A graphical method for testing the equality of several variances

SUMMARY The problem of testing the equality of several variances arises in many areas. For testing the equality of variances, several tests are available in the literature which demonstrate only the statistical significance of the variances. In this paper, a graphical method is presented for testing the equality of variances. This method simultaneously demonstrates the statistical and engineering significance. Two examples are given to illustrate the proposed graphical method, and the conclusions obtained are compared with the existing tests.     

Comparison of Nonparametric Regression Curves by Spline Smoothing

In this article, procedures are proposed to test the hypothesis of equality of two or more regression functions. Tests are proposed by p-values, first under homoscedastic regression model, which are derived using fiducial method based on cubic spline interpolation. Then, we construct a test in the heteroscedastic case based on Fisher's method of combining independent tests. We study the behaviors of the tests by simulation experiments, in which comparisons with other tests are also given. The proposed tests have good performances. Finally, an application to a data set are given to illustrate the usefulness of the proposed test in practice.     

Smooth Tests of Copula Specifications

We present a family of smooth tests for the goodness of fit of semiparametric multivariate copula models. The proposed tests are distribution free and can be easily implemented. They are diagnostic and constructive in the sense that when a null distribution is rejected, the test provides useful pointers to alternative copula distributions. We then propose a method of copula density construction, which can be viewed as a multivariate extension of Efron and Tibshirani. We further generalize our methods to the semiparametric copula-based multivariate dynamic models. We report extensive Monte Carlo simulations and three empirical examples to illustrate the effectiveness and usefulness of our method.     

DATA‐DRIVEN SMOOTH TESTS AND A DIAGNOSTIC TOOL FOR LACK‐OF‐FIT FOR CIRCULAR DATA

Two contributions to the statistical analysis of circular data are given. First we construct data‐driven smooth goodness‐of‐fit tests for the circular von Mises assumption. As a second method, we propose a new graphical diagnostic tool for the detection of lack‐of‐fit for circular distributions. We illustrate our methods on two real datasets.     

Improved likelihood-based inference for the MA(1) model

An improved likelihood-based method is proposed to test for the significance of the first-order moving average model. Compared with commonly used tests which depend on the asymptotic properties of the maximum likelihood estimate and the likelihood ratio statistic, the proposed method has remarkable accuracy. Application of the method to a data set on book sales is presented to demonstrate the implementation of the method. Simulation studies are subsequently performed to illustrate the accuracy of the method compared to the traditional methods. Additionally, a simple and effective correction is used to deal with the boundary problem.     

A sequential conditional probability ratio test procedure for comparing diagnostic tests

In this paper, we derive sequential conditional probability ratio tests to compare diagnostic tests without distributional assumptions on test results. The test statistics in our method are nonparametric weighted areas under the receiver-operating characteristic curves. By using the new method, the decision of stopping the diagnostic trial early is unlikely to be reversed should the trials continue to the planned end. The conservatism reflected in this approach to have more conservative stopping boundaries during the course of the trial is especially appealing for diagnostic trials since the end point is not death. In addition, the maximum sample size of our method is not greater than a fixed sample test with similar power functions. Simulation studies are performed to evaluate the properties of the proposed sequential procedure. We illustrate the method using data from a thoracic aorta imaging study.     

Testing the rate ratio under inverse sampling based on gradient statistic

Inverse sampling is widely applied in studies with dichotomous outcomes, especially when the subjects arrive sequentially or the response of interest is difficult to obtain. In this paper, we investigate the rate ratio test problem under inverse sampling based on gradient statistic with the asymptotic method and parametric bootstrap technique. The gradient statistic has many advantages, for example, it is simple to calculate and competitive with Wald-type, score and likelihood ratio tests in terms of local power. Numerical studies are carried out to evaluate the performance of our gradient test and the existing tests, namely Wald-type, score and likelihood ratio tests. The simulation results suggest that the gradient test based on the parametric bootstrap method has excellent type I error control and large powers even in small sample design. Two real examples, from a heart disease study and a drug comparison study, are applied to illustrate our methods.     

A Note on a Partition of the Likelihood Ratio Test for Autoregressive Covariance Structure

A sequence of nested hypotheses is presented for the examination of the assumption of autoregressive covariance structure in, for example, a repeated measures experiment. These hypotheses arise naturally by specifying the joint density of the underlying vector random variable as a product of conditional densities and the density of a subset of the vector random variable. The tests for all but one of the nested hypotheses are well known procedures, namely analysis of variance F-tests and Bartlett's test of equality of variances. While the procedure is based on tests of hypotheses, it may be viewed as an exploratory tool which can lead to model identification. An example is presented to illustrate the method.     

Robust tests in nonlinear regression models

Robust tests for testing subhypotheses in nonlinear models are developed. These are drop-in-dispersion testing procedures, score-type and Wald-type testing procedures. The asymptotic properties and influence functions are obtained. Robust tests that perform well in the presence of heteroscedasticity are also developed. Simulation results are provided to illustrate these procedures.     

Graphical evaluation of the adequacy of the method of unweighted means

A graphical technique is introduced to assess the adequacy of the method of unweighted means in providing approximate F -tests for an unbalanced random model. These tests are similar to those obtained under a balanced ANOVA. The proposed technique is simple and can easily be used to determine the effects of imbalance and values of the variance components on the adequacy of the approximation. The one-way and two-way random models are used to illustrate the proposed methodology. Extensions to higher-order models are also mentioned.     

