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.     

Robust ANCOVA: Some Small-sample Results when there are Multiple Groups and Multiple Covariates

Numerous methods have been proposed for dealing with the serious practical problems associated with the conventional analysis of covariance method, with an emphasis on comparing two groups when there is a single covariate. Recently, Wilcox (2005a: section 11.8.2) outlined a method for handling multiple covariates that allows nonlinearity and heteroscedasticity. The method is readily extended to multiple groups, but nothing is known about its small-sample properties. This paper compares three variations of the method, each method based on one of three measures of location: means, medians and 20% trimmed means. The methods based on a 20% trimmed mean or median are found to avoid Type I error probabilities well above the nominal level, but the method based on medians can be too conservative in various situations; using a 20% trimmed mean gave the best results in terms of Type I errors. The methods are based in part on a running interval smoother approximation of the regression surface. Included are comments on required sample sizes that are relevant to the so-called curse of dimensionality.     

Rank-based empirical likelihood inference on medians of k populations

We propose a nonparametric method, called rank-based empirical likelihood (REL), for making inferences on medians and cumulative distribution functions (CDFs) of k populations. The standard distribution-free approach to testing the equality of k medians requires that the k population distributions have the same shape. Our REL-ratio (RELR) test for this problem requires fewer assumptions and can effectively use the symmetry information when the distributions are symmetric. Furthermore, our RELR statistic does not require estimation of variance, and achieves asymptotic pivotalness implicitly. When the k populations have equal medians we show that the REL method produces valid inferences for the common median and CDFs of k populations. Simulation results show that the REL approach works remarkably well in finite samples. A real data example is used to illustrate the proposed REL method.     

Calculating the simplex median

While much attention has recently focussed on the use of multivariate medians for estimating the centre of a data set, a particular median based upon random simplexes proposed by Liu has been overshadowed by progress on other multivariate medians. The purpose of this paper is to redress the balance, and to show that Liu's simplex median is tractable, and has distinctive desirable properties that recommend it for use in data analysis.     

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model.

We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.     

Prior distribution elicitation for generalized linear and piecewise-linear models

An elicitation method is proposed for quantifying subjective opinion about the regression coefficients of a generalized linear model. Opinion between a continuous predictor variable and the dependent variable is modelled by a piecewise-linear function, giving a flexible model that can represent a wide variety of opinion. To quantify his or her opinions, the expert uses an interactive computer program, performing assessment tasks that involve drawing graphs and bar-charts to specify medians and other quantiles. Opinion about the regression coefficients is represented by a multivariate normal distribution whose parameters are determined from the assessments. It is practical to use the procedure with models containing a large number of parameters. This is illustrated through practical examples and the benefit from using prior knowledge is examined through cross-validation.     

Comparing two independent groups via the lower and upper quantiles

The most common strategy for comparing two independent groups is in terms of some measure of location intended to reflect the typical observation. However, it can be informative and important to compare the lower and upper quantiles as well, but when there are tied values, extant techniques suffer from practical concerns reviewed in the paper. For the special case where the goal is to compare the medians, a slight generalization of the percentile bootstrap method performs well in terms of controlling Type I errors when there are tied values [Wilcox RR. Comparing medians. Comput. Statist. Data Anal. 2006;51:1934–1943]. But our results indicate that when the goal is to compare the quartiles, or quantiles close to zero or one, this approach is highly unsatisfactory when the quantiles are estimated using a single order statistic or a weighted average of two order statistics. The main result in this paper is that when using the Harrell–Davis estimator, which uses all of the order statistics to estimate a quantile, control over the Type I error probability can be achieved in simulations, even when there are tied values, provided the sample sizes are not too small. It is demonstrated that this method can also have substantially higher power than the distribution free method derived by Doksum and Sievers [Plotting with confidence: graphical comparisons of two populations. Biometrika 1976;63:421–434]. Data from two studies are used to illustrate the practical advantages of the method studied here.     

On simultaneous closeness probabilities of order statistics from odd sample sizes to the population median

In this note, we present alternative derivations for the probability that an individual order statistic is closest to the target parameter among all order statistics from a complete random sample. This approach is simpler than the geometric arguments used earlier. We also provide a simple direct proof for the symmetry property of the simultaneous closeness probabilities among order statistics for the estimation of percentiles from a symmetric family. Finally, we offer an alternative simpler proof for the result that sample medians from larger odd sample sizes are Pitman closer to the population median than sample medians from smaller odd sample sizes.     

On the Behrens–Fisher problem from the spatial median point of view

We present a non-parametric affine-invariant test for the multivariate Behrens–Fisher problem. The proposed method based on the spatial medians is asymptotic and does not require normality of the data. To improve its finite sample performance, we apply a correction of the type which was already used in a similar test based on trimmed means, however, our simulations show that in the case of heavy-tailed distributions our method performs better. Also in a simulation comparison with a recently published rank-based test our test yields satisfactory results.     

Distribution-Free Confidence Intervals for Difference and Ratio of Medians

The classic nonparametric confidence intervals for a difference or ratio of medians assume that the distributions of the response variable or the log-transformed response variable have identical shapes in each population. Asymptotic distribution-free confidence intervals for a difference and ratio of medians are proposed which do not require identically shaped distributions. The new asymptotic methods are easy to compute and simulation results show that they perform well in small samples.     

Variable Selection for Naive Bayes Semisupervised Learning

This article deals with a semisupervised learning based on naive Bayes assumption. A univariate Gaussian mixture density is used for continuous input variables whereas a histogram type density is adopted for discrete input variables. The EM algorithm is used for the computation of maximum likelihood estimators of parameters in the model when we fix the number of mixing components for each continuous input variable. We carry out a model selection for choosing a parsimonious model among various fitted models based on an information criterion. A common density method is proposed for the selection of significant input variables. Simulated and real datasets are used to illustrate the performance of the proposed method.     

