Interval Estimation Procedures Research Articles

A Better Alternative to the Generalized Bilal Distribution: A New Model and Applications

In 2017, A. M. Abd-Elrahman [A new two-parameter lifetime distribution with decreasing, increasing or upside-down bathtub-shaped failure rate, Commun. Stat. - Theory Methods 46 (2017) 8865–8880, doi:10.1080/03610926.2016.1193198] introduced a generalization of the Bilal distribution, where a new two-parameter distribution, generalized Bilal distribution (GBD), was presented. He showed that its failure rate function can be upside-down bathtub-shaped. The failure rate can either be decreasing or increasing due to some mathematical and statistical reasons, which will be given below. In this paper, we introduce a simple and better alternative to the GBD, which will be denoted by WMD. We show that the WMD is a two-parameter distribution which can fit five different types of data sets with respect to their empirical hazard rate functions. Most properties of the WMD are investigated. Point and interval estimation procedures for the two unknown parameters are presented. The existence and uniqueness of the maximum likelihood estimates are proved. The moment estimates are obtained and we showed that one of these estimates is the minimum variance unbiased estimate (MVUE) for its corresponding parameter. A simulation study is provided and the paper is motivated by applications to four different real data sets. A detailed analysis for the Meeker and Escobar data is provided by the book of Meeker and Escobar [Statistical Methods for Reliability Data, 2nd edn. (John Wiley, 1998)]. The results may show that the new distribution provides a better fit than some other most recent existing and already known distributions in the literature. Finally, some concluding remarks are presented.

Dynamic response analysis for the eastern North Pacific gray whale population: an alternative approach

Gerrodette and DeMaster (1990) conclude that dynamic response analysis indicates that the gray whale population passed through its maximum net productivity level (MNPL, approximately equivalent to MSY level) between 1967 and 1980. Their conclusion is examined using models for population trends which permit a point of inflection; these are fitted globally to the time series of census estimates available up to 1987-88. A cubic and a logistic model are used. The cubic model results indicate with almost 100% confidence that the population passed through MNPL within two years of 1973-74. However, both this conclusion and that of Gerrodette and DeMaster are considered to be unreliable. This is because the curves fitted by both analyses correspond to markedly decreasing population sizes over parts of the periods to which they apply. This is inconsistent with plausible population dynamics behaviour, which is itself an underlying pre-requisite for dynamic response analysis methodology. A suggestion is made as to how applications of dynamic response analysis methodology such as that of Boveng et al. (1988) could be adapted to ensure the necessary respect of such constraints. Results of a parametric bootstrap procedure for confidence interval estimation applied to the logistic model indicate that the probability that the population passed through MNPL during the period of the censuses is not large. The census data are scarcely adequate to allow for reliable estimates of the curvature of the population trajectory to be made. The logistic model dynamic response analysis indicates that there is a somewhat greater likelihood that the gray whale population was below rather than above its MNPL in 1990, given the data available at the time.

Sensitivity and Uncertainty Analysis for Two-stream Capture-Recapture Methods in Disease Surveillance.

Capture-recapture methods are widely applied in estimating the number ( ) of prevalent or cumulatively incident cases in disease surveillance. Here, we focus the bulk of our attention on the common case in which there are 2 data streams. We propose a sensitivity and uncertainty analysis framework grounded in multinomial distribution-based maximum likelihood, hinging on a key dependence parameter that is typically nonidentifiable but is epidemiologically interpretable. Focusing on the epidemiologically meaningful parameter unlocks appealing data visualizations for sensitivity analysis and provides an intuitively accessible framework for uncertainty analysis designed to leverage the practicing epidemiologist's understanding of the implementation of the surveillance streams as the basis for assumptions driving estimation of . By illustrating the proposed sensitivity analysis using publicly available HIV surveillance data, we emphasize both the need to admit the lack of information in the observed data and the appeal of incorporating expert opinion about the key dependence parameter. The proposed uncertainty analysis is a simulation-based approach designed to more realistically acknowledge variability in the estimated associated with uncertainty in an expert's opinion about the nonidentifiable parameter, together with the statistical uncertainty. We demonstrate how such an approach can also facilitate an appealing general interval estimation procedure to accompany capture-recapture methods. Simulation studies illustrate the reliable performance of the proposed approach for quantifying uncertainties in estimating in various contexts. Finally, we demonstrate how the recommended paradigm has the potential to be directly extended for application to data from >2 surveillance streams.

