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The Behrens-Fisher problem refers to a statistical challenge in comparing the mean vectors of two normally distributed p-variate populations when the covariance matrices of these populations are assumed to be unequal. This problem can be addressed using Hotelling's T² test that requires equality of covariance matrices. However, when the assumption of equality between the two covariance matrices is violated, the performance of this test can be affected, leading to incorrect conclusions. This article presents the implementation of 11 alternative tests proposed in the statistical literature for the Behrens-Fisher problem. These tests are hosted in the stests library in R, and any of these tests can be used through a single function. Additionally, this article conducted a Monte Carlo simulation study in which factors such as sample size, the distance between the mean vectors, and a scaling factor between the covariance matrices were examined. The results found that the rejection rate of the null hypothesis (H0: µ1 = µ2) increases when there is a greater discrepancy between the two mean vectors and when the sample size increases. The results demonstrate that all the tests developed in the stests package, which address the multivariate Behrens-Fisher problem, are plausible for comparing two mean vectors. [ABSTRACT FROM AUTHOR]

This article presents a comparison of multivariate normal mean vectors under covariance positive definite matrices. We introduce an improved parametric bootstrap (IPB) approach for addressing the multivariate Behrens-Fisher problem, specifically focusing on cases with unequal covariance matrices. Additionally, we evaluate the performance of the IPB test by comparing it with three existing tests: the parametric bootstrap (PB) test, the generalized variable (GV) test, and the Johansen test. Through Monte Carlo simulation, our results demonstrate that both the IPB test and the PB test exhibit superior control over Type I error rates compared to the GV and Johansen tests. Notably, the IPB test outperforms the PB test in terms of controlling Type I error rates. Consequently, our study concludes that the IPB test represents a preferred statistical method for testing the equality of mean vectors in the multivariate Behrens-Fisher problem. [ABSTRACT FROM AUTHOR]

A practical solution based on the method of variance estimates recovery is proposed for the Behrens–Fisher problem. Unlike the classical solution, the proposed test is not based on the central limit theorem and, therefore, it can control Type I error well for small or moderate sample sizes. In this sense, it performs better than the most commonly used Welch's approximate t-test, which may be liberal for small or moderate sample sizes. On the other hand, the powers of the proposed test appears to be close to that of the Welch's approximate t-test and better than those of the Fisher's solution and the generalized p-value solution; the latter two have no fixed p-value and need to simulate. The proposed test has a meaningful interpretation within and not just across experiments. It can be a competitive alternative solution. [ABSTRACT FROM AUTHOR]

In this paper, the problem of testing the equality of the mean vectors of k populations with possibly unknown and unequal covariance matrices is investigated in high-dimensional settings. The null distributions of most existing tests are asymptotically normal which inevitably imposes strong conditions on covariance matrices. However, we assume here only mild additional conditions on the proposed test, which offers much flexibility in practical applications. Additionally, the Welch–Satterthwaite χ 2 -approximation we adopted can automatically mimic the shape of the null distribution of the proposed test statistic, while the normal approximation cannot achieve the adaptivity. Finally, an extensive simulation study shows that the proposed test has better performance on both size and power compared with existing methods. [ABSTRACT FROM AUTHOR]

For high-dimensional two-sample Behrens–Fisher problems, several non-scale-invariant and scale-invariant tests have been proposed. Most of them impose strong assumptions on the underlying group covariance matrices so that their test statistics are asymptotically normal. However, in practice, these assumptions may not be satisfied or hardly be checked so that these tests may not be able to maintain the nominal size well in practice. To overcome this difficulty, in this paper, a normal reference scale-invariant test is proposed and studied. It works well by neither imposing strong assumptions on the underlying group covariance matrices nor assuming their equality. It is shown that under some regularity conditions and the null hypothesis, the proposed test and a chi-square-type mixture have the same normal and non-normal limiting distributions. It is then justifiable to approximate the null distribution of the proposed test using that of the chi-square-type mixture. The distribution of the chi-square type mixture can be well approximated by the Welch–Satterthwaite chi-square-approximation with the approximation parameter consistently estimated from the data. The asymptotic power of the proposed test is established. Numerical results demonstrate that the proposed test has much better size control and power than several well-known non-scale-invariant and scale-invariant tests. [ABSTRACT FROM AUTHOR]

