Short confidence intervals for variance components

Four approximate methods are proposed to construct confidence intervals for the estimation of variance components in unbalanced mixed models. The first three methods are modifications of the Wald, arithmetic and harmonic mean procedures, see Harville and Fenech (1985), while the fourth is an adaptive approach, combining the arithmetic and harmonic mean procedures. The performances of the proposed methods were assessed by a Monte Carlo simulation study. It was found that the intervals based on Wald's method maintained the nominal confidence levels across all designs and values of the parameters under study. On the other hand, the arithmetic (harmonic) mean method performed well for small (large) values of the variance component, relative to the error variance component. The adaptive procedure performed rather well except for extremely unbalanced designs. Further, compared with equal tails intervals, the intervals which use special tables, e.g., Table 678 of Tate and Klett (1959), provided adequate coverage while having much shorter lengths and are thus recommended for use in practice.     

Reliable confidence intervals for assessing normal process incapability

This study aims to provide a reliable confidence interval for assessing the process incapability index [C_pp]. The concept of the generalized pivotal quantities is utilized for constructing the generalized confidence interval for [C_pp]. And, simulations are performed for demonstrating our proposed method and one existent method. The results show that the empirical confidences of these two methods are significantly affected by the degree of process departure. Therefore, we suggest the practitioners to select proper one for capability testing purpose based on the information of degree of process departure.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Repeated confidence intervals and prediction intervals using stochastic curtailment

Jennlson and Turnbull (1984,1989) proposed procedures for repeated confidence intervals for parameters of interest In a clinical trial monitored with group sequential methods. These methods are extended for use with stochastic curtailment procedures for two samples in the estimation of differences of means, differences of proportions, odds ratios, and hazard ratios. Methods are described for constructing 1) confidence intervals for these estimates at repeated times In the course of a trial, and 2) prediction intervals for predicted estimates at the end of a trial. Specific examples from several clinical trials are presented.     

Convergence rates for uniform confidence intervals based on local polynomial regression estimators

We investigate the convergence rates of uniform bias-corrected confidence intervals for a smooth curve using local polynomial regression for both the interior and boundary region. We discuss the cases when the degree of the polynomial is odd and even. The uniform confidence intervals are based on the volume-of-tube formula modified for biased estimators. We empirically show that the proposed uniform confidence intervals attain, at least approximately, nominal coverage. Finally, we investigate the performance of the volume-of-tube based confidence intervals for independent non-Gaussian errors.     

Comparing confidence intervals for Goodman and Kruskal's gamma coefficient

This study was motivated by the question which type of confidence interval (CI) one should use to summarize sample variance of Goodman and Kruskal's coefficient gamma. In a Monte-Carlo study, we investigated the coverage and computation time of the Goodman–Kruskal CI, the Cliff-consistent CI, the profile likelihood CI, and the score CI for Goodman and Kruskal's gamma, under several conditions. The choice for Goodman and Kruskal's gamma was based on results of Woods [Consistent small-sample variances for six gamma-family measures of ordinal association. Multivar Behav Res. 2009;44:525–551], who found relatively poor coverage for gamma for very small samples compared to other ordinal association measures. The profile likelihood CI and the score CI had the best coverage, close to the nominal value, but those CIs could often not be computed for sparse tables. The coverage of the Goodman–Kruskal CI and the Cliff-consistent CI was often poor. Computation time was fast to reasonably fast for all types of CI.     

Shortest confidence and prediction intervals for the log-normal

We provide the shortest prediction interval for X, and the shortest confidence interval for the median of X, when X has the log-normal distribution for both the case σ², the variance of log X, known and unknown. Tables are given to assist the practitioner in constructing these intervals. A real-life example is provided to illustrate the results.     

Two new confidence intervals for the coefficient of variation in a normal distribution

In this article we introduce an approximately unbiased estimator for the population coefficient of variation, τ, in a normal distribution. The accuracy of this estimator is examined by several criteria. Using this estimator and its variance, two approximate confidence intervals for τ are introduced. The performance of the new confidence intervals is compared to those obtained by current methods.     

A quadratic programming approach to the construction of simultaneous confidence intervals

Richmond (1982) uses a linear programming approach to the construction of simultaneous confidence intervals for a set of linear estimable parametric functions of the normal mean vector. We present a quadratic programming approach which constructs narrower confidence intervals than the linear programming approach given by Richmond (1982).     

Comparing confidence intervals for the logarithmic series distribution

It is assumed that a small random sample of fixed size n is drawn from a logarithmic series distribution with parameter θ and that it is desired to estimate θ by means of a two-sided confidence interval. In this note Crow's system of confidence intervals is compared, in shortness of intervals, with Clopper and Pearson's, and the corresponding randomized counterparts.     

On the lengths of t-based confidence intervals

Confidence interval is a basic type of interval estimation in statistics. When dealing with samples from a normal population with the unknown mean and the variance, the traditional method to construct t-based confidence intervals for the mean parameter is to treat the n sampled units as n groups and build the intervals. Here we propose a generalized method. We first divide them into several equal-sized groups and then calculate the confidence intervals with the mean values of these groups. If we define “better” in terms of the expected length of the confidence interval, then the first method is better because the expected length of the confidence interval obtained from the first method is shorter. We prove this intuition theoretically. We also specify when the elements in each group are correlated, the first method is invalid, while the second can give us correct results in terms of the coverage probability. We illustrate this with analytical expressions. In practice, when the data set is extremely large and distributed in several data centers, the second method is a good tool to get confidence intervals, in both independent and correlated cases. Some simulations and real data analyses are presented to verify our theoretical results.     