Testing Constancy for Isotonic Regressions

Abstract.  In this paper, we propose a bootstrap method for testing the constancy of an isotonic regression. The technique we develop is completely non-parametric and enlarges the appli-cability of the classical chi-bar-squared tests, which require normality assumptions. We prove that our procedure is asymptotically correct and consistent. Moreover, by means of simulations we show that it behaves suitably in practice, and similarly to the chi-bar-squared tests under normality. Finally, we illustrate the method with the study of a real case that is well known in the related literature.     

A class of tests of proportional hazards assumption for left-truncated and right-censored data

In this paper, we proposed a class of tests of proportional hazards assumption for left-truncated and right-censored data based on a pair of estimators of the hazard ratio constant. Using counting process and martingale theory, the asymptotically normal distribution of the test statistic is derived and a family of consistent estimators of variance are also provided. Extensive simulation studies were conducted to evaluate the performance of the proposed test statistics under finite sample situations. Two real data sets are analyzed to illustrate our method.     

A class of exact UMP unbiased tests for conditional symmetry in small-sample square contingency tables

Testing conditional symmetry against various alternative diagonals-parameter symmetry models often provides a point of departure in studies of square contingency tables with ordered categories. Typically, chi-square or likelihood-ratio tests are used for such purposes. Since these tests depend on the validity of asymptotic approximation, they may be inappropriate in small-sample situations where exact tests are required. In this paper, we apply the theory of UMP unbiased tests to develop a class of exact tests for conditional symmetry in small samples. Oesophageal cancer and longitudinal income data are used to illustrate the approach.     

Direct estimation of the percentile 'p-value' for the one-sample median test

In this paper we outline and illustrate an easy-to-use inference procedure for directly calculating the approximate bootstrap percentile-type p-value for the one-sample median test, i.e. we calculate the bootstrap p -value without resampling, by using a fractional order statistics based approach. The method parallels earlier work on fractionalorder-statistics-based non-parametric bootstrap percentile-type confidence intervals for quantiles. Monte Carlo simulation studies are performed, which illustrate that the fractional-order-statistics-based approach to the one-sample median test has accurate type I error control for small samples over a wide range of distributions; is easy to calculate; and is preferable to the sign test in terms of type I error control and power. Furthermore, the fractional-order-statistics-based median test is easily generalized to testing that any quantile has some hypothesized value; for example, tests for the upper or lower quartile may be performed using the same framework.     

ROC curve and covariates: extending induced methodology to the non-parametric framework

Continuous diagnostic tests are often used to discriminate between diseased and healthy populations. The receiver operating characteristic (ROC) curve is a widely used tool that provides a graphical visualisation of the effectiveness of such tests. The potential performance of the tests in terms of distinguishing diseased from healthy people may be strongly influenced by covariates, and a variety of regression methods for adjusting ROC curves has been developed. Until now, these methodologies have assumed that covariate effects have parametric forms, but in this paper we extend the induced methodology by allowing for arbitrary non-parametric effects of a continuous covariate. To this end, local polynomial kernel smoothers are used in the estimation procedure. Our method allows for covariate effect not only on the mean, but also on the variance of the diagnostic test. We also present a bootstrap-based method for testing for a significant covariate effect on the ROC curve. To illustrate the method, endocrine data were analysed with the aim of assessing the performance of anthropometry for predicting clusters of cardiovascular risk factors in an adult population in Galicia (NW Spain), duly adjusted for age. The proposed methodology has proved useful for providing age-specific thresholds for anthropometric measures in the Galician community.     

Tests of goodness of fit based on Phi-divergence

In this paper, we introduce a general goodness of fit test based on Phi-divergence. Consistency of the proposed test is established. We then study some special cases of tests for normal, exponential, uniform and Laplace distributions. Through Monte Carlo simulations, the power values of the proposed tests are compared with some known competing tests under various alternatives. Finally, some numerical examples are presented to illustrate the proposed procedure.     

Equivalence of two tests in varying coefficient partially linear errors in variable model with missing responses

This paper considers two tests on varying coefficient partially linear errors-in-variables models (VCPLM-EV) with missing responses under the linear constraint. The restricted estimator for the parametric component is derived and proven to share asymptotically normal distribution. In order to test the linear constraint, two statistics based on the profile Lagrange multiplier method and the corrected residual sum of squares method respectively, are proposed. It is of interest to obtain that the magnitudes of the two statistics are equal exactly and follow the asymptotical chi-square distribution. This reveals a new type of Wilk’s phenomenon in VCPLM-EV models with missing response. Finally, some numerical examples are carried out to illustrate relevant performances.     

A basis approach to goodness-of-fit testing in recurrent event models

A class of tests for the hypothesis that the baseline hazard function in Cox's proportional hazards model and for a general recurrent event model belongs to a parametric family C identical with {lambda(0)(.; xi): xi in Xi} is proposed. Finite properties of the tests are examined via simulations, while asymptotic properties of the tests under a contiguous sequence of local alternatives are studied theoretically. An application of the tests to the general recurrent event model, which is an extended minimal repair model admitting covariates, is demonstrated. In addition, two real data sets are used to illustrate the applicability of the proposed tests.     

非参数检验在市场调查中的应用

本文分析了非参数检验在市场调查中应用,并结合一些市场调查的数据资料对几种常用的非参数检验方法做了实征分析。