Confidence Intervals for the Scaled Half-Logistic Distribution under Progressive Type-II Censoring

Confidence interval construction for the scale parameter of the half-logistic distribution is considered using four different methods. The first two are based on the asymptotic distribution of the maximum likelihood estimator (MLE) and log-transformed MLE. The last two are based on pivotal quantity and generalized pivotal quantity, respectively. The MLE for the scale parameter is obtained using the expectation-maximization (EM) algorithm. Performances are compared with the confidence intervals proposed by Balakrishnan and Asgharzadeh via coverage probabilities, length, and coverage-to-length ratio. Simulation results support the efficacy of the proposed approach

On the Type-I Half-logistic Distribution and Related Contributions: A Review

The half-logistic (HL) distribution is a widely considered statistical model for studying lifetime phenomena arising in science, engineering, finance, and biomedical sciences. One of its weaknesses is that it has a decreasing probability density function and an increasing hazard rate function only. Due to that, researchers have been modifying the HL distribution to have more functional ability. This article provides an extensive overview of the HL distribution and its generalization (or extensions). The recent advancements regarding the HL distribution have led to numerous results in modern theory and statistical computing techniques across science and engineering. This work extended the body of literature in a summarized way to clarify some of the states of knowledge, potentials, and important roles played by the HL distribution and related models in probability theory and statistical studies in various areas and applications. In particular, at least sixty-seven flexible extensions of the HL distribution have been proposed in the past few years. We give a brief introduction to these distributions, emphasizing model parameters, properties derived, and the estimation method. Conclusively, there is no doubt that this summary could create a consensus between various related results in both theory and applications of the HL-related models to develop an interest in future studies

Inference for the jones and faddy's skewed t-distribution based on progressively type-II censored samples

In this paper, the location and the scale parameters of Jones and Faddy’s skewed t (JFST) distribution are estimated based on progressively Type-II censored samples. We obtain maximum likelihood (ML) and modified maximum likelihood (MML) estimators of unknown parameters. Then, confidence intervals for the estimators of μ and σ are obtained. The performances of proposed methodologies are compared via Monte-Carlo simulation study. It is concluded that the ML and MML estimators are close, especially for moderate and large sample sizes. At the end of the study, real life data is analyzed for illustrative proposes

Pivotal-based inference for a Pareto distribution under the adaptive progressive Type-II censoring scheme

This paper proposes an inference approach based on a pivotal quantity under the adaptive progressive Type-II censoring scheme. To exemplify the proposed methodology, an extensively employed distribution, a Pareto distribution, is utilized. This distribution has limitations in estimating confidence intervals for unknown parameters from classical methods such as the maximum likelihood and bootstrap methods. For example, in the maximum likelihood method, the asymptotic variance-covariance matrix does not always exist. In addition, both classical methods can yield confidence intervals that do not satisfy nominal levels when a sample size is not large enough. Our approach resolves these limitations by allowing us to construct exact intervals for unknown parameters with computational simplicity. Aside from this, the proposed approach leads to closed-form estimators with properties such as unbiasedness and consistency. To verify the validity of the proposed methodology, two approaches, a Monte Carlo simulation and a real-world data analysis, are conducted. The simulation testifies to the superior performance of the proposed methodology as compared to the maximum likelihood method, and the real-world data analysis examines the applicability and scalability of the proposed methodology

STATISTICAL INTERVALS FOR VARIOUS DISTRIBUTIONS BASED ON DIFFERENT INFERENCE METHODS

Statistical intervals (e.g., confidence, prediction, or tolerance) are widely used to quantify uncertainty, but complex settings can create challenges to obtain such intervals that possess the desired properties. My thesis will address diverse data settings and approaches that are shown empirically to have good performance. We first introduce a focused treatment on using a single-layer bootstrap calibration to improve the coverage probabilities of two-sided parametric tolerance intervals for non-normal distributions. We then turn to zero-inflated data, which are commonly found in, among other areas, pharmaceutical and quality control applications. However, the inference problem often becomes difficult in the presence of excess zeros. When data are semicontinuous, the log-normal and gamma distributions are often considered for modeling the positive part of the model. The problems of constructing a confidence interval for the mean and calculating an upper tolerance limit of a zero-inflated gamma population are considered using generalized fiducial inference. Furthermore, we use generalized fiducial inference on the problem of constructing confidence intervals for the population mean of zero-inflated Poisson distribution. Birnbaum–Saunders distribution is widely used as a failure time distribution in reliability applications to model failure times. Statistical intervals for Birnbaum–Saunders distribution are not well developed. Moreover, we utilize generalized fiducial inference to obtain the upper prediction limit and upper tolerance limit for Birnbaum–Saunders distribution. Simulation studies and real data examples are used to illustrate the effectiveness of the proposed methods

On estimating the reliability in a multicomponent system based on progressively-censored data from Chen distribution

This research deals with classical, Bayesian, and generalized estimation of stress-strength reliability parameter, Rs;k = Pr(at least s of (X1;X2; :::;Xk) exceed Y) = Pr(Xks+1:k \u3eY) of an s-out-of-k : G multicomponent system, based on progressively type-II right-censored samples with random removals when stress and strength are two independent Chen random variables. Under squared-error and LINEX loss functions, Bayes estimates are developed by using Lindley’s approximation and Markov Chain Monte Carlo method. Generalized estimates are developed using generalized variable method while classical estimates - the maximum likelihood estimators, their asymptotic distributions, asymptotic confidence intervals, bootstrap-based confidence intervals - are also developed. A simulation study and a real-world data analysis are provided to illustrate the proposed procedures. The size of the test, adjusted and unadjusted power of the test, coverage probability and expected lengths of the confidence intervals, and biases of the estimators are also computed, compared and contrasted

Improved approximate confidence intervals for censored data

This dissertation includes three papers. The first paper compares different procedures to compute confidence intervals for parameters and quantiles of the Weibull distribution for Type I censored data. The methods can be classified into three groups. The first group contains methods based on the commonly-used normal approximation for the distribution of (possibly transformed) studentized maximum likelihood estimators. The second group contains methods based on the likelihood ratio statistic and its modifications. The methods in the third group use a parametric bootstrap approach, including the use of bootstrap-type simulation to calibrate the procedures in the first two groups. We use the Monte Carlo simulation to investigate the finite sample properties of these procedures. Exceptional cases, which are due to problems caused by the Type I censoring, are noted;The second paper extends the results from Jensen (1993) and show that the distribution of signed squared root likelihood ratio statistics can be approximated by its bootstrap distribution up to second order accuracy when data are censored. Similar results apply to likelihood ratio Statistics and Probability; Our simulation study based on Type I censored data and the two parameter Weibull model shows that the bootstrap signed square root likelihood ratio statistics and its modification outperform the other methods like bootstrap-t and BCa in constructing one-sided confidence bounds;The third paper describes existing methods and develops new methods for constructing simultaneous confidence bands for a cumulative distribution function (cdf). Our results are built on extensions of previous work by Cheng and Iles (1983, 1988). A general approach is presented for construction of two-sided simultaneous confidence band for a continuous parametric model cdf from complete and censored data using standard large-sample approximations and then extending and comparing these to corresponding simulation or bootstrap calibrated versions of the same methods. Both two-sided and one-sided simultaneous confidence bands for location-scale parameter model are discussed in detail including situations with complete and censored data. A simulation for the Weibull distribution and Type I censored data is given. We illustrate the implementation of the methods with an application to estimate probability of detection (POD) used to assess nondestructive evaluation (NDE) capability