Studentized bootstrap confidence intervals based on M-estimates

This article reviews and applies saddlepoint approximations to studentized confidence intervals based on robust M-estimates. The latter are known to be very accurate without needing standard theory assumptions. As examples, the classical studentized statistic, the studentized versions of Huber's M-estimate of location, of its initially MAD scaled version and of Huber's proposal 2 are considered. The aim is to know whether the studentized statistics yield robust confidence intervals with coverages close to nominal, with short intervals. The results of an extensive simulation study and the recommendations for practical use given in this article may fill gaps in the current literature and stimulate further discussion and research.     

A Note on Simultaneous Confidence Intervals for Ratio of Variances

An explicit formula for confidence intervals for ratios of variances of several populations is presented. The intervals are based on jackknife statistics and the critical point of the studentized range distribution. The asymptotic probability of coverage is not less than the nominal value provided that the distributions of the sampled populations belong to a location-scale family of probabilities with finite fourth moment.     

RCEV heteroscedasticity test based on the studentized residuals

All the usual heteroscedasticity tests in the statistics and econometrics literature are based on raw residuals. Although the raw residuals are heteroscedastic, studentized residuals can still be homoscedastic. In this study, the version of Çelik’s RCEV heteroscedasticity test which is based on studentized residuals is introduced.     

An L 2 Comparison Between the Bootstrap and the Empirical Edgeworth Expansion

The Bootstrap estimate for studentized statistics is more accurate than both the normal approximation and the two-term empirical Edgeworth expansion. In this article, it will be shown that the three-term empirical Edgeworth expansion for studentized statistics compares well with the bootstrap. It is also shown that the three-term Edgeworth expansion is superior to the bootstrap in some cases, using more efficient estimators than sample moments in the Edgeworth expansion, such as using maximum likelihood estimators in the one-parameter exponential family.     

Gauss-laguerre dynamics:an efficient method of computing integrals in multiple comparisons

Computing with Gauss-Laguerre quadratures is a common practice when computing the distribution of the studentized version of many statistics used in normal theory multiple comparisons. We study the dynamics of the points associated with such quadrature and apply the results to the simple example of the studentized maximum modulus. Because most distributions used in multiple comparisons require relatively expensive numerical quadrature for evaluation of their non-studentized integrand, the efficiency of the suggested method is clear as it requires fewer integrand evaluations than the standard Gauss-Laguerre quadrature.     

Regression residuals and test statistics: Assessing naive outlier deletion

The authors derive the joint distributions of a studentized deleted residual and various regression quantities, calculated with all the data or with one case deleted. They show that the correlation between the studentized deleted residual and the deleted test statistic has an interesting interpretation in terms of well‐known regression quantities. These results allow them to examine the effect of applying some naive outlier deletion methods before making inferences about a regression parameter.     

Likelihood Ratio Tests for the Non Equivalence of Means

We derive likelihood ratio (LR) tests for the null hypothesis of equivalence that the normal means fall into a practical indifference zone. The LR test can easily be constructed and applied to k ≥ 2 treatments. Simulation results indicate that the LR test might be slightly anticonservative statistically, but when the sample sizes are large, it always produces the nominal level for mean configurations under the null hypothesis. More powerful than the studentized range test, the LR test is a straightforward application that requires only current existing statistical tables, with no complicated computations.     

On the sensitivity to non-normality of a test for outliers in linear models

In this note we use the class of exponential power distributions to assess the robustness to non-normality of the test for outliers based on the maximum absolute studentized residual. We find that the significance levels can be quite markedly affected by even moderate departures from normality of the error distribution in a regression model when the sample size is moderately large.     

Critical point and power calculations for the studentized range test for generally correlated means

The studentized range test is a widely applied statistical procedure to compare several normal means within the analysis of variance. However, up to now no general methodology is available to perform the all-pair comparisons precisely, such as the computation of p-values or quantiles in the simple unbalanced one-way layout. Instead, a variety of approximations have been proposed in the past. This article focuses on exact computations of simultaneous confidence intervals and exact sample size determinations for all-pair comparisons in the analysis of variance involving arbitrary variance-covariance matrices. General power expressions in closed form are developed and numerical issues concerning the arising multivariate central and noncentral t-distributions are discussed. An application to the usual fixed effects analysis of covariance illustrates the use of the obtained results.     

Some remarks on bootstrap techniques for constructing confidence intervals

As techniques of constructing confidence intervals for a parameter, the classical, the studentized and the prepivoted bootstrap are discussed and compared in a Monte-Carlo study.     

History and role of order statistics

This paper starts with an itroduction which gives definitions of order statistics,makes a clear distinction between order statistics and rank statistics, and gives an overview of the problems in which methods based on order statstics are useful.The following sections deal with the history and role of order sta¬tistics in selected areas: central tendency, dispersion, and regression; treatment of outliers and robust estimation; maximum likelihood estimators; best linear unbiased estimators; extreme-values; multiple comparisons and studentized range. The last two sections discuss a basic coverage property and some of its conse¬quences and describe the author's chronological annotated bibliography of order statistics. Finally, there is a list of references.     