Semiparametric estimation of treatment effect with density ratio model

In this study we propose a unified semiparametric approach to estimate various indices of treatment effect under the density ratio model, which connects two density functions by an exponential tilt. For each index, we construct two estimating functions based on the model and apply the generalized method of moments to improve the estimates. The estimating functions are allowed to be non smooth with respect to parameters and hence make the proposed method more flexible. We establish the asymptotic properties of the proposed estimators and illustrate the application with several simulations and two real data sets.     

Bernstein polynomial model for grouped continuous data

Grouped data are commonly encountered in applications. All data from a continuous population are grouped due to rounding of the individual observations. The Bernstein polynomial model is proposed as an approximate model in this paper for estimating a univariate density function based on grouped data. The coefficients of the Bernstein polynomial, as the mixture proportions of beta distributions, can be estimated using an EM algorithm. The optimal degree of the Bernstein polynomial can be determined using a change-point estimation method. The rate of convergence of the proposed density estimate to the true density is proved to be almost parametric by an acceptance–rejection argument used for generating random numbers. The proposed method is compared with some existing methods in a simulation study and is applied to the Chicken Embryo Data.     

Testing the equality of quantiles for several normal populations

Many procedures exist for testing equality of means or medians to compare several independent distributions. However, the mean or median do not determine the entire distribution. In this article, we propose a new small-sample modification of the likelihood ratio test for testing the equality of the quantiles of several normal distributions. The merits of the proposed test are numerically compared with the existing tests—a generalized p-value method and likelihood ratio test—with respect to their sizes and powers. The simulation results demonstrate that proposed method is satisfactory; its actual size is very close to the nominal level. We illustrate these approaches using two real examples.     

The Kumaraswamy Gumbel distribution

The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. We propose a generalization—referred to as the Kumaraswamy Gumbel distribution—and provide a comprehensive treatment of its structural properties. We obtain the analytical shapes of the density and hazard rate functions. We calculate explicit expressions for the moments and generating function. The variation of the skewness and kurtosis measures is examined and the asymptotic distribution of the extreme values is investigated. Explicit expressions are also derived for the moments of order statistics. The methods of maximum likelihood and parametric bootstrap and a Bayesian procedure are proposed for estimating the model parameters. We obtain the expected information matrix. An application of the new model to a real dataset illustrates the potentiality of the proposed model. Two bivariate generalizations of the model are proposed.     

On rules based on sample medians for selection of the largest location parameter

The problem of selection of a subset containing the largest of several location parameters is considered, and a Gupta-type selection rule based on sample medians is investigated for normal and double exponential populations. Numerical comparisons between rules based on medians and means of small samples are made for normal and contaminated normal populations, assuming the popula-tion means to be equally spaced. It appears that the rule based on sample means loses its superiority over the rule based on sample medians in case the samples are heavily contaminated. The asymptotic relative efficiency (ARE) of the medians procedure relative to the means procedure is also computed, assuming the normal means to be in a slippage configuration. The means proce-dure is found to be superior to the median procedure in the sense of ARE. As in the small sample case, the situation is reversed if the normal populations are highly contaminate.     

Using Adjacent-category Logits Procedure for Estimating Receiver Operating Characteristic Surface

This article presents a semiparametric method for estimating receiver operating characteristic surface under density ratio model. The construction of the proposed method is based on the adjacent-category logit model and the empirical likelihood approach. A bootstrap approach for the VUS estimator inference is presented. In a simulation study, the proposed estimator is compared with the existing parametric and nonparametric estimators in terms of bias, standard error, and mean square error. Finally, a real data example and some discussions on the proposed method are provided.     

An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions

The Inverse Gaussian (IG) distribution is commonly introduced to model and examine right skewed data having positive support. When applying the IG model, it is critical to develop efficient goodness-of-fit tests. In this article, we propose a new test statistic for examining the IG goodness-of-fit based on approximating parametric likelihood ratios. The parametric likelihood ratio methodology is well-known to provide powerful likelihood ratio tests. In the nonparametric context, the classical empirical likelihood (EL) ratio method is often applied in order to efficiently approximate properties of parametric likelihoods, using an approach based on substituting empirical distribution functions for their population counterparts. The optimal parametric likelihood ratio approach is however based on density functions. We develop and analyze the EL ratio approach based on densities in order to test the IG model fit. We show that the proposed test is an improvement over the entropy-based goodness-of-fit test for IG presented by Mudholkar and Tian (2002). Theoretical support is obtained by proving consistency of the new test and an asymptotic proposition regarding the null distribution of the proposed test statistic. Monte Carlo simulations confirm the powerful properties of the proposed method. Real data examples demonstrate the applicability of the density-based EL ratio goodness-of-fit test for an IG assumption in practice.     

Modeling Data with Flat Densities

In this article, a simple probability distribution for modeling data with flat densities is introduced and used to model real world assessment data. The new distribution is a mixture of three distributions, two truncated normals, and a uniform. The parameters are estimated by using the sample percentiles because the likelihood and the method of moment approaches result in complicated forms. The proposed method is mainly based on the shape of the empirical density. The proposed method looks promising for modeling flat densities that are similar to the one used in the study.     

Powerful Test of Two Proportions by Assuming a Registered Prior Density

A method for constructing powerful significance tests for the equivalence of two proportions is proposed by assuming prior density values. Recent changes in the medical research environment emphasize the need for choice of a prior density in advance of any study. The proposed test is based on the posterior probability of the alternative model and preserves the significance level with minimal reduction of power. The new test performs better than the familiar mid-p test under the uniform prior density condition. In addition, the computational burden is low. Potential extensions of the proposed test to related problems are also discussed.