Coverage accuracy for a mean without third moment

For constructing a confidence interval for the mean of a random variable with a known variance, one may prefer the sample mean standardized by the true standard deviation to the Student's t-statistic since the information of knowing the variance is used in the former way. In this paper, by comparing the leading error term in the expansion of the coverage probability, we show that the above statement is not true when the third moment is infinite. Our theory prefers the Student's t-statistic either when one-sided confidence intervals are considered for a heavier tail distribution or when two-sided confidence intervals are considered. Unlike other existing expansions for the Student's t-statistic, the derived explicit expansion for the case of infinite third moment can be used to estimate the coverage error so that bias correction becomes possible.     

One-sided confidence intervals for population variances of skewed distributions

In this article we study the coverage accuracy of one-sided bootstrap-t confidence intervals for the population variances combined with Hall's and Johnson's transformation. We compare the coverage accuracy of all suggested intervals and intervals based on the Chi-square statistic for variances of positively skewed distributions. In addition, we describe and discuss an application of the presented methods for measuring and analyzing revenue variability within the food retail industry. The results show that both Hall's transformation and Johnson's transformation approaches yield good coverage accuracy of the lower endpoint confidence intervals, better than method based on the Chi-square statistic. For the upper endpoint confidence intervals Hall's bootstrap-t method yields the best coverage accuracy when compared with other methods.     

On exact confidence intervals for the common mean of several normal populations

In this paper we consider the problem of constructing exact confidence intervals for the common mean of several normal populations with unknown and possibly unequal variances. Several procedures based on pivots and P-values are discussed and compared.     

The Interpretation of R 2 in Autoregressive-Moving Average Time Series Models

It is demonstrated that factors needed to conduct tests and form confidence intervals for the ratio of two normal variances can be found using one of the new desk calculators which compute F probabilities.     

Confidence Intervals for Difference Between Two Poisson Rates

In this article, we develop four explicit asymptotic two-sided confidence intervals for the difference between two Poisson rates via a hybrid method. The basic idea of the proposed method is to estimate or recover the variances of the two Poisson rate estimates, which are required for constructing the confidence interval for the rate difference, from the confidence limits for the two individual Poisson rates. The basic building blocks of the approach are reliable confidence limits for the two individual Poisson rates. Four confidence interval estimators that have explicit solutions and good coverage levels are employed: the first normal with continuity correction, Rao score, Freeman and Tukey, and Jeffreys confidence intervals. Using simulation studies, we examine the performance of the four hybrid confidence intervals and compare them with three existing confidence intervals: the non-informative prior Bayes confidence interval, the t confidence interval based on Satterthwait's degrees of freedom, and the Bayes confidence interval based on Student's t confidence coefficient. Simulation results show that the proposed hybrid Freeman and Tukey, and the hybrid Jeffreys confidence intervals can be highly recommended because they outperform the others in terms of coverage probabilities and widths. The other methods tend to be too conservative and produce wider confidence intervals. The application of these confidence intervals are illustrated with three real data sets.     

Comparison op some two sample tests for equality of variances

The F-test, F max-test and Bartlett's test are compared on the basis of power for the purpose of testing the equality of variances in two normal populations. The power of each test is expressed as a linear combination of F-probabilities. Bartlett's test is noted to be unbiased, UMPU, consistent against all alterna¬tives and the test which yields minimum length confidence intervals on the ratio of the variancesλ=σ₁ ² /σ² ₂ The two samples Bartlett critical values, although not recognized as such, are found in the works of other authors. Tables of the powers of each test are given for various values of λ, levels of significance a and the respective sample sizes, n₁ and n₂.     

NON-EXISTENCE OF AN EXACT β-CONFIDENCE INTERVAL FOR A SCALE PARAMETER*

Abstract There are given k (≥22) independent distributions with c.d.f.'s F(x;θ_j) indexed by a scale parameter θ_j, j = 1,…, k. Let θ_[i] (i = 1,…, k) denote the ith smallest one of θ₁,…, θ_k. In this paper we wish to show that, under some regularity conditions, there does not exist an exact β-level (0≤β1) confidence interval for the ith smallest scale parameter θ_i based on k independent samples. Since the log transformation method may not yield the desired results for the scale parameter problem, we will treat the scale parameter case directly without transformation. Application is considered for normal variances. Two conservative one-sided confidence intervals for the ith smallest normal variance and the percentage points needed to actually apply the intervals are provided.     

