be used for modeling over-dispersed as well as equi-dispersed count data. A number of important statistical properties of the proposed count model, such as the probability generating function, the moment generating function, the moments, the survival function and the hazard rate function have been studied in detail. Monotonic properties such as the log concavity are also studied. Stochastic ordering for the proposed model is also investigated in detail. Method of moments and the maximum-likelihood estimators of the parameters of the proposed model are found to work well in simulation experiments as well as in the real-life data analysis. Asymptotic confidence intervals and the coverage probabilities of the parameters are also studied through extensive simulation experiments. Two real life modelling applications demonstrate that the proposed distribution may prove to be useful for the practitioners for modelling over-dispersed count data compared to its closest competitors.

Estimation of Reliability under Conditional Stress - Strength Setup based on Weibull Distribution

The Weibull distribution has been extensively studied and applied across various fields due to its versatility in modeling a wide range of phenomena, especially in reliability engineering, survival analysis, and lifetime modeling. The concept of Rl a,b , which represents a system's reliability in a conditional stress-strength setup, was proposed by Sabre and Khorshidian (2021). In this research, the problem of estimating reliability of the component is considered when strength variable X and stress variable Y follow independent Weibull distributions with common shapes and different scale parameters under conditional stress-strength setup. The maximum likelihood estimator, asymptotic confidence interval, Bootstrap estimators, Boot-p estimators, and Bayes estimator under-squared error loss function with associated highest posterior density interval are constructed for conditional stress-strength reliability. Simulation study is conducted to estimate mean square error (MSE) of estimator of conditional stress-strength reliability. The real data analysis is also carried out.

On the study of the stress-strength reliability in Weibull-$F$ Models

In this paper, the problem of inferencing on the stress-strength reliability under Weibull-$F$ Models when the stress and strength systems belong to the different families of distributions from the Weibull-$F$ Model is considered. Some stochastic comparisons between the survival distribution functions of this model are obtained. Also, the asymptotic and several bootstrap confidence intervals of stress-strength reliability are studied. The efficiency of asymptotic and bootstrap confidence intervals are analyzed by simulation. The numerical example based on real-life data is displayed as an illustration.

Exponentiated half‐logistic Weibull distribution with reliability inference

AbstractThis paper presents an in‐depth analysis of the exponentiated half‐logistic Weibull (EHLW) distribution, investigating its fundamental statistical properties and exploring parameters estimation using both maximum likelihood and Bayesian approaches. Specifically, our focus centers on the analysis of stress–strength reliability, denoted as , where the strength variable follows EHLW distribution, and the stress variable follows either Weibull distribution or the EHLW distribution. The estimation of the parameter is discussed in both scenarios, employing both Maximum Likelihood and Bayesian approaches. Furthermore, we calculate that the asymptotic confidence interval, percentile bootstrap interval, and the highest probability density credible interval are obtained for the stress–strength parameter . In order to assess the efficiency of these estimation techniques, simulation studies are conducted, providing valuable insights into the performance of each estimation approach. Finally, the proposed reliability model is applied to real datasets, highlighting its practical significance.

Statistical inference of inverted Nadarajah–Haghighi distribution under type-II generalized hybrid censoring competing risks data

Tahir et al. (J Stat Comput Simul 88(14):2775–2798, 2018) introduced the inverse Nadarajah–Haghighi distribution (INHD) and demonstrated its ability to model positive real data sets with decreasing and upside-down bathtub hazard rate shapes. This article focuses on the inference of unknown parameters using a generalized Type-II hybrid censoring scheme (GT-II HCS) for the INHD in the presence of competing risks. The maximum likelihood (ML) and Bayes approaches are used to estimate the model parameters. Based on the squared error loss function, we compute Bayes estimates using Markov Chain Monte Carlo (MCMC) by applying Metropolis-Hasting (M-H) algorithm. Furthermore, the asymptotic confidence intervals, bootstrap confidence intervals (BCIs) and the highest posterior density (HPD) credible intervals are constructed. Using real data sets and simulation studies, we examined the introduced methods of inference with different sample sizes.

Randomly Censored Kumaraswamy Distribution

In this paper, inferential procedures based on classical and Bayesian framework for the Kumaraswamy distribution under random censoring model are studied. We first propose estimators for the distribution parameters, reliability function, failure rate function, and Mean time to system failure based on the maximum likelihood estimation method. Then, we calculate asymptotic confidence intervals for the parameters based on the observed Fisher’s information matrix. Also, for the parameters and reliability characteristics, Bayesian estimates are derived using the importance sampling and Gibbs sampling procedures. Highest posterior density credible intervals for the parameters are constructed using Markov Chain Monte Carlo method. Expected time on test of experiment with random censoring is also calculated. A simulation study is conducted to compare the efficiency of the derived estimates. Finally, the analysis of a real data set is presented for the illustration purpose.