Empirical Likelihood Ratio Test (ELRT) is a powerful nonparametric test for mean. Three versions of ELRT are available for testing equality of mean of two groups. When the variances are unknown and are equal, the asymptotic null distribution of all the three-test statistics is central chi-square with 1 degrees of freedom. When variances are unknown and unequal, Vikas and Rao (2019) derived the asymptotic null distribution of a version of the ELRT statistic and showed that it is scaled chi-squared with 1 degrees of freedom. In this paper, we derived the asymptotic null distribution of the remaining two versions of the ELRT and showed that it is scaled chi-squared distribution with 1 degree of freedom and the scaling factor is the same as derived by Vikas and Rao (2019). One of the versions of ELRT is computationally tedious. This paper studies the finite sample property of popular form of ELRT as a solution to Behrens-Fisher (BF) problem. The test and its corrected version maintain level of significance for moderate sample size. The test has got good power property and it is comparable with some of the popular solutions of the BF problem. The test is applied to a real-life data, which is used in many research papers. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of comparing the population means of two groups of independent observations. An approximate randomization test procedure based on the test statistic of Chen & Qin (2010) is proposed. The asymptotic behaviour of the test statistic, as well as the randomized statistic, is studied under weak conditions. In our theoretical framework, observations are not assumed to be identically distributed even within groups. No condition on the eigenstructure of the covariance matrices is imposed. Furthermore, the sample sizes of the two groups are allowed to be unbalanced. Under general conditions, all possible asymptotic distributions of the test statistic are obtained. We derive the asymptotic level and local power of the approximate randomization test procedure. Our theoretical results show that the proposed test procedure can adapt to all possible asymptotic distributions of the test statistic, always has the correct test level asymptotically and has good power behaviour. Our numerical experiments show that the proposed test procedure has favourable performance compared with several alternative test procedures. [ABSTRACT FROM AUTHOR]

The Behrens–Fisher problem occurs when testing the equality of means of two normal distributions without the assumption that the two variances are equal. This paper presents approaches based on the exact and near-exact distributions for the test statistic of the Behrens–Fisher problem, depending on different combinations of even or odd sample sizes. We present the exact distribution when both sample sizes are odd and the near-exact distribution when one or both sample sizes are even. The near-exact distributions are based on a finite mixture of generalized integer gamma (GIG) distributions, used as an approximation to the exact distribution, which consists of an infinite series. The proposed tests, based on the exact and the near-exact distributions, are compared with Welch's t-test through Monte Carlo simulations, in particular for small and unbalanced sample sizes. The results show that the proposed approaches are competent solutions to the Behrens–Fisher problem, exhibiting precise sizes and better powers than Welch's approach for those cases. Numerical studies show that the Welch's t-test tends to be a bit more conservative than the test statistics based on the exact or near-exact distribution, in particular when sample sizes are small and unbalanced, situations in which the proposed exact or near-exact distributions obtain higher powers than Welch's t-test. [ABSTRACT FROM AUTHOR]

In the context of high-throughput data, the differences in continuous markers between two groups are usually assessed by ordering the p-values obtained from the two-sample pooled t-test or Wilcoxon-Mann-Whitney test and choosing a stringent cutoff such as 10-8 to control the family-wise error rate (FWER) or false discovery rate (FDR). All markers with p-values below the cutoff are declared to be significantly associated with the phenotype. This inherently assumes that the test procedure provides valid type I error estimates in extreme tails of the null distribution. The aforementioned tests assume homoscedasticity in the two groups, and the t-test further assumes underlying distributions to be normally distributed. Cao et al. (Biometrika, 2013, 100, 495-502) have shown that in the context of multiple hypotheses testing the approach based on FDR may not be valid under non-normality and/or heteroscedasticity. Therefore, having a test statistic that is robust to these violations is needed. In this study, we propose a robust analog of Behrens-Fisher statistic based on trimmed means, conduct an extensive simulation study to compare its performance with other competing approaches, and demonstrate its usefulness by applying it to DNA methylation data used by Teschendorff et al. (Genome Res., 2010, 20, 440-446). An R program to implement the proposed method is provided in the Supplementary Material. [ABSTRACT FROM AUTHOR]