Evaluating the Discrepancy Between Scale Reliability and Cronbach’s Coefficient Alpha Using Latent Variable Modeling

ABSTRACT This article outlines a readily applicable procedure for point and interval estimation of the population discrepancy between reliability and the popular Cronbach’s coefficient alpha for unidimensional multi-component measuring instruments with uncorrelated errors, which are widely used in behavioral and social research. The method is developed within the latent variable modeling framework and can be used to evaluate the degree to which coefficient alpha underestimates scale reliability in empirical measurement research employing such instruments. The approach is straight-forwardly utilized with readily available software and is illustrated using a numerical example.

Reliability inference and remaining useful life prediction for the doubly accelerated degradation model based on Wiener process

<abstract><p>Degradation data are an important source of products' reliability information. Though stochastic degradation models have been widely used for fitting degradation data, there is a lack of efficient and accurate methods to get their confidence intervals, especially in small sample cases. In this paper, based on the Wiener process, a doubly accelerated degradation test model is proposed, in which both the drift and diffusion parameters are affected by the stress level. The point estimates of model parameters are derived by constructing a regression model. Furthermore, based on the point estimates of model parameters, the interval estimation procedures are developed for the proposed model by constructing generalized pivotal quantities. First, the generalized confidence intervals of model parameters are developed. Second, based on the generalized pivotal quantities of model parameters, using the substitution method the generalized confidence intervals for some interesting quantities, such as the degradation rate $ \mu_0 $, the diffusion parameter $ \sigma_0^2 $, the reliability function $ R(t_0) $ and the mean lifetime $ E(T) $, are obtained. In addition, the generalized prediction intervals for degradation amount $ X_0(t) $ and remaining useful life at the normal use stress level are also developed. Extensive simulations are conducted to investigate the performances of the proposed generalized confidence intervals in terms of coverage percentage and average interval length. Finally, a real data set is given to illustrate the proposed model.</p></abstract>

Estimation for Akshaya Failure Model with Competing Risks under Progressive Censoring Scheme with Analyzing of Thymic Lymphoma of Mice Application

In several experiments of survival analysis, the cause of death or failure of any subject may be characterized by more than one cause. Since the cause of failure may be dependent or independent, in this work, we discuss the competing risk lifetime model under progressive type-II censored where the removal follows a binomial distribution. We consider the Akshaya lifetime failure model under independent causes and the number of subjects removed at every failure time when the removal follows the binomial distribution with known parameters. The classical and Bayesian approaches are used to account for the point and interval estimation procedures for parameters and parametric functions. The Bayes estimate is obtained by using the Markov Chain Monte Carlo (MCMC) method under symmetric and asymmetric loss functions. We apply the Metropolis–Hasting algorithm to generate MCMC samples from the posterior density function. A simulated data set is applied to diagnose the performance of the two techniques applied here. The data represented the survival times of mice kept in a conventional germ-free environment, all of which were exposed to a fixed dose of radiation at the age of 5 to 6 weeks, which was used as a practice for the model discussed. There are 3 causes of death. In group 1, we considered thymic lymphoma to be the first cause and other causes to be the second. On the base of mice data, the survival mean time (cumulative incidence function) of mice of the second cause is higher than the first cause.

Evaluation of Polytomous Item Locations in Multicomponent Measuring Instruments: A Note on a Latent Variable Modeling Procedure.

