A new generalized confidence interval for the among-group variance in the heteroscedastic one-way random effects model

Based on the generalized inference idea, a new kind of generalized confidence intervals is derived for the among-group variance component in the heteroscedastic one-way random effects model. We construct structure equations of all variance components in the model based on their minimal sufficient statistics; meanwhile, the fiducial generalized pivotal quantity (FGPQ) can be obtained through solving an implicit equation of the parameter of interest. Then, the confidence interval is derived naturally from the FGPQ. Simulation results demonstrate that the new procedure performs very well in terms of both empirical coverage probability and average interval length.     

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.     

Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model

The studied topic is motivated by the problem of interlaboratory comparisons. This paper focuses on the confidence interval estimation of the between group variance in the unbalanced heteroscedastic one-way random effects model. Several interval estimators are proposed and compared by means of the simulation study. The most recommended (safest) is the confidence interval based on Bonferroni's inequality.     

A Monte Carlo simulation study of two-sided tolerance intervals in balanced one-way random effects model for non-normal errors

Recently, Ong and Mukerjee [Probability matching priors for two-sided tolerance intervals in balanced one-way and two-way nested random effects models. Statistics. 2011;45:403–411] developed two-sided Bayesian tolerance intervals, with approximate frequentist validity, for a future observation in balanced one-way and two-way nested random effects models. These were obtained using probability matching priors (PMP). On the other hand, Krishnamoorthy and Lian [Closed-form approximate tolerance intervals for some general linear models and comparison studies. J Stat Comput Simul. 2012;82:547–563] studied closed-form approximate tolerance intervals by the modified large-sample (MLS) approach. We compare the performances of these two approaches for normal as well as non-normal error distributions. Monte Carlo simulation methods are used to evaluate the resulting tolerance intervals with regard to achieved confidence levels and expected widths. It turns out that PMP tolerance intervals are less conservative for data with large number of classes and small number of observations per class and the MLS procedure is preferable for smaller sample sizes.     

Tests and Confidence Intervals for an Extended Variance Component Using the Modified Likelihood Ratio Statistic

Abstract.  The large deviation modified likelihood ratio statistic is studied for testing a variance component equal to a specified value. Formulas are presented in the general balanced case, whereas in the unbalanced case only the one-way random effects model is studied. Simulation studies are presented, showing that the normal approximation to the large deviation modified likelihood ratio statistic gives confidence intervals for variance components with coverage probabilities very close to the nominal confidence coefficient.     

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.     

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.     

Noninformative priors for the between-group variance in the unbalanced one-way random effects model with heterogeneous error variances

For the unbalanced one-way random effects model with heterogeneous error variances, we propose the non-informative priors for the between-group variance and develop the first- and second-order matching priors. It turns out that the second-order matching priors do not exist and the reference prior and Jeffreys prior do not satisfy a first-order matching criterion. We also show that the first-order matching prior meets the frequentist target coverage probabilities much better than the Jeffreys prior and reference prior through simulation study, and the Bayesian credible intervals based on the matching prior and reference prior give shorter intervals than the existing confidence intervals by examples.     

Confidence intervals for the between group variance in the unbalanced one-way random effects model of anaylsis of variance

A confidence interval for the between group variance is proposed which is deduced from Wald'sexact confidence interval for the rtio of the two variance components in the one-way random effects model and the exact confidence interval for the error variance resp.an unbiased estimator of the error variance. In a simulation study the confidence coeffecients for these two intervals are compared with the confidence coefficients of two other commonly used confidence intervals. There the confidence interval derived here yields confidence coefficiends which are always greater than the prescriped level.     

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.     

Confidence intervals for variance components in unbalanced one-way random effects model using non-normal distributions

In scenarios where the variance of a response variable can be attributed to two sources of variation, a confidence interval for a ratio of variance components gives information about the relative importance of the two sources. For example, if measurements taken from different laboratories are nine times more variable than the measurements taken from within the laboratories, then 90% of the variance in the responses is due to the variability amongst the laboratories and 10% of the variance in the responses is due to the variability within the laboratories. Assuming normally distributed sources of variation, confidence intervals for variance components are readily available. In this paper, however, simulation studies are conducted to evaluate the performance of confidence intervals under non-normal distribution assumptions. Confidence intervals based on the pivotal quantity method, fiducial inference, and the large-sample properties of the restricted maximum likelihood (REML) estimator are considered. Simulation results and an empirical example suggest that the REML-based confidence interval is favored over the other two procedures in unbalanced one-way random effects model.     

Harmonic mean approach to uunbalanced random effects models under heteroscedasticity

In this paper we consider unbalanced random effects models under heteroscedastic variances. By using' the harmonic mean approach, it is shown that the problems are analogous to those from balanced random effects models under horaoscedastic variances. Thus, by using the harmonic mean approach, statistical inferences about variance components are derived by using procedures from balanced models under homoscedastic variances. Laguerre polynomial expansion is used to approximate the sampling distributions of relevant statistics.     