Comparison of multivariate mean of more than two groups is generally known as multivariate variance analysis (MANOVA) problem. However, classical test statistics used to solve MANOVA problems are highly affected by assumption violations. In addition, multivariate test statistics are insufficient in cases where the number of dependent variables is greater than the number of observations. This study aims to compare some proposed test statistics to determine whether mean vectors are equal in high-dimensional Behrens-Fisher problems. For this purpose, a simulation analysis was carried out for four different test statistics in cases where variance-covariance matrices are heterogeneous. Three different variancecovariance models were used in the study. In addition, different experimental conditions were taken into account for the number of dependent variables and the number of observations. The results of the study showed that the performances of the test statistics were generally comparable. However, it has been observed that the performance of test statistics varies according to experimental conditions. [ABSTRACT FROM AUTHOR]

In this article, we have proposed a three-stage procedure for the estimation of the difference of the means of two multivariate normal populations having unknown and unequal variances. Point as well as confidence region estimation is done for the same. Here, we have used the concept of classical Behrens-Fisher problem. Second-order approximations are obtained in both the cases, i.e., point estimation and confidence region estimation. [ABSTRACT FROM AUTHOR]

When fitting a Gaussian mixture regression model to observed data, estimating a between-group contrast can be a practical issue. One can use the estimate to compare the effects of a particular covariate or a set of covariates across different subpopulations. By applying fiducial generalized pivotal quantities, a small-sample solution is proposed in this paper to obtain interval estimates of between-group contrasts. Specifically, a Markov chain Monte Carlo sampler, which takes the membership uncertainty of each individual into account, is designed to generate realizations from the target distributions for computing the required interval estimates. A plant virus transmission study is first introduced as a motivating example for the present study. Next, the observed data are analyzed to illustrate the proposed method. Based on the simulation results, it is further shown that the proposed method can maintain the empirical coverage rates sufficiently close to the nominal level. Supplementary materials accompanying this paper appear on-line. [ABSTRACT FROM AUTHOR]

A unified testing framework is presented for large-dimensional mean vectors of one or several populations which may be non-normal with unequal covariance matrices. Beginning with one-sample case, the construction of tests, underlying assumptions and asymptotic theory, is systematically extended to multi-sample case. Tests are defined in terms of U-statistics-based consistent estimators, and their limits are derived under a few mild assumptions. Accuracy of the tests is shown through simulations. Real data applications, including a five-sample unbalanced MANOVA analysis on count data, are also given. [ABSTRACT FROM AUTHOR]

Analogs of the classical one way MANOVA model have recently been suggested that do not assume that population covariance matrices are equal or that the error vector distribution is known. These tests are based on the sample mean and sample covariance matrix corresponding to each of the p populations. We show how to extend these tests using other measures of location such as the trimmed mean or coordinatewise median. These new bootstrap tests can have some outlier resistance, and can perform better than the tests based on the sample mean if the error vector distribution is heavy tailed. [ABSTRACT FROM AUTHOR]

We consider statistical procedures for hypothesis testing of real valued functionals of matched pairs with missing values. In order to improve the accuracy of existing methods, we propose a novel multiplication combination procedure. Dividing the observed data into dependent (completely observed) pairs and independent (incompletely observed) components, it is based on combining separate results of adequate tests for the two sub data sets. Our methods can be applied for parametric as well as semiparametric and nonparametric models and make use of all available data. In particular, the approaches are flexible and can be used to test different hypotheses in various models of interest. This is exemplified by a detailed study of mean- as well as rank-based approaches under different missingness mechanisms with different amount of missing data. Extensive simulations show that in most considered situations, the proposed procedures are more accurate than existing competitors particularly for the nonparametric Behrens-Fisher problem. A real data set illustrates the application of the methods. [ABSTRACT FROM AUTHOR]

Multiregional clinical trials have been accepted in recent years as a useful means of accelerating the development of new drugs and abridging their approval time. The statistical properties of multiregional clinical trials are being widely discussed. In practice, variance of a continuous response may be different from region to region, but it leads to the assessment of the efficacy response falling into a Behrens-Fisher problem-there is no exact testing or interval estimator for mean difference with unequal variances. As a solution, this study applies interval estimations of the efficacy response based on Howe's, Cochran-Cox's, and Satterthwaite's approximations, which have been shown to have well-controlled type I error rates. However, the traditional sample size determination cannot be applied to the interval estimators. The sample size determination to achieve a desired power based on these interval estimators is then presented. Moreover, the consistency criteria suggested by the Japanese Ministry of Health, Labour and Welfare guidance to decide whether the overall results from the multiregional clinical trial obtained via the proposed interval estimation were also applied. A real example is used to illustrate the proposed method. The results of simulation studies indicate that the proposed method can correctly determine the required sample size and evaluate the assurance probability of the consistency criteria. [ABSTRACT FROM AUTHOR]