On testing the bioequivalence of several treatments using the measure of distance

A studentized range test is proposed to test the hypothesis of bioequivalence of normal means in terms of a standardized distance among means. A least favourable configuration (LFC) of means to guarantee the maximum level at a null hypothesis and an LFC of means to guarantee the minimum power at an alternative hypothesis are obtained. This level and power of the test are fully independent of the unknown means and variances. For a given level, the critical value of the test under a null hypothesis can be determined. Furthermore, if the power under an alternative is also required at a given level, then both the critical value and the required sample size for an experiment can be simultaneously determined. In situations where the common population variance is unknown and the bioequivalence is the actual distance between means without standardization, a two-stage sampling procedure can be employed to find these solutions.     

Book Reviews

The diagnostic tools examined in this article are applicable to regressions estimated with panel data or cross-sectional data drawn from a population with grouped structure. The diagnostic tools considered include (a) tests for the existence of group effects under both fixed and random effects models, (b) checks for outlying groups, and (c) specification tests for comparing the fixed and random effects models. A group-specific counterpart to the studentized residual is introduced. The methods are illustrated using a hedonic housing price regression.     

Critical values for testing for a single outlier in a nonlinear regression model

The maximum absolute studentized residual is commonly used for testing for a single outlier in a linear regression model. This test statistic, however, is seldom discussed in a nonlinear regression setting. We simulate the critical values for the tests under various nonlinear models. The associated critical values are found to be very close to one another. Moreover, they are very well approximated using the critical values obtained from F-distributions based on the Bonferroni equations in linear models. The results are promising even in samples of size 6.     

On the relative performance of the block bootstrap for dependent data

In the independent setting, both Efron's bootstrap and “empiricai Edgeworth expansion” (E.E-expansion) give second-order accurate approximations to distributions of standardized and studentized statistics in the smooth function model. As a result, Efron's bootstrap was often regarded as roughly equivalent to the one-term E.E-expansion. However, a more detailed analysis shows that Efron's bootstrap outperforms the E.E-expansion in terms of loss functions by Bhattacharya and Qumsiyeh (1989) and in terms of probabilities for large deviations by Hall (1990) and Jing et a1 (1994). in this paper, we shall study the performances of the block bootstrap and the E.E-expansion for the weakly dependent data. It turns out that similar properties hold:both perform equally well at the center of the distribution but the block bootstrap provides accurate approximations even in the tails of the distributions. The study is focued on the simple case of standardized and studentized sample mean, but the conclusions can be easily extended to the smooth function of multivariate means.     

A Nonparametric Test for the Parallelism of Two First-Order Autoregressive Processes

A nonparametric testing procedure for the parallelism of two first-order autoregressive processes is presented. This paper discuss the Mann–Whitney statistic, its natural competitor two-sample t -test, and the bootstrap method. It studies the asymptotic efficacies of the studentized Mann–Whitney statistic and the t -test statistic with their relative efficiency. Simulation results for comparing the powers of these test statistics are also presented.     

The studentized location linear discriminant function

The location linear discriminant function is used in a two-population classification problem when the available data are generated from both binary and continuous random variables. Asymptotic distribution of the studentized location linear discriminant function is derived directly without the inversion of the corresponding characteristic function. The resulting plug-in estimate of the overall error of misclassification consists of the estimate based on the limiting distribution of the discriminant plus a correction term up to the second order. By comparison, our estimate avoids exact knowledge of the Mahalanobis distances which is necessary when the expansions of Vlachonikolis (1985) are used in the case of an arbitrary cut-off point. An example is re-examined and analysed in the present context.     

Influence Diagnostics on Testing Linear Hypothesis in Linear Models with Correlated Errors

To assess the influence of observations on the parameter estimates, case deletion diagnostics are commonly used in linear regression models. For linear models with correlated errors we study the influence of observations on testing a linear hypothesis using single and multiple case deletions. The change in likelihood ratio test and F test theoretically is derived and it is shown these tests to be completely determined by two proposed generalized externally studentized residuals. An illustrative example of a real data set is also reported.     

The performance of univariate goodness-of-fit tests for normality based on the empirical characteristic function in large samples

A test based on the studentized empirical characteristic function calculated in a single point is derived. An empirical power comparison is made between this test and tests like the Epps–Pulley, Shapiro–Wilks, Anderson–Darling and other tests for normality. It is shown to outperform the more complicated Epps-Pulley test based on the empirical characteristic function and a Cramér-von Mises type expression in a simulation study. The test performs especially good in large samples and the derived test statistic has an asymptotic normal distribution which is easy to apply.     

A sequential testing procedure for outliers and structural change

In this paper a test for outliers based on externally studentized residuals is shown to be related to a test for predictive failure. The relationships between a test for outliers, a test for a correlated mean shift and a test for an intercept shift are developed. A sequential testing procedure for outliers and structural change is shown to be independent, so that the overall size of the joint test can be determined exactly. It is established that a joint test for outliers and constancy of variances cannot be performed.