An adjusted likelihood-ratio approach to the behrens-fisher problem

In many situations it is necessary to test the equality of the means of two normal populations when the variances are unknown and unequal. This paper studies the celebrated and controversial Behrens-Fisher problem via an adjusted likelihood-ratio test using the maximum likelihood estimates of the parameters under both the null and the alternative models. This procedure allows the significance level to be adjusted in accordance with the degrees of freedom to balance the risk due to the bias in using the maximum likelihood estimates and the risk due to the increase of variance. A large scale Monte Carlo investigation is carried out to show that -2 InA has an empirical chi-square distribution with fractional degrees of freedom instead of a chi-square distribution with one degree of freedom. Also Monte Carlo power curves are investigated under several different conditions to evaluate the performances of several conventional procedures with that of this procedure with respect to control over Type I errors and power.     

Efficient sample size allocation with cost constraints for heterogeneous-variance group comparison

When conducting research with controlled experiments, sample size planning is one of the important decisions that researchers have to make. However, current methods do not adequately address this issue with regard to variance heterogeneity with some cost constraints for comparing several treatment means. This paper proposes a sample size allocation ratio in the fixed-effect heterogeneous analysis of variance when group variances are unequal and in cases where the sampling and/or variable cost has some constraints. The efficient sample size allocation is determined for the purpose of minimizing total cost with a designated power or maximizing the power with a given total cost. Finally, the proposed method is verified by using the index of relative efficiency and the corresponding total cost and the total sample size needed. We also apply our method in a pain management trial to decide an efficient sample size. Simulation studies also show that the proposed sample size formulas are efficient in terms of statistical power. SAS and R codes are provided in the appendix for easy application.     

A solution to the multivariate behrens-fisher problem

Some new algebra on pattern and transition matrices is used to determine the degrees of freedom and the parameter matrix, if the distribution of a linear sum of Wishart matrices is approximated by a single Wishart distribution. This approximation is then used to find a solution to the multivariate Behrens-Fisher problem similar to the Welch (1947) solution in the univariate case.     

Behrens-Fisher问题的正态逼近

 本文提出用基于得分检验的正态逼近方法来解决Behrens-Fisher问题,即比较方差比未知时两正态总体的均值。模拟结果显示：在所有的研究情况下,我们的方法都能很好地控制第一类错误,检验功效也不差;而最常用的Welch近似t检验在样本量不等时大多数情况都不能控制第一类错误。     

A four-moment approach and other practical solutions to the behrens-f1sher problem

Fixed sample size approximately similar tests for the Behrens-Fisher problem are studied and compared with various other tests suggested in current sttistical methodelogy texts. Several fourmoment approxiamtely similar tests are developed and offered as alternatives. These tests are shown to be good practical solutions which are easily implemented in practice.     

A fitted test for the behrens-fisher problem

In 1975, Lee and Gurland proposed a solution to the Behrens-Fisher problem. It had excellent control of size and power and was relatively simple to use. However it requires extensive special tables. This article proposes a modification of this approach. It replaces the tables with easily computed functions of the sample sizes and a standard t table. Control of size and power are equivalent to that obtained by Lee and Gurland. Furthermore, the test is also compared with the Welch's approximate t test and shows better control of size, with similar power curves when sample sizes are at least four from each of the two normal populations.     

A modification of the kruskal-wallis statistic for the generalized behrens-fisher problem

The problem of testing the equality of the medians of several populations is considered. Standard distribution-free procedures for this problem require that the populations have the same shape in order to maintain their nominal significance level, ever asymptotically, under the null hypothesis of equal medians , A modification of the Kruskal-Wallis test statistic is proposed which is exactly distribution-free under the usual nonparanetric asswnption that the continuous populations are identical with any shape. It is asymptotically distribution-free when the Continuous populations are asswned to be syrmnetric with equal medians.     

Modifications of the Mack-Wolfe umbrella tests for a generalized Behrens-Fisher problem

We are concerned with testing procedures for umbrella alternatives in the k-sample location problem without making the assumption that the underlying populations have the same shape. Modifications of the Mack-Wolfe tests are proposed for the cases when the peak of the umbrella is known or unknown. The proposed procedures are exactly distribution-free when the continuous populations have the same shape. The modified test for peak-known umbrella alternatives remains asymptotically distribution-free when the continuous populations are symmetric, but not necessarily with the same shape.     

Optimizing the wald-wolfowitz runs statistic using a linkage tolerance: guidelines based on computer simulation

When performing the Wald-Wolfowitz runs test, observations from two samples are combined and ordered, and the test statistic is the number of sequences of observations from the same sample. This test statistic is equivalent to the number of links between observations from different samples, if we consider each observation to be linked to the next higher and next lower observations. While it is known that the Wald-Wolfowitz runs test is not very powerful, what would be the effect on the power of the Wald-Wolfowitz runs test if all observations within a specified Euclidean distance or “tolerance” were linked instead? This question is motivated by the simulation results of Whaley and Quade (1985), who found that for normal data, the power of the multi-dimensional runs test using a linkage tolerance compared favorably to Hotelling's T² in some instances. The results of a similar simulation procedure show that the power of the Wald-Wolfowitz runs test does indeed improve when observations are linked using a tolerance. The results also suggest that a better large sample approximation to the distribution of the test statistic needs to be found.     

