Inferences on the Among-Group Variance Component in Unbalanced Heteroscedastic One-Fold Nested Design

This article studies the hypothesis testing and interval estimation for the among-group variance component in unbalanced heteroscedastic one-fold nested design. Based on the concepts of generalized p-value and generalized confidence interval, tests and confidence intervals for the among-group variance component are developed. Furthermore, some simulation results are presented to compare the performance of the proposed approach with those of existing approaches. It is found that the proposed approach and one of the existing approaches can maintain the nominal confidence level across a wide array of scenarios, and therefore are recommended to use in practical problems. Finally, a real example is illustrated.     

Inferences on the intraclass correlation coefficients in the unbalanced two-way random effects model with interaction

In this paper, the hypothesis testing and interval estimation for the intraclass correlation coefficients are considered in a two-way random effects model with interaction. Two particular intraclass correlation coefficients are described in a reliability study. The tests and confidence intervals for the intraclass correlation coefficients are developed when the data are unbalanced. One approach is based on the generalized p-value and generalized confidence interval, the other is based on the modified large-sample idea. These two approaches simplify to the ones in Gilder et al. [2007. Confidence intervals on intraclass correlation coefficients in a balanced two-factor random design. J. Statist. Plann. Inference 137, 1199–1212] when the data are balanced. Furthermore, some statistical properties of the generalized confidence intervals are investigated. Finally, some simulation results to compare the performance of the modified large-sample approach with that of the generalized approach are reported. The simulation results indicate that the modified large-sample approach performs better than the generalized approach in the coverage probability and expected length of the confidence interval.     

Construction of Simultaneous Confidence Intervals for Ratios of Means of Lognormal Distributions

For constructing simultaneous confidence intervals for ratios of means for lognormal distributions, two approaches using a two-step method of variance estimates recovery are proposed. The first approach proposes fiducial generalized confidence intervals (FGCIs) in the first step followed by the method of variance estimates recovery (MOVER) in the second step (FGCIs–MOVER). The second approach uses MOVER in the first and second steps (MOVER–MOVER). Performance of proposed approaches is compared with simultaneous fiducial generalized confidence intervals (SFGCIs). Monte Carlo simulation is used to evaluate the performance of these approaches in terms of coverage probability, average interval width, and time consumption.     

Confidence intervals on intraclass correlation coefficients in a balanced two-factor random design

A modified large-sample (MLS) approach and a generalized confidence interval (GCI) approach are proposed for constructing confidence intervals for intraclass correlation coefficients. Two particular intraclass correlation coefficients are considered in a reliability study. Both subjects and raters are assumed to be random effects in a balanced two-factor design, which includes subject-by-rater interaction. Computer simulation is used to compare the coverage probabilities of the proposed MLS approach (GiTTCH) and GCI approaches with the Leiva and Graybill [1986. Confidence intervals for variance components in the balanced two-way model with interaction. Comm. Statist. Simulation Comput. 15, 301–322] method. The competing approaches are illustrated with data from a gauge repeatability and reproducibility study. The GiTTCH method maintains at least the stated confidence level for interrater reliability. For intrarater reliability, the coverage is accurate in several circumstances but can be liberal in some circumstances. The GCI approach provides reasonable coverage for lower confidence bounds on interrater reliability, but its corresponding upper bounds are too liberal. Regarding intrarater reliability, the GCI approach is not recommended because the lower bound coverage is liberal. Comparing the overall performance of the three methods across a wide array of scenarios, the proposed modified large-sample approach (GiTTCH) provides the most accurate coverage for both interrater and intrarater reliability.     

Assessing Occupational Exposure via the Unbalanced One-Way Random Effects Model

In this article, an unbalanced one-way random effects model is considered for the log-transformed shift-long exposure measurements. Exact test and confidence interval for the proportion of workers whose mean exposure exceeds the occupational exposure limit are developed based on the concepts of generalized p-value and generalized confidence interval. Some simulation results to compare the performance of the proposed test with that of the existing method are reported. The simulation results indicate that the proposed method appears to have significant gain in the size and power.     

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.     

ON CONFIDENCE INTERVALS FOR GENERALIZED ADDITIVE MODELS BASED ON PENALIZED REGRESSION SPLINES

Generalized additive models represented using low rank penalized regression splines, estimated by penalized likelihood maximisation and with smoothness selected by generalized cross validation or similar criteria, provide a computationally efficient general framework for practical smooth modelling. Various authors have proposed approximate Bayesian interval estimates for such models, based on extensions of the work of Wahba, G. (1983) [Bayesian confidence intervals for the cross validated smoothing spline. J. R. Statist. Soc. B 45 , 133–150] and Silverman, B.W. (1985) [Some aspects of the spline smoothing approach to nonparametric regression curve fitting. J. R. Statist. Soc. B 47 , 1–52] on smoothing spline models of Gaussian data, but testing of such intervals has been rather limited and there is little supporting theory for the approximations used in the generalized case. This paper aims to improve this situation by providing simulation tests and obtaining asymptotic results supporting the approximations employed for the generalized case. The simulation results suggest that while across‐the‐model performance is good, component‐wise coverage probabilities are not as reliable. Since this is likely to result from the neglect of smoothing parameter variability, a simple and efficient simulation method is proposed to account for smoothing parameter uncertainty: this is demonstrated to substantially improve the performance of component‐wise intervals.     