Two variants of the empirical likelihood ratio test for testing equality of means were in existence for some time. We unify these two approaches and propose a new variant of the empirical likelihood ratio test for testing equality of means. The main aim of this paper is to propose these empirical likelihood ratio tests as distribution free solutions to the Behrens-Fisher problem in the case of normal distribution. Finite sample performance of these tests along with Wilcoxon Mann Whitney rank sum test is studied through extensive Monte-Carlo simulation. The results indicate that for moderate to large sample sizes the corrected version of the proposed test maintains the level of significance and has got good power property. For small sample sizes, the test is slightly liberal but is more powerful compared to other tests and following the idea of Emerson (2009), we advocate this test as a solution to the Behrens-Fisher problem. The simulation results also indicate the influence of the unknown variances on type I error rate of the empirical likelihood ratio tests and the Wilcoxon Mann Whitney test. The data on driving time given in Lehmann (1975; p. 83) is reanalysed using empirical likelihood ratio tests and the results are compared with other well known solutions to the Behrens-Fisher problem. [ABSTRACT FROM AUTHOR]

The comparison of two means is one of the most commonly applied statistical procedures in psychology. The independent samples t-test corrected for unequal variances is commonly known as Welchs test, and is widely considered to be a robust alternative to the independent samples t-test. The properties of Welchs test that make it Type I error robust are examined. The degrees of freedom used in Welchs test are a random variable, the distributions of which are examined using simulation. It is shown how the distribution for the degrees of freedom is dependent on the sample sizes and the variances of the samples. The impact of sample variances on the degrees of freedom, the resultant critical value and the test statistic is considered, and hence gives an insight into why Welchs test is Type I error robust under normality.

The Wilcoxon-Mann-Whitney (WMW) test is a popular rank-based two-sample testing procedure for the strong null hypothesis that the two samples come from the same distribution. A modified WMW test, the Fligner-Policello (FP) test, has been proposed for comparing the medians of two populations. A fact that may be under-appreciated among some practitioners is that the FP test can also be used to test the strong null like the WMW. In this article, we compare the power of the WMW and FP tests for testing the strong null. Our results show that neither test is uniformly better than the other and that there can be substantial differences in power between the two choices. We propose a new, modified WMW test that combines the WMW and FP tests. Monte Carlo studies show that the combined test has good power compared to either the WMW and FP test. We provide a fast implementation of the proposed test in an open-source software. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Suppose there are two independent random samples from two populations or groups. A common multivariate two-sample test of hypotheses isversuswhereis a population location measure of theith population fori= 1, 2. The two-sample Hotelling’sT2test is the classical method, and is a special case of the one way MANOVA model if the two populations are assumed to have the same population covariance matrix. This paper suggests using a recent bootstrap technique to develop analogs of Hotelling’sT2test. The new tests can have considerable outlier resistance, and the tests do not need the population covariance matrices to be equal. [ABSTRACT FROM PUBLISHER]

Non-normality and heteroscedasticity are common in applications. For the comparison of two samples in the non-parametric Behrens-Fisher problem, different tests have been proposed, but no single test can be recommended for all situations. Here, we propose combining two tests, the Welch t test based on ranks and the Brunner-Munzel test, within a maximum test. Simulation studies indicate that this maximum test, performed as a permutation test, controls the type I error rate and stabilizes the power. That is, it has good power characteristics for a variety of distributions, and also for unbalanced sample sizes. Compared to the single tests, the maximum test shows acceptable type I error control. [ABSTRACT FROM AUTHOR]

The receiver operating characteristic (ROC) curve is frequently used as a measure of accuracy of continuous markers in diagnostic tests. The area under the ROC curve (AUC) is arguably the most widely used summary index for the ROC curve. Although the small sample size scenario is common in medical tests, a comprehensive study of small sample size properties of various methods for the construction of the confidence/credible interval (CI) for the AUC has been by and large missing in the literature. In this paper, we describe and compare 29 non-parametric and parametric methods for the construction of the CI for the AUC when the number of available observations is small. The methods considered include not only those that have been widely adopted, but also those that have been less frequently mentioned or, to our knowledge, never applied to the AUC context. To compare different methods, we carried out a simulation study with data generated from binormal models with equal and unequal variances and from exponential models with various parameters and with equal and unequal small sample sizes. We found that the larger the true AUC value and the smaller the sample size, the larger the discrepancy among the results of different approaches. When the model is correctly specified, the parametric approaches tend to outperform the non-parametric ones. Moreover, in the non-parametric domain, we found that a method based on the Mann-Whitney statistic is in general superior to the others. We further elucidate potential issues and provide possible solutions to along with general guidance on the CI construction for the AUC when the sample size is small. Finally, we illustrate the utility of different methods through real life examples. [ABSTRACT FROM AUTHOR]