On the use of the t-statistic in preliminary testing procedures for the Behrens-Fisher problem

Preliminary testing procedures for the two means problem traditionally employ the pooled variance t-statistic. In this paper we show that bias of the t-statistic under conditions of heterogeneity of variance may be increased if use of the t-statistic is conditional on an affirmative F-test. For this reason we conclude that use of the t-statistic in preliminary testing procedures is inappropriate.     

Simulated Power Curves for some Location Tests

In this paper we compare the power properties of some location tests. The most widely used such test is Student's t. Recently bootstrap-based tests have received much attention in the literature. A bootstrap version of the t-test will be included in our comparison. Finally, the nonparametric tests based on the idea of permuting the signs will be represented in our comparison. Again, we will initially concentrate on a version of that test based on the mean. The permutation tests predate the bootstrap by about fourty years. Theoretical results of Pitman (1937) and Bickel & Freedman (1981) show that these three methods are asymptotically equivalent if the underlying distribution is symmetric and has finite second moment. In the modern literature, the use of the nonparametric techniques is advocated on the grounds that the size of the test would be either exact, or more nearly exact. In this paper we report on a simulation study that compares the power curves and we show that it is not necessary to use resampling tests with a statistic based on the mean of the sample.     

Testing the equality of two normal percentiles

Two statistics are suggested for testing the equality of two normal percentiles where population means and variances are unknown. The first is based on the generalized likelihood ratio test (LRT), the second on Cochran's statistic used in the Behrens-Fisher problem. Size and power comparisons are made by using simulation and asympototic theory.     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient     

Inferences on regression coefficients in a regression model under heteroscedasticity and robustness with respect to departure from normality

In this paper we consider a simple linear regression model under heteroscedasticity and nonnormality. A statistical test for testing the regression coefficient is then derived by assuming normality for the random disturbances and by applying Welch's method. Some Monte Carlo studies are generated for assessing robustness of this test. By combining Tiku's robust procedure with the new test, a robust but more powerful test is developed.     

An efficient algorithm for computing the noncentrality parameters of chi-squared tests

Use of Newton's method for computing the noncentrality parameter based on the specified power in sample size problems of chi-squared tests requires that we evaluate both the noncentral chi-squareddistribution function and its derivative with respect to the noncentrality parameter. A close relationship between computing formulas for them is revealed, by which their evaluations can be performed jointly. This property greatly reduces the amount of computation involved. The corresponding algorithm is provided in a step-by-step form.     

Rank tests for the k-sample problem with restricted alternatives

An asymptotically maximin most powerful rank test among somewhere asymptotically most powerful linear rank tests with scores generating function cf> is derived for each of the simple order alternative, the simple loop alternative and the simple tree alternative in the k-sample problem. The comparisons of the tests obtained with the rank analogues of the Bartholomew's xv tests are made in terms of local asymptotic relative efficiency. It is found that our tests are better than the rank analogues of the xk tests. Furthermore, the asymptotic equivalence of the ranking by the pooled sample to the ranking in pairs are discuss¬ed and the tests which are asymptotically equivalent to ours are given.     

Detecting a Change in the Correlation Coefficient in a Sequence of Bivariate Normal Variables

We consider the problem of testing the hypothesis that the correlation coefficient is stable in a sequence of n observations of independent, bivariate normal random variables against the alternative that the correlation coefficient changes after an unknown point t(t < n). We propose an estimate of the changepoint t and report on power comparisons between the commonly used test for this problem and our proposed test. Some applications to finance are discussed.     

Levels of significance of some two-sample tests wkem observations are from compound normal distributions

This paper presents an investigation of the behavior of the levels of significance of the two-sample t and its related tests and the Mann-Whitney test when the samples are randomly drawn from mixtures of two normal populations (compound normals) and when the sample sizes are small (combined sample sizes ? 15). The use of the compound normal allows for investigation when the underlying populations are unequal, nonnormal, heterogeneous in variances, unimodal or bimodal, possessing smaller than normal kurtosis or containing contamination. The exact distribution of the t and its related tests are given. However, they are not readily amenable to calculations. Most of the numerical results presented were obtained by simulations     

A smoothed em algorithm for the solution of wicksell's corpuscle problem

In this paper a new method called the EMS algorithm is used to solve Wicksell's corpuscle problem, that is the determination of the distribution of the sphere radii in a medium given the radii of their profiles in a random slice. The EMS algorithm combines the EM algorithm, a procedure for obtaining maximum likelihood estimates of parameters from incomplete data, with simple smoothing. The method is tested on simulated data from three different sphere radii densities, namely a bimodal mixture of Normals, a Weibull and a Normal. The effect of varying the level of smoothing, the number of classes in which the data is binned and the number of classes for which the estimated density is evaluated, is investigated. Comparisons are made between these results and those obtained by others in this field.     

Nonparametric tests of hypotheses for umbrella alternatives

The author proposes a general method for constructing nonparametric tests of hypotheses for umbrella alternatives. Such alternatives are relevant when the treatment effect changes in direction after reaching a peak. The author's class of tests is based on the ranks of the observations. His general approach consists of defining two sets of rankings: the first is induced by the alternative and the other by the data itself. His test statistic measures the distance between the two sets. The author determines the asymptotic distribution for some special cases of distances under both the null and the alternative hypothesis when the location of the peak is known or unknown. He shows the good power of his tests through a limited simulation study