Confidence intervals for the mean and a percentile based on zero-inflated lognormal data

The problems of estimating the mean and an upper percentile of a lognormal population with nonnegative values are considered. For estimating the mean of a such population based on data that include zeros, a simple confidence interval (CI) that is obtained by modifying Tian's [Inferences on the mean of zero-inflated lognormal data: the generalized variable approach. Stat Med. 2005;24:3223—3232] generalized CI, is proposed. A fiducial upper confidence limit (UCL) and a closed-form approximate UCL for an upper percentile are developed. Our simulation studies indicate that the proposed methods are very satisfactory in terms of coverage probability and precision, and better than existing methods for maintaining balanced tail error rates. The proposed CI and the UCL are simple and easy to calculate. All the methods considered are illustrated using samples of data involving airborne chlorine concentrations and data on diagnostic test costs.     

Inferences on the Coefficients of Variation in a Multivariate Normal Population

In this article, the problem of testing the equality of coefficients of variation in a multivariate normal population is considered, and an asymptotic approach and a generalized p-value approach based on the concepts of generalized test variable are proposed. Monte Carlo simulation studies show that the proposed generalized p-value test has good empirical sizes, and it is better than the asymptotic approach. In addition, the problem of hypothesis testing and confidence interval for the common coefficient variation of a multivariate normal population are considered, and a generalized p-value and a generalized confidence interval are proposed. Using Monte Carlo simulation, we find that the coverage probabilities and expected lengths of this generalized confidence interval are satisfactory, and the empirical sizes of the generalized p-value are close to nominal level. We illustrate our approaches using a real data.     

A note on fiducial generalized pivots for in one-way heteroscedastic ANOVA with random effects

The paper deals with generalized confidence intervals for the between-group variance in one-way heteroscedastic (unbalanced) ANOVA with random effects. The approach used mimics the standard one applied in mixed linear models with two variance components, where interval estimators are based on a minimal sufficient statistic derived after an initial reduction by the principle of invariance. A minimal sufficient statistic under heteroscedasticity is found to resemble its homoscedastic counterpart and further analogies between heteroscedastic and homoscedastic cases lead us to two classes of fiducial generalized pivots for the between-group variance. The procedures suggested formerly by Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346] and Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390] are found to belong to these two classes. We comment briefly on some of their properties that were not mentioned in the original papers. In addition, properties of another particular generalized pivot are considered.     

Confidence limits for stress–strength reliability involving Weibull models

The problem of interval estimation of the stress–strength reliability involving two independent Weibull distributions is considered. An interval estimation procedure based on the generalized variable (GV) approach is given when the shape parameters are unknown and arbitrary. The coverage probabilities of the GV approach are evaluated by Monte Carlo simulation. Simulation studies show that the proposed generalized variable approach is very satisfactory even for small samples. For the case of equal shape parameter, it is shown that the generalized confidence limits are exact. Some available asymptotic methods for the case of equal shape parameter are described and their coverage probabilities are evaluated using Monte Carlo simulation. Simulation studies indicate that no asymptotic approach based on the likelihood method is satisfactory even for large samples. Applicability of the GV approach for censored samples is also discussed. The results are illustrated using an example.     

Efficient estimation in partially linear single‐index models for longitudinal data

In this paper, we consider the estimation of both the parameters and the nonparametric link function in partially linear single‐index models for longitudinal data that may be unbalanced. In particular, a new three‐stage approach is proposed to estimate the nonparametric link function using marginal kernel regression and the parametric components with generalized estimating equations. The resulting estimators properly account for the within‐subject correlation. We show that the parameter estimators are asymptotically semiparametrically efficient. We also show that the asymptotic variance of the link function estimator is minimized when the working error covariance matrices are correctly specified. The new estimators are more efficient than estimators in the existing literature. These asymptotic results are obtained without assuming normality. The finite‐sample performance of the proposed method is demonstrated by simulation studies. In addition, two real‐data examples are analyzed to illustrate the methodology.     

A new confidence interval in errors-in-variables model with known error variance

This paper considers constructing a new confidence interval for the slope parameter in the structural errors-in-variables model with known error variance associated with the regressors. Existing confidence intervals are so severely affected by Gleser–Hwang effect that they are subject to have poor empirical coverage probabilities and unsatisfactory lengths. Moreover, these problems get worse with decreasing reliability ratio which also result in more frequent absence of some existing intervals. To ease these issues, this paper presents a fiducial generalized confidence interval which maintains the correct asymptotic coverage. Simulation results show that this fiducial interval is slightly conservative while often having average length comparable or shorter than the other methods. Finally, we illustrate these confidence intervals with two real data examples, and in the second example some existing intervals do not exist.     

