Approximate confidence intervals for the difference in proportions for partially observed binary data

We consider partially observed binary matched-pair data. We assume that the incomplete subjects are missing at random. Within this missing framework, we propose an EM-algorithm based approach to construct an interval estimator of the proportion difference incorporating all the subjects. In conjunction with our proposed method, we also present two improvements to the interval estimator through some correction factors. The performances of the three competing methods are then evaluated through extensive simulation. Recommendation for the method is given based on the ability to preserve type-I error for various sample sizes. Finally, the methods are illustrated in two real-world data sets. An R-function is developed to implement the three proposed methods.

Estimating the natural indirect effect and the mediation proportion via the product method

Background The natural indirect effect (NIE) and mediation proportion (MP) are two measures of primary interest in mediation analysis. The standard approach for mediation analysis is through the product method, which involves a model for the outcome conditional on the mediator and exposure and another model describing the exposure–mediator relationship. The purpose of this article is to comprehensively develop and investigate the finite-sample performance of NIE and MP estimators via the product method. Methods With four common data types with a continuous/binary outcome and a continuous/binary mediator, we propose closed-form interval estimators for NIE and MP via the theory of multivariate delta method, and evaluate its empirical performance relative to the bootstrap approach. In addition, we have observed that the rare outcome assumption is frequently invoked to approximate the NIE and MP with a binary outcome, although this approximation may lead to non-negligible bias when the outcome is common. We therefore introduce the exact expressions for NIE and MP with a binary outcome without the rare outcome assumption and compare its performance with the approximate estimators. Results Simulation studies suggest that the proposed interval estimator provides satisfactory coverage when the sample size ≥500 for the scenarios with a continuous outcome and sample size ≥20,000 and number of cases ≥500 for the scenarios with a binary outcome. In the binary outcome scenarios, the approximate estimators based on the rare outcome assumption worked well when outcome prevalence less than 5% but could lead to substantial bias when the outcome is common; in contrast, the exact estimators always perform well under all outcome prevalences considered. Conclusions Under samples sizes commonly encountered in epidemiology and public health research, the proposed interval estimator is valid for constructing confidence interval. For a binary outcome, the exact estimator without the rare outcome assumption is more robust and stable to estimate NIE and MP. An R package is developed to implement the methods for point and variance estimation discussed in this paper.

Imprecise credibility theory

The classical credibility theory is a cornerstone of experience rating, especially in the field of property and casualty insurance. An obstacle to putting the credibility theory into practice is the conversion of available prior information into a precise choice of crucial hyperparameters. In most real-world applications, the information necessary to justify a precise choice is lacking, so we propose an imprecise credibility estimator that honestly acknowledges the imprecision in the hyperparameter specification. This results in an interval estimator that is doubly robust in the sense that it retains the credibility estimator’s freedom from model specification and fast asymptotic concentration, while simultaneously being insensitive to prior hyperparameter specification.

Inference on reliability of stress-strength model with Peng-Yan extended Weibull distributions

In this paper we estimate R = PfX < Yg when X and Y are independent random variables following the Peng-Yan extended Weibull distribution. We find maximum likelihood estimator of R and its asymptotic distribution. This asymptotic distribution is used to construct asymptotic confidence intervals. In the case of equal shape parameters, we derive the exact confidence intervals, too. A procedure for deriving bootstrap-p confidence intervals is presented. The UMVUE of R and the UMVUE of its variance are derived and also the Bayes point and interval estimator of R for conjugate priors are obtained. Finally, we perform a simulation study in order to compare these estimators and provide a real data example.

Performance of a Robust Confidence Interval for the Process Capability Index Cp based on the Modified Trimmed Standard Deviation Under Non-Normal Distributions

In this paper, a robust interval estimator for the classical process capability index (Cp) based on the modified trimmed standard deviation (MTSD = ST ∗ ) is considered under both normal and non-normal distributions. The performance of the newly proposed process capability index interval estimator over the existing method is compared using a simulation study. As a performance criterion, we consider both simulated coverage probability and average width. Simulation results evident that the proposed confidence interval based on the robust estimator performed well for most of cases. For illustration purposes, two real-life data from industry are analyzed which supported our simulation results to some extent. As a result, the proposed method can be recommend to be used by the practitioners in various fields of industry, engineering and physical sciences.

Clustering of Extremes in Financial Returns: A Study of Developed and Emerging Markets

This paper investigates the clustering or dependency of extremes in financial returns by estimating the extremal index value, in which smaller values of the extremal index correspond to more clustering. We apply the interval estimator method to determine the extremal index for a range of threshold values in the developed and emerging markets from 2007–2017. The indices we used to represent developed markets are from France, Germany, Italy, Japan, USA, UK, Spain, and Sweden. For the emerging markets, we use indices from China, Brazil, India, Malaysia, Russia, Saudi Arabia, and Portugal. The results show that clustering occurs in the emerging and developed markets under several threshold values. This study will shed light on the dependency structure of financial returns data and the proprieties of the extremes returns. Moreover, understanding clustering of extremes in these markets can help investors reduce the exposure to extreme financial events, such as the financial crisis.

A Classroom Approach to Illustrate Transformation and Bootstrap Confidence Interval Techniques Using the Poisson Distribution

The Poisson distribution is here used to illustrate transformation and bootstrap techniques in order to construct a confidence interval for a mean. A comparison is made between the derived intervals and the Wald and score confidence intervals. The discussion takes place in a classroom, where the teacher and the students have previously discussed and evaluated the Wald and score confidence intervals. While step by step interactively getting acquainted with new techniques, the students will learn about the effects of e.g. bias and asymmetry and ways of dealing with such phenomena. The primary purpose of this teacher-student communication is therefore not to find the best possible interval estimator for this particular case, but rather to provide a study displaying a teacher and her/his students interacting with each other in an efficient and rewarding way. The teacher has a strategy of encouraging the students to take initiatives. This is accomplished by providing the necessary background of the problem and some underlying theory after which the students are confronted with questions and problem solving. From this the learning process starts. The teacher has to be flexible according to how the students react. The students are supposed to have studied mathematical statistics for at least two semesters.