Stress-strength reliability of exponentiated Fréchet distributions based on Type-II censored data

In this paper, we consider inference of the stress-strength parameter, R, based on two independent Type-II censored samples from exponentiated Fréchet populations with different index parameters. The maximum likelihood and uniformly minimum variance unbiased estimators, exact and asymptotic confidence intervals and hypotheses testing for R are obtained. We conduct a Monte Carlo simulation study to evaluate the performance of these estimators and confidence intervals. Finally, two real data sets are analysed for illustrative purposes.     

Approximate Bayesianity of Frequentist Confidence Intervals for a Binomial Proportion

The well-known Wilson and Agresti–Coull confidence intervals for a binomial proportion p are centered around a Bayesian estimator. Using this as a starting point, similarities between frequentist confidence intervals for proportions and Bayesian credible intervals based on low-informative priors are studied using asymptotic expansions. A Bayesian motivation for a large class of frequentist confidence intervals is provided. It is shown that the likelihood ratio interval for p approximates a Bayesian credible interval based on Kerman’s neutral noninformative conjugate prior up to O(n^{? 1}) in the confidence bounds. For the significance level α ? 0.317, the Bayesian interval based on the Jeffreys’ prior is then shown to be a compromise between the likelihood ratio and Wilson intervals. Supplementary materials for this article are available online.     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

The two-sample -test with a known ratio of variances

We consider the two-sample t-test where error variances are unknown but with known relationships between them. This situation arises, for example, when two measuring instruments average different number of replicates to report the response. In particular we compare our procedure with the usual Satterthwaite approximation in the two sample t-test with variances unequal. Our procedure uses the knowledge of a known ratio of variances while the Satterthwaite approximation assumes only that the two variances are unequal. Simulations show that our procedure has both better size and better power than the Satterthwaite approximation. Finally, we consider an extension of our results to the General Linear Model.     

Simultaneous confidence sets and confidence intervals for multiple ratios

This paper deals with the problem of simultaneously estimating multiple ratios. In the simplest case of only one ratio parameter, Fieller's theorem (J. Roy. Statist. Soc. Ser. B 16 (1954) 175) provides a confidence interval for the single ratio. For multiple ratios, there is no method available to construct simultaneous confidence intervals that exactly satisfy a given familywise confidence level. Many of the methods in use are conservative since they are based on probability inequalities. In this paper, first we consider exact simultaneous confidence sets based on the multivariate t-distribution. Two approaches of determining the exact simultaneous confidence sets are outlined. Second, approximate simultaneous confidence intervals based on the multivariate t-distribution with estimated correlation matrix and a resampling approach are discussed. The methods are applied to ratios of linear combinations of the means in the one-way layout and ratios of parameter combinations in the general linear model. Extensive Monte Carlo simulation is carried out to compare the performance of the various methods with respect to the stability of the estimated critical points and of the coverage probabilities.     

Inferences on the ratio of two generalized variances: independent and correlated cases

Statistical inferences about the dispersion of multivariate population are determined by generalized variance. In this article, we consider constructing a confidence interval and testing the hypotheses about the ratio of two independent generalized variances, and the ratio of two dependent generalized variances in two multivariate normal populations. In the case of independence, we first propose a computational approach and then obtain an approximate approach. In the case of dependence, we give an approach using the concepts of generalized confidence interval and generalized p value. In each case, simulation studies are performed for comparing the methods and we find satisfactory results. Practical examples are given for each approach.     