Reliability analysis at usual operating settings for Weibull Constant-stress model with improved adaptive Type-Ⅱ progressively censored samples

<abstract><p>An improved adaptive Type-Ⅱ progressive censoring scheme was recently introduced to ensure that the examination duration will not surpass a specified threshold span. Employing this plan, this paper aimed to investigate statistical inference using Weibull constant-stress accelerated life tests. Two classical setups, namely maximum likelihood and maximum product of spacings, were explored to estimate the scale, shape, and reliability index under normal use conditions as well as their asymptotic confidence intervals. Through the same suggested classical setups, the Bayesian estimation methodology via the Markov chain Monte Carlo technique based on the squared error loss was considered to acquire the point and credible estimates. To compare the efficiency of the various offered approaches, a simulation study was carried out with varied sample sizes and censoring designs. The simulation findings show that the Bayesian approach via the likelihood function provides better estimates when compared with other methods. Finally, the utility of the proposed techniques was illustrated by analyzing two real data sets indicating the failure times of a white organic light-emitting diode and a pump motor.</p></abstract>

Analysis of a new jointly hybrid censored Rayleigh populations

<abstract><p>When a researcher wants to perform a life-test comparison study of items made by two separate lines inside the same institution, joint censoring strategies are particularly important. In this paper, we present a new joint Type-Ⅰ hybrid censoring that enables an experimenter to stop the investigation as soon as a pre-specified number of failures or time is first achieved. In the context of newly censored data, the estimates of the unknown mean lifetimes of two different Rayleigh populations are acquired using maximum likelihood and Bayesian inferential techniques. The normality characteristic of classical estimators is used to offer asymptotic confidence interval bounds for each unknown parameter. Against gamma conjugate priors, the Bayes estimators and related credible intervals are gathered about symmetric and asymmetric loss functions. Since classical and Bayes estimators are acquired in closed form, simulation tests can be easily made to evaluate the effectiveness of the proposed methodologies. The efficiency of the suggested approaches is examined in terms of four metrics, namely: Root mean squared error, average relative absolute bias, average confidence length, and coverage probability. To demonstrate the applicability of the offered approaches to real events, two real applications employing data sets from the engineering area are analyzed. As a result, when the experimenter's primary goal is to complete the test as soon as the total number of failures or the threshold period is recorded, the numerical results reveal that the recommended strategy is adaptable and very helpful in completing the study.</p></abstract>

Inference and other aspects for $ q- $Weibull distribution via generalized order statistics with applications to medical datasets

<abstract><p>This work utilizes generalized order statistics (GOSs) to study the $ q $-Weibull distribution from several statistical perspectives. First, we explain how to obtain the maximum likelihood estimates (MLEs) and utilize Bayesian techniques to estimate the parameters of the model. The Fisher information matrix (FIM) required for asymptotic confidence intervals (CIs) is generated by obtaining explicit expressions. A Monte Carlo simulation study is conducted to compare the performances of these estimates based on type Ⅱ censored samples. Two well-established measures of information are presented, namely extropy and weighted extropy. In this context, the order statistics (OSs) and sequential OSs (SOSs) for these two measures are studied based on this distribution. A bivariate $ q $-Weibull distribution based on the Farlie-Gumbel-Morgenstern (FGM) family and its relevant concomitants are studied. Finally, two concrete instances of medical real data are ultimately provided.</p></abstract>

Estimation of multicomponent stress–strength reliability for exponentiated Gumbel distribution

In this paper, the stress–strength reliability R s , k of a multicomponent s-out-of-k system for exponentiated Gumbel distribution is considered. An s-out-of-k system means a system with total k components and the system can survive only when atleast s of the total components function properly. The ability of the system to overcome the experiencing stress with its strength is termed as its stress–strengh reliability. The maximum likelihood estimator and Bayes estimator for R s , k are obtained. The Bayes estimators are obtained using Markov chain Monte Carlo(MCMC) method under both symmetric and asymmetric loss functions. The loss functions we considered are squared error loss function, LINEX loss function and entropy loss function. The asymptotic, bootstrap and highest posterior density(HPD) confidence intervals for R s , k are also obtained. A simulation study is conducted for evaluating the efficiency of the estimators derived in this paper. Real data sets are also considered for illustration.

Stochastic differential equations mixed model for individual growth with the inclusion of genetic characteristics

In early work we have studied a class of stochastic differential equation (SDE) models, for which the Gompertz and the Bertalanffy-Richards stochastic models are particular cases, to describe individual growth in random environments, and applied it to model cattle weight evolution using real data. We have started to work on these type of models considering that the model parameters are fixed, i.e. the same for all the animals. Aiming to incorporate variability among individuals, we consider that the model parameters can be random variables, resulting in SDE mixed models. In additon, here we consider SDE mixed models, allowing the parameters to be random and propose to incorporate each animal's genetic characteristics considering the transformed animal's weight at maturity to be a function of its genetic values. The main objective is to extend the SDE mixed model to the more realistic case where the individual genetic value becomes an important component in the estimated growth curve. For the estimation of the model parameters we have used maximum likelihood estimation theory. Estimates and asymptotic confidence intervals of the parameters are presented. A comparison with SDE non-mixed model and SDE mixed model without the inclusion of genetic characteristics is shown with the conclusion that the incorporation of some genetic characteristics in the model parameters improves the estimation of the animal's growth curve.