This note is concerned with evaluation of location parameters for polytomous items in multiple-component measuring instruments. A point and interval estimation procedure for these parameters is outlined that is developed within the framework of latent variable modeling. The method permits educational, behavioral, biomedical, and marketing researchers to quantify important aspects of the functioning of items with ordered multiple response options, which follow the popular graded response model. The procedure is routinely and readily applicable in empirical studies using widely circulated software and is illustrated with empirical data.

Inverse Gaussian process based reliability analysis for constant-stress accelerated degradation data

The constant stress accelerated degradation test has been widely used to evaluate product reliability when degradation measurements of product can be observed. This paper considers interval estimation procedures of the constant stress accelerated degradation model based on the inverse Gaussian process. The exact confidence interval is established for the shape parameter of the inverse Gaussian accelerated degradation model. Using the generalized pivotal quantity procedure, the generalized confidence intervals of other model parameters and some quantities of interest such as the pth percentile and reliability function of lifetime at the normal using stress level are derived. In addition, the generalized prediction interval is developed for the future degradation levels at the normal stress level used. The bootstrap confidence intervals of the proposed inverse Gaussian degradation model are also discussed, and the two interval estimation methods are compared. The performances of the proposed interval estimation procedures are assessed by Monte Carlo simulation in terms of coverage percentage and average interval length. Finally, an example is provided to illustrate the proposed procedures.

Risk prediction with imperfect survival outcome information from electronic health records.

Readily available proxies for the time of disease onset such as the time of the first diagnostic code can lead to substantial risk prediction error if performing analyses based on poor proxies. Due to the lack of detailed documentation and labor intensiveness of manual annotation, it is often only feasible to ascertain for a small subset the current status of the disease by a follow-up time rather than the exact time. In this paper, we aim to develop risk prediction models for the onset time efficiently leveraging both a small number of labels on the current status and a large number of unlabeled observations on imperfect proxies. Under a semiparametric transformation model for onset and a highly flexible measurement error model for proxy onset time, we propose the semisupervised risk prediction method by combining information from proxies and limited labels efficiently. From an initially estimator solely based on the labeled subset, we perform a one-step correction with the full data augmenting against a mean zero rank correlation score derived from the proxies. We establish the consistency and asymptotic normality of the proposed semisupervised estimator and provide a resampling procedure for interval estimation. Simulation studies demonstrate that the proposed estimator performs well in a finite sample. We illustrate the proposed estimator by developing a genetic risk prediction model for obesity using data from Mass General Brigham Healthcare Biobank.

Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints

Modern systems are operating under dynamic environments and the components therein often exhibit positively correlated lifetimes. Moreover, due to various practical reasons such as the load sharing mechanism, it is not uncommon that lifetimes of some components dominate the others within the same system. In this study, we first propose a statistical model for system reliability evaluation by jointly considering the correlated component lifetimes and the lifetime ordering constraints. In specific, the effects of the dynamic environments are incorporated by modelling the cumulative hazard function as an exponential dispersion process and the lifetime ordering constraints are modelled by truncating the support of the joint lifetime distribution. We then discuss the statistical inference based on the proposed model. The point estimates of the model parameters as well as the lifetime quantiles are obtained by the maximum likelihood method, and the confidence intervals are constructed by using the generalized pivots. Extensive simulation show that the proposed interval estimation procedures can achieve accurate coverage even under small sample sizes. Two real examples are used to illustrate the proposed modelling and estimation framework.

Symposium Title: Cognitive Resources in Affective States and After Brain Damage

The conditions for economic motivation for cost-effective use of resources are a prime requirement if an organization needs to implement energy-saving technologies, regardless of its purpose and type of activity. Energy-saving methods must be applied both at the technical level, and the system (management) level. The basis of energy saving is the systematic implementation of a set of technical and technological measures, which should be preceded by the optimization of organizations’ energy consumption at the system level. The chapter uses a rank analysis to show the efficiency of approximating the electricity consumption at the units of mining enterprises. The interval estimation procedure allowed for the identification and grading the mining enterprise units, revealing those with the abnormal electricity consumption. The trend of time series of relative electricity consumption for each facility is calculated. The chapter uses the results of calculations containing the health index figures for electricity consumption to establish facilities that require a priority in-depth energy inspection. This chapter focuses on the development of a methodology for timely identification of priority units for an in-depth energy survey for the implementation of energy saving projects and increasing the efficiency of electricity use in mining enterprises as objects of technocenosis.