Ellipsoidal confidence regions for a normal covariance matrix

We obtain an asymptotic expansion of the confidence coefficient for an ellipsoidal confidence region on the elements of a normal covariance matrix. This leads to simultaneous confidence intervals on all linear functions of the elements of this matrix, which are compared with those of Roy (1954).     

Distribution-free confidence intervals for quantiles and tolerance intervals in terms of k-records

In this paper, we consider the problem of determining non-parametric confidence intervals for quantiles when available data are in the form of k-records. Distribution-free confidence intervals as well as lower and upper confidence limits are derived for fixed quantiles of an arbitrary unknown distribution based on k-records of an independent and identically distributed sequence from that distribution. The construction of tolerance intervals and limits based on k-records is also discussed. An exact expression for the confidence coefficient of these intervals are derived. Some tables are also provided to assist in choosing the appropriate k-records for the construction of these confidence intervals and tolerance intervals. Some simulation results are presented to point out some of the features and properties of these intervals. Finally, the data, representing the records of the amount of annual rainfall in inches recorded at Los Angeles Civic Center, are used to illustrate all the results developed in this paper and also to demonstrate the improvements that they provide on those based on either the usual records or the current records.     

Bootstrap confidence intervals in mixtures of discrete distributions

The problem of building bootstrap confidence intervals for small probabilities with count data is addressed. The law of the independent observations is assumed to be a mixture of a given family of power series distributions. The mixing distribution is estimated by nonparametric maximum likelihood and the corresponding mixture is used for resampling. We build percentile-t and Efron percentile bootstrap confidence intervals for the probabilities and we prove their consistency in probability. The new theoretical results are supported by simulation experiments for Poisson and geometric mixtures. We compare percentile-t and Efron percentile bootstrap intervals with eight other bootstrap or asymptotic theory based intervals. It appears that Efron percentile bootstrap intervals outperform the competitors in terms of coverage probability and length.     

Nonparametric standard errors and confidence intervals

We investigate several nonparametric methods; the bootstrap, the jackknife, the delta method, and other related techniques. The first and simplest goal is the assignment of nonparametric standard errors to a real-valued statistic. More ambitiously, we consider setting nonparametric confidence intervals for a real-valued parameter. Building on the well understood case of confidence intervals for the median, some hopeful evidence is presented that such a theory may be possible.     

A comparative study of confidence intervals to assess biosimilarity from analytical data

Assessment of analytical similarity of tier 1 quality attributes is based on a set of hypotheses that tests the mean difference of reference and test products against a margin adjusted for standard deviation of the reference product. Thus, proper assessment of the biosimilarity hypothesis requires statistical tests that account for the uncertainty associated with the estimations of the mean differences and the standard deviation of the reference product. Recently, a linear reformulation of the biosimilarity hypothesis has been proposed, which facilitates development and implementation of statistical tests. These statistical tests account for the uncertainty in the estimation process of all the unknown parameters. In this paper, we survey methods for constructing confidence intervals for testing the linearized reformulation of the biosimilarity hypothesis and also compare the performance of the methods. We discuss test procedures using confidence intervals to make possible comparison among recently developed methods as well as other previously developed methods that have not been applied for demonstrating analytical similarity. A computer simulation study was conducted to compare the performance of the methods based on the ability to maintain the test size and power, as well as computational complexity. We demonstrate the methods using two example applications. At the end, we make recommendations concerning the use of the methods.     

Small sample performance of jackknife confidence intervals for the james-stein estimator

The primary goal of this paper is to examine the small sample coverage probability and size of jackknife confidence intervals centered at a Stein-rule estimator. A Monte Carlo experiment is used to explore the coverage probabilities and lengths of nominal 90% and 95% delete-one and infinitesimal jackknife confidence intervals centered at the Stein-rule estimator; these are compared to those obtained using a bootstrap procedure.     

Approximate confidence intervals for a ratio of two variance components

New approximate confidence intervals for the ratio of two variance components in an unbalanced mixed .linear model with a single set of random effects are proposed. Contrary to the confidence intervals known in the literature the new intervals preserve the confidence coefficient and cover the exact confidence interval which, however, is not easy to establish as it requires the solution of complicated nonlinear equations.     

Simultaneous confidence intervals by iteratively adjusted alpha for relative effects in the one-way layout

A bootstrap based method to construct 1−α simultaneous confidence intervals for relative effects in the one-way layout is presented. This procedure takes the stochastic correlation between the test statistics into account and results in narrower simultaneous confidence intervals than the application of the Bonferroni correction. Instead of using the bootstrap distribution of a maximum statistic, the coverage of the confidence intervals for the individual comparisons are adjusted iteratively until the overall confidence level is reached. Empirical coverage and power estimates of the introduced procedure for many-to-one comparisons are presented and compared with asymptotic procedures based on the multivariate normal distribution.