GEEs for repeated categorical responses based on generalized residuals

Clustered or correlated samples of categorical response data arise frequently in many fields of application. The method of generalized estimating equations (GEEs) introduced in Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] is often used to analyse this type of data. GEEs give consistent estimates of the regression parameters and their variance based upon the Pearson residuals. Park et al. [Alternative GEE estimation procedures for discrete longitudinal data, Comput. Stat. Data Anal. 28 (1998), pp. 243–256] considered a modification of the GEE approach using the Anscombe residual and the deviance residual. In this work, we propose to extend this idea to a family of generalized residuals. A wide simulation study is conducted for binary and Poisson correlated outcomes and also two numerical illustrations are presented.     

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.     

The present status of confidence interval estimation for one-factor random models

This paper summarizes the present status of confidence interval estimation for one-factor random models. We provide both closed-form intervals and generalized confidence intervals for several parameters. Simple numerical examples are provided to illustrate computations. In order to limit the focus of this review, we consider only one-at-a-time (non-simultaneous) intervals with homoscedastic normal error terms, and do not include any Bayesian or bootstrap methods.     

Harmonic mean approach to unbalanced mixed effects models under heteroscedastic variances

In this paper we consider unbalanced mixed models (Scheffe's model) under heteroscedastic variances. By using the harmonic mean approach, It is shown that the problems appear to be anologous to those problems from balanced mixed models under homoscedastic variance. Thus, by using harmonic mean approach, statistical inferences about fixed effects and variance components are derived by using those from balanced models under homoscedastic variance. Laguerre polynomial expansion is used Lo approximate sampling distributions of relevant statistics.     

Pareto analysis based on records

Estimation of the parameters of an exponential distribution based on record data has been treated by Samaniego and Whitaker [On estimating population characteristics from record-breaking observations, I. Parametric results, Naval Res. Logist. Q. 33 (1986), pp. 531–543] and Doostparast [A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. Recently, Doostparast and Balakrishnan [Optimal record-based statistical procedures for the two-parameter exponential distribution, J. Statist. Comput. Simul. 81(12) (2011), pp. 2003–2019] obtained optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning location and scale parameters based on record data from a two-parameter exponential model. In this paper, we derive optimal statistical procedures including point and interval estimation as well as most powerful tests based on record data from a two-parameter Pareto model. For illustrative purpose, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.     

On improvement in estimating the population mean in simple random sampling

Kadilar and Cingi [Ratio estimators in simple random sampling, Appl. Math. Comput. 151 (3) (2004), pp. 893–902] introduced some ratio-type estimators of finite population mean under simple random sampling. Recently, Kadilar and Cingi [New ratio estimators using correlation coefficient, Interstat 4 (2006), pp. 1–11] have suggested another form of ratio-type estimators by modifying the estimator developed by Singh and Tailor [Use of known correlation coefficient in estimating the finite population mean, Stat. Transit. 6 (2003), pp. 655–560]. Kadilar and Cingi [Improvement in estimating the population mean in simple random sampling, Appl. Math. Lett. 19 (1) (2006), pp. 75–79] have suggested yet another class of ratio-type estimators by taking a weighted average of the two known classes of estimators referenced above. In this article, we propose an alternative form of ratio-type estimators which are better than the competing ratio, regression, and other ratio-type estimators considered here. The results are also supported by the analysis of three real data sets that were considered by Kadilar and Cingi.     

Comparing Equal-Tail Probability and Unbiased Confidence Intervals for the Intraclass Correlation Coefficient

The conventional confidence interval for the intraclass correlation coefficient assumes equal-tail probabilities. In general, the equal-tail probability interval is biased and other interval procedures should be considered. Unbiased confidence intervals for the intraclass correlation coefficient are readily available. The equal-tail probability and unbiased intervals have exact coverage as they are constructed using the pivotal quantity method. In this article, confidence intervals for the intraclass correlation coefficient are built using balanced and unbalanced one-way random effects models. The expected length of confidence intervals serves as a tool to compare the two procedures. The unbiased confidence interval outperforms the equal-tail probability interval if the intraclass correlation coefficient is small and the equal-tail probability interval outperforms the unbiased interval if the intraclass correlation coefficient is large.     

A note on testing the nullity of the between group variance in the one-way random effects model under variance heterogeneity

In an unbalanced and heteroscedastic one-way random effects model, we compare, by way of simulation, several test statistics for testing the null hypothesis that the variance of the random effects, also named the between group variance, is zero. These tests are the classical F-test, the test proposed by Jeyaratnam & Othman, the Welch test, and a modified version of Welch's test.