Interval estimation for concentration in the ELISA setting

Enzyme-linked immunosorbent assays (ELISAs) are often used to quantify the concentration of biological substances. In a typical analysis only a point estimate of the concentration will be presented as interval estimation continues to present challenges for non-linear dose-response models. In this setting, interval estimates calculated using a Wald approach can suffer from poor coverage and have limits that fall outside parameter boundaries. Here we compare profile likelihood interval estimation procedures to Wald type intervals for the interval estimation of a concentration in the ELISA setting. Through a comprehensive simulation study, it is shown that profile likelihood methods result in interval estimates with superior coverage and that are more robust to differences in assay design when compared to Wald based approaches.

Interval estimation procedures for true scores of a test composed of polytomous items: An application of the multinomial error model.

When a person takes alternative forms of the same test across replications of the testing procedure, the test taker's observed scores on the alternative forms are rarely identical. In educational and psychological measurement, inconsistencies in a test taker's scores that are irrelevant to the construct being measured are attributed to errors of measurement. Typically, errors of measurement are summarized as the standard deviation of a test taker's observed scores over replication of the same testing procedure. Assuming that errors of measurement follow a multinomial distribution (i.e., multinomial error model), the main goal of this study was to propose two interval estimation procedures, which are referred to as the score-like and Perks procedures, for true scores of a test with polytomous items. The performance of the score-like and Perks procedures was compared with that of two normal approximation procedures under the multinomial error model and a procedure based on item response theory (IRT) through simulation. In general, the score-like and Perks procedures outperformed the other three procedures when data were generated under the multinomial error theory framework and showed reasonable results when data were generated under the IRT framework. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Interval Estimation of Item Response Probabilities Along Studied Latent Dimensions

ABSTRACT An interval estimation procedure is discussed that can be used to evaluate the probability of a particular response for a binary or binary scored item at a pre-specified point along an underlying latent continuum. The item is assumed to: (a) be part of a unidimensional multi-component measuring instrument that may contain also polytomous items, and (b) to follow a one-parameter or two-parameter logistic model. The approach is developed within the framework of latent variable modeling, and can be used in answering theoretically and empirically relevant questions concerning specific response probabilities at certain distances from the latent population mean. The method is readily and widely applicable with popular statistical software, and is illustrated using an empirical example.

Accurate confidence intervals for inter-laboratory calibration and common mean estimation

For a calibration problem that arises in analytical chemistry, the Rocke-Lorenzato model and some generalizations are considered, so as to capture the near constant variability at very low levels of the analyte, and the increasing measurement variation with increased concentration of the analyte. For the common mean problem, a heteroscedastic one-way random effects model is often used to model the data and will be considered herein. Existing interval estimation procedures for these problems are likelihood based large sample procedures, and are not satisfactory, especially for small samples. A modified likelihood methodology, which can be numerically implemented, is shown to perform very well for the above problems. The analysis of inter-laboratory data is reported in order to illustrate the application of the proposed methods.

Statistical inference for component lifetime distribution from coherent system lifetimes under a proportional reversed hazard model

Proportional reversed hazard model and exponentiated distributions have received considerable attention in the statistical literature due to its flexibility. In this paper, we develop the tools for statistical inference of the lifetime distribution of components in a n-component coherent system while the system lifetimes are observed, the system structure is known and the component lifetime follows the proportional reversed hazard model. Different point and interval estimation procedures based on frequentist and Bayesian approaches are developed. The existence and uniqueness of the maximum likelihood estimator are discussed. In addition, two statistical testing procedures, a pivotal quantity approach and a likelihood ratio test, to test whether the exponentiated parameter equals to a particular value are proposed. A numerical example is used to illustrate the methodologies developed in this paper and a Monte Carlo simulation study is employed to evaluate the performance of the statistical inferential procedures.