In recent years, high-dimensional data has become increasingly prevalent with rapid development of data collecting technologies. Much work has been done for hypothesis testing on mean vectors, especially for high-dimensional two-sample problems. Rather than considering a specific problem, we are interested in a general linear hypothesis testing problem on mean vectors of several populations, which includes many existing hypotheses about mean vectors as its special cases. We propose a test statistic based on a linear combination of U-statistics which can be quickly calculated without using computationally intensive U-statistics. Asymptotic normality and power of our test are derived under mild conditions without requiring an explicit relation between the data dimension and the sample size. Our test is applicable to non-normal multi-sample high-dimensional data without assuming a common covariance matrix among different samples. It also works well even when different samples follow different distributions, provided that some moment conditions are satisfied. A simple, computationally-efficient, and ratio-consistent estimator of the unknown variance of our test statistic is also provided which is not based on U-statistic calculations. Simulation studies and a real data example are presented to demonstrate the good performance of our test. [ABSTRACT FROM AUTHOR]

Comparisons of different treatments or production processes are the goals of a significant fraction of applied research. Unsurprisingly, two-sample problems play a main role in statistics through natural questions such as "Is the the new treatment significantly better than the old?" However, this is only partially answered by some of the usual statistical tools for this task. More importantly, often practitioners are not aware of the real meaning behind these statistical procedures. We analyze these troubles from the point of view of the order between distributions, the stochastic order, showing evidence of the limitations of the usual approaches, paying special attention to the classical comparison of means under the normal model. We discuss the unfeasibility of statistically proving stochastic dominance, but show that it is possible, instead, to gather statistical evidence to conclude that slightly relaxed versions of stochastic dominance hold. [ABSTRACT FROM AUTHOR]

The Behrens--Fisher problem concerns the inferences for the difference between means of two independent normal populations without the assumption of equality of variances. In this article, we compare three approximate confidence intervals and a generalized confidence interval for the Behrens--Fisher problem. We also show how to obtain simultaneous confidence intervals for the three population case (analysis of variance, ANOVA) by the Bonferroni correction factor. We conduct an extensive simulation study to evaluate thesemethods in respect to their type I error rate, power, expected confidence interval width, and coverage probability. Finally, the considered methods are applied to two real dataset. [ABSTRACT FROM AUTHOR]

This paper defends the fiducial argument. In particular, an interpretation of the fiducial argument is defended in which fiducial probability is treated as being subjective and the role taken by pivots in a more standard interpretation is taken by what are called primary random variables, which in fact form a special class of pivots. The resulting methodology, which is referred to as subjective fiducial inference, is outlined in the first part of the paper. This is followed by a defence of this methodology arranged in a series of criticisms and responses. These criticisms reflect objections that are often raised against standard fiducial inference and incorporate more specific concerns that are likely to exist with respect to subjective fiducial inference. It is hoped that the responses to these criticisms clarify the contribution that a system of fiducial reasoning can make to statistical inference. [ABSTRACT FROM AUTHOR]

The essence of the generalised multivariate Behrens-Fisher problem (BFP) is how to test the null hypothesis of equality of mean vectors for two or more populations when their dispersion matrices differ. Solutions to the BFP usually assume variables are multivariate normal and do not handle high-dimensional data. In ecology, species' count data are often high-dimensional, non-normal and heterogeneous. Also, interest lies in analysing compositional dissimilarities among whole communities in non-Euclidean (semi-metric or non-metric) multivariate space. Hence, dissimilarity-based tests by permutation (e.g., PERMANOVA, ANOSIM) are used to detect differences among groups of multivariate samples. Such tests are not robust, however, to heterogeneity of dispersions in the space of the chosen dissimilarity measure, most conspicuously for unbalanced designs. Here, we propose a modification to the PERMANOVA test statistic, coupled with either permutation or bootstrap resampling methods, as a solution to the BFP for dissimilarity-based tests. Empirical simulations demonstrate that the type I error remains close to nominal significance levels under classical scenarios known to cause problems for the un-modified test. Furthermore, the permutation approach is found to be more powerful than the (more conservative) bootstrap for detecting changes in community structure for real ecological datasets. The utility of the approach is shown through analysis of 809 species of benthic soft-sediment invertebrates from 101 sites in five areas spanning 1960 km along the Norwegian continental shelf, based on the Jaccard dissimilarity measure. [ABSTRACT FROM AUTHOR]