Generalized Confidence Interval for the Scale Parameter of the Power-Law Process

The power-law process is widely used in the analysis of repairable system reliability. In this article, interval estimation for the scale parameter is investigated under some general conditions. A procedure to derive a generalized confidence interval for the scale parameter is presented. We also study the accuracy of the generalized confidence interval by Monte Carlo simulation. Finally, two examples are shown to illustrate the proposed procedure.     

Interval estimation for the generalized inverted exponential distribution under progressive first failure censoring

In this paper, we consider the inferential procedures for the generalized inverted exponential distribution under progressive first failure censoring. The exact confidence interval for the scale parameter is derived. The generalized confidence intervals (GCIs) for the shape parameter and some commonly used reliability metrics such as the quantile and the reliability function are explored. Then the proposed procedure is extended to the prediction interval for the future measurement. The GCIs for the reliability of the stress-strength model are discussed under both equal scale and unequal scale scenarios. Extensive simulations are used to demonstrate the performance of the proposed GCIs and prediction interval. Finally, an example is used to illustrate the proposed methods.     

Inference for the common mean of several Birnbaum–Saunders populations

The Birnbaum–Saunders distribution is a widely used distribution in reliability applications to model failure times. For several samples from possible different Birnbaum–Saunders distributions, if their means can be considered as the same, it is of importance to make inference for the common mean. This paper presents procedures for interval estimation and hypothesis testing for the common mean of several Birnbaum–Saunders populations. The proposed approaches are hybrids between the generalized inference method and the large sample theory. Some simulation results are conducted to present the performance of the proposed approaches. The simulation results indicate that our proposed approaches perform well. Finally, the proposed approaches are applied to analyze a real example on the fatigue life of 6061-T6 aluminum coupons for illustration.     

Inference of stress-strength for the Type-II generalized logistic distribution under progressively Type-II censored samples

This article studies the estimation of the reliability R = P[Y < X] when X and Y come from two independent generalized logistic distributions of Type-II with different parameters, based on progressively Type-II censored samples. When the common scale parameter is unknown, the maximum likelihood estimator and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Bayes estimator of R and the corresponding credible interval using the Gibbs sampling technique have been proposed too. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are extracted. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real dataset is given for illustrative purposes. Finally, methods are extended for proportional hazard rate models.     

Generalized confidence intervals for the process capability indices in general random effect model with balanced data

In this paper, we consider the interval estimation problem on the process capability indices in general random effect model with balanced data. The confidence intervals for three commonly used process capability indices are developed by using the concept of generalized confidence interval. Furthermore, some simulation results on the coverage probability and expected value of the generalized lower confidence limits are reported. The simulation results indicate that the proposed confidence intervals do provide quite satisfactory coverage probabilities.     

Inferences on a linear combination of K multivariate normal mean vectors

In this paper, the hypothesis testing and confidence region construction for a linear combination of mean vectors for K independent multivariate normal populations are considered. A new generalized pivotal quantity and a new generalized test variable are derived based on the concepts of generalized p-values and generalized confidence regions. When only two populations are considered, our results are equivalent to those proposed by Gamage et al. [Generalized p-values and confidence regions for the multivariate Behrens–Fisher problem and MANOVA, J. Multivariate Aanal. 88 (2004), pp. 117–189] in the bivariate case, which is also known as the bivariate Behrens–Fisher problem. However, in some higher dimension cases, these two results are quite different. The generalized confidence region is illustrated with two numerical examples and the merits of the proposed method are numerically compared with those of the existing methods with respect to their expected areas, coverage probabilities under different scenarios.     

Generalized simultaneous confidence regions for regression coefficients and their ratios

This study constructs a simultaneous confidence region for two combinations of coefficients of linear models and their ratios based on the concept of generalized pivotal quantities. Many biological studies, such as those on genetics, assessment of drug effectiveness, and health economics, are interested in a comparison of several dose groups with a placebo group and the group ratios. The Bonferroni correction and the plug-in method based on the multivariate-t distribution have been proposed for the simultaneous region estimation. However, the two methods are asymptotic procedures, and their performance in finite sample sizes has not been thoroughly investigated. Based on the concept of generalized pivotal quantity, we propose a Bonferroni correction procedure and a generalized variable (GV) procedure to construct the simultaneous confidence regions. To address a genetic concern of the dominance ratio, we conduct a simulation study to empirically investigate the probability coverage and expected length of the methods for various combinations of sample sizes and values of the dominance ratio. The simulation results demonstrate that the simultaneous confidence region based on the GV procedure provides sufficient coverage probability and reasonable expected length. Thus, it can be recommended in practice. Numerical examples using published data sets illustrate the proposed methods.