Statistical inference for decision curve analysis, with applications to cataract diagnosis.

Statistical learning methods are widely used in medical literature for the purpose of diagnosis or prediction. Conventional accuracy assessment via sensitivity, specificity, and ROC curves does not fully account for clinical utility of a specific model. Decision curve analysis (DCA) becomes a novel complement as it incorporates a clinical judgment of the relative value of benefits (treating a true positive case) and harms (treating a false positive case) associated with prediction models. The preference of a patient or a policy-maker is formulated statistically as the underlying threshold probability, above which the patient would choose to be treated. Net benefit is then calculated for possible threshold probability, which places benefits and harms on the same scale. We consider the inference problems for DCA in this paper. Interval estimation procedure and inference methodology are provided after we derive the relevant asymptotic properties. Our formulation can accommodate the classification problems with multiple categories. We carry out numerical studies to assess the performance of the proposed methods. An eye disease dataset is analyzed to illustrate our proposals.

Improving Hall’s Accelerated Sequential Procedure: Generalized Multistage Fixed-Width Confidence Intervals for a Normal Mean

We have developed an innovative generalized multistage fixed-width confidence interval (FWCI) estimation procedure for an unknown normal mean when the variance also remains unknown, which can be viewed as a mixture of sequential sampling and batch sampling procedures. Requiring the same expected sample size as Hall’s accelerated sequential sampling procedure, it enables the number of sampling operations to be further reduced by a predetermined factor. Extensive sets of simulations are provided to highlight the remarkable performance of the new multistage FWCI estimation procedure, and real data illustrations are included to demonstrate the practical applicability.

What Makes Action and Outcome Temporally Close to Each Other: A Systematic Review and Meta-Analysis of Temporal Binding

Temporal binding refers to the subjective compression of the temporal interval between a voluntary action and its external sensory consequences. While empirical evidence and theoretical accounts have indicated the potential linkage between temporal binding and action outcome prediction mechanisms, several questions regarding the underlying processes and the fundamental nature of temporal binding remain unanswered. Based on the sophisticated classification of predictive processes proposed by Hughes et al. (2013), we conducted a systematic, quantitative review of the binding effect as measured with two representative procedures, i.e., Libet clock procedure and interval estimation procedure. Although both procedures were designed to measure the same phenomenon, we revealed a larger effect size and higher sensitivity to perceptual moderators in binding observed with the clock procedure than with the interval estimation. Moreover, in the former, we observed different characteristics for the two perceptual shifts that comprise temporal binding. Action shifts depended more on whether one can control outcome onsets with voluntary actions. In contrast, outcome shifts depended more on the degree to which participants could predict, rather than control, the action outcome onset. These results indicate that action shift occurs based on the activation of learned action–outcome associations by planning and executing actions, while outcome shift occurs based on comparing predicted and observed outcomes. By understanding the nature of each experimental procedure and each shift, future research can use optimal methods depending on the goal. We discuss, as an example, the implications for the underlying disorders of agency in schizophrenia.

Notes on the two-stage procedure under an incomplete block two-period crossover design

Under an incomplete block crossover design with two periods, we derive the least-squares estimators for the period effect, treatment effects and carry-over effects in explicit formulae based on within-patient differences. Using the commonly-used strategy of searching a base model for making inferences in regression analysis, we define a two-stage test procedure in studying treatment effects. On the basis of Monte Carlo simulation, we evaluate the performance of the two-stage procedure for hypothesis testing, point and interval estimation of treatment effects in a variety of situations. We note that use of the two-stage procedure can be potentially misleading and hence one should not apply a test procedure to exclusively determine whether he/she needs to account for the carry-over effect in studying treatment effects. We use the double-blind crossover trial comparing two different doses of formoterol with placebo on the forced expiratory volume in 1 second (FEV1) readings to illustrate the use of the two-stage procedure, as well as the distinction between use of two-stage procedure and the approach with assuming no carry-over effects based on one's subjective knowledge.