Small Sample Confidence Intervals for the Odds Ratio

Abstract

Numerous methods—based on exact and asymptotic distributions—can be used to obtain confidence intervals for the odds ratio in 2 × 2 tables. We examine ten methods for generating these intervals based on coverage probability, closeness of coverage probability to target, and length of confidence intervals. Based on these criteria, Cornfield’s method, without the continuity correction, performed the best of the methods examined here. A drawback to use of this method is the significant possibility that the attained coverage probability will not meet the nominal confidence level. Use of a mid-P value greatly improves methods based on the “exact” distribution. When combined with the Wilson rule for selection of a rejection set, the resulting method is a procedure that performed very well. Crow’s method, with use of a mid-P, performed well, although it was only a slight improvement over the Wilson mid-P method. Its cumbersome calculations preclude its general acceptance. Woolf's (logit) method—with the Haldane–Anscombe correction— performed well, especially with regard to length of confidence intervals, and is recommended based on ease of computation.     

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.     

Effect size for comparing two or more normal distributions based on maximal contrasts in outcomes

Effect size is a concept that can be especially useful in bioequivalence and studies designed to find important and not just statistically significant differences among responses to treatments based on independent random samples. We develop and explore a new effect size related to a maximal superiority ordering for assessing the separation among two or more normal distributions, possibly having different means and different variances. Confidence intervals and tests of hypothesis for this effect size are developed using a p value obtained by averaging over a distribution on variances. Since there is almost always some difference among treatments, instead of the usual hypothesis test of exactly no effect, researchers should consider testing that an appropriate effect size has at least, or at most, some meaningful magnitude, when one is available, possibly established using the framework developed here. A simulation study of type I error rate, power and interval length is presented. R-code for constructing the confidence intervals and carrying out the tests here can be downloaded from Author’s website.     

Coverage‐adjusted Confidence Intervals for a Binomial Proportion

We consider the classic problem of interval estimation of a proportion p based on binomial sampling. The ‘exact’ Clopper–Pearson confidence interval for p is known to be unnecessarily conservative. We propose coverage adjustments of the Clopper–Pearson interval that incorporate prior or posterior beliefs into the interval. Using heatmap‐type plots for comparing confidence intervals, we show that the coverage‐adjusted intervals have satisfying coverage and shorter expected lengths than competing intervals found in the literature.     

Classical and Bayesian estimation of stress-strength reliability in type II censored Pareto distributions

ABSTRACT

In this paper, the stress-strength reliability, R, is estimated in type II censored samples from Pareto distributions. The classical inference includes obtaining the maximum likelihood estimator, an exact confidence interval, and the confidence intervals based on Wald and signed log-likelihood ratio statistics. Bayesian inference includes obtaining Bayes estimator, equi-tailed credible interval, and highest posterior density (HPD) interval given both informative and non-informative prior distributions. Bayes estimator of R is obtained using four methods: Lindley's approximation, Tierney-Kadane method, Monte Carlo integration, and MCMC. Also, we compare the proposed methods by simulation study and provide a real example to illustrate them.     

Confidence intervals for nonparametric quantile regression: an emphasis on smoothing splines approach

In this paper we consider the problem of constructing confidence intervals for nonparametric quantile regression with an emphasis on smoothing splines. The mean‐based approaches for smoothing splines of Wahba (1983) and Nychka (1988) may not be efficient for constructing confidence intervals for the underlying function when the observed data are non‐Gaussian distributed, for instance if they are skewed or heavy‐tailed. This paper proposes a method of constructing confidence intervals for the unknown τth quantile function (0<τ<1) based on smoothing splines. In this paper we investigate the extent to which the proposed estimator provides the desired coverage probability. In addition, an improvement based on a local smoothing parameter that provides more uniform pointwise coverage is developed. The results from numerical studies including a simulation study and real data analysis demonstrate the promising empirical properties of the proposed approach.     

Jackknife empirical likelihood method for testing the equality of two variances

In this paper, we propose a nonparametric method based on jackknife empirical likelihood ratio to test the equality of two variances. The asymptotic distribution of the test statistic has been shown to follow χ² distribution with the degree of freedom 1. Simulations have been conducted to show the type I error and the power compared to Levene's test and F test under different distribution settings. The proposed method has been applied to a real data set to illustrate the testing procedure.