Inferences on the reliability in balanced and unbalanced one-way random models

In this article, the hypothesis testing and interval estimation for the reliability parameter are considered in balanced and unbalanced one-way random models. The tests and confidence intervals for the reliability parameter are developed using the concepts of generalized p-value and generalized confidence interval. Furthermore, some simulation results are presented to compare the performances between the proposed approach and the existing approach. For balanced models, the simulation results indicate that the proposed approach can provide satisfactory coverage probabilities and performs better than the existing approaches across the wide array of scenarios, especially for small sample sizes. For unbalanced models, the simulation results show that the two proposed approaches perform more satisfactorily than the existing approach in most cases. Finally, the proposed approaches are illustrated using two real examples.     

Generalized confidence interval for the slope in linear measurement error model

A generalized confidence interval for the slope parameter in linear measurement error model is proposed in this article, which is based on the relation between the slope of classical regression model and the measurement error model. The performance of the confidence interval estimation procedure is studied numerically through Monte Carlo simulation in terms of coverage probability and expected length.     

Empirical likelihood for generalized linear models with fixed and adaptive designs

Empirical likelihood inference for generalized linear models with fixed and adaptive designs is considered. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter and the resulting estimator is shown to be asymptotically normal. Some simulations are conducted to illustrate the proposed method.     

Bootstrapping probability-proportional-to-size samples via calibrated empirical population

A collection of six novel bootstrap algorithms, applied to probability-proportional-to-size samples, is explored for variance estimation, confidence interval and p-value production. Developed according to bootstrap fundamentals such as the mimicking principle and the plug-in rule, these algorithms make use of an empirical bootstrap population informed by sampled units each with assigned weight. Starting from the natural choice of Horvitz–Thompson (HT)-type weights, improvements based on calibration to known population features are fostered. Focusing on the population total as the parameter to be estimated and on the distribution of the HT estimator as the target of bootstrap estimation, simulation results are presented with the twofold objective of checking practical implementation and of investigating the statistical properties of the bootstrap estimates supplied by the algorithms explored.     

A case study of green tree frog population size estimation by repeated capture–mark–recapture method with individual tagging: a parametric bootstrap method vs Jolly–Seber method

This paper deals with estimation of a green tree frog population in an urban setting using repeated capture–mark–recapture (CMR) method over several weeks with an individual tagging system which gives rise to a complicated generalization of the hypergeometric distribution. Based on the maximum likelihood estimation, a parametric bootstrap approach is adopted to obtain interval estimates of the weekly population size which is the main objective of our work. The method is computation-based; and programming intensive to implement the algorithm for re-sampling. This method can be applied to estimate the population size of any species based on repeated CMR method at multiple time points. Further, it has been pointed out that the well-known Jolly–Seber method, which is based on some strong assumptions, produces either unrealistic estimates, or may have situations where its assumptions are not valid for our observed data set.     

Ratio estimates and fieller's theorem in regression modelling

This paper presents applications of statistical linear models in which a confidence interval is required for the ratio of linear combinations of the model's parameters, Fieller's theorem is used to obtain the solution.     

Construction of fixed width confidence intervals for a Bernoulli success probability using sequential sampling: a simulation study

This article considers the construction of level 1?α fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229–1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than ‘exact’ for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119–126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209–212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101–133] have close to 1?α confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.     

Empirical likelihood for generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.     

A new confidence interval in measurement error model with the reliability ratio known

For the slope parameter of the measurement error model with the reliability ratio known, this article constructs a fiducial generalized confidence interval (FGCI) which is proved to have correct asymptotic coverage. Simulation results demonstrate that the FGCI often outperforms the existing intervals in terms of empirical coverage probability, average interval length, and false parameter coverage rate. Two examples are also provided to illustrate our approach.     

An improved score interval with a modified midpoint for a binomial proportion

One of the most basic and important problems in statistical inference is the construction of the confidence interval (CI). In this paper, we propose a novel CI for a binomial proportion by modifying the midpoint of the score interval. The proposed modified interval can solve the ‘downward spikes’ problem of the score interval without enlarging the interval length. Simulation studies are carried out to illustrate the performance of the modified interval. With regard to the criterions of coverage probability, mean absolute error and expected length, our method is competitive among the several commonly used methods for constructing a CI. A real data example is also presented to show the application of our method.