The paper derives a class of non-informative, probability matching priors and of default, data-dependent priors for the difference in two normal means when the variance(s) are unknown. These distributions are useful to get a Bayesian model in the situation where one has to assess compatibility of data obtained by a laboratory with the specified values and uncertainty estimate given in the certificate of analysis. The latter is then interpreted as the parameter assignment for measurand’s normal distribution. The Bayes rules arising from these prior distributions are compared via the frequentist coverage probability of corresponding credible intervals for lab’s bias. [ABSTRACT FROM AUTHOR]

We consider profile analysis with unequal covariance matrices under multivariate normality. In particular, we discuss this problem for high-dimensional data where the dimension is larger than the sample size. We propose three test statistics based on Bennett's (1951) transformation and the Dempster trace criterion proposed by Dempster (1958). We derive the null distributions as well as the nonnull distributions of the test statistics. Finally, in order to investigate the accuracy of the proposed statistics, we perform Monte Carlo simulations for some selected values of parameters. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find a procedure to test equal means that is robust at the significance level. A simulation study is conducted to compare the performance of different strategies, including unconditionally applying the bootstrap ANOVA and twelve adaptive tests that take in pre-testing normality, homoscedasticity and skewness. Various final tests of equal means have been considered, like the ANOVA, bootstrap ANOVA, Welch, Brown-Forsythe and bootstrap James test. Our simulation results reveal that the usual adaptive test used by applied researchers (based on testing normality and homoscedasticity to choose from the ANOVA, Welch and Kruskal-Wallis tests) performs poorly. The simulation results show that preliminary tests may improve the performance of a test, and that this depends on the pre-tests chosen. In particular, we find that using decisions on normality to select the right homoscedasticity test and then choosing between the Brown-Forsythe test and the bootstrap ANOVA leads to controlling the Type I error rate in all of the settings studied. [ABSTRACT FROM AUTHOR]

We consider the problem of comparingkregression models, when the variances are not assumed to be equal. For this problem, the classical F test can lead to misleading results, and there is no simple test which adequately controls the size when the sample sizes are small. Fork= 2, the most widely used test is the “weighted F test,” also known as the “asymptotic Chow test.” But this test does not work well for small samples, and various modifications have been proposed in the literature. Fork> 2, few tests are available and only the parametric-bootstrap (PB) test of Tian et al. (2009) controls the size fairly adequately. In this article, we propose three fairly simple F tests which can easily be applied in the general case,k⩾ 2, and avoid the complications of the PB test. Our simulations indicate that these tests have satisfactory performance. Also, our simulations confirm that the power properties of our proposed tests are similar to the PB test. Therefore, our proposed tests provide simple alternatives to the PB test, which can easily be used by practitioners who may not be familiar with the PB. [ABSTRACT FROM AUTHOR]

We consider statistical inference for a vector-valued parameter of interest in a regular asymptotic model with a finite-dimensional nuisance parameter. We use highly accurate likelihood theory to derive a directional test, in which the p-value is obtained by one-dimensional numerical integration. This extends the results of Davison et al. (2014) for linear exponential families to nonlinear parameters of interest and to more general models. Examples and simulations provide comparisons with the likelihood ratio test and adjusted versions of the likelihood ratio test. The directional approach gives extremely accurate inference, even in high-dimensional settings where the likelihood ratio versions can fail catastrophically. [ABSTRACT FROM AUTHOR]

The Birnbaum-Saunders distribution is widely used in fields such as survival analysis. In this paper, we use a new method to construct confidence intervals for the shape parameter of the Birnbaum-Saunders distribution and find it superior to some existing methods. Our method can be called the 'joint pivot method' as we obtain the confidence interval by uniting two pivots. Furthermore, we find that the joint pivot method can also be used to solve the Behrens-Fisher problem. Simulation studies show that the joint pivot method is comparable with some commonly practiced ones. Finally, we analyse two real data sets to illustrate the proposed procedure. [ABSTRACT FROM AUTHOR]