Outlier detection in contingency tables using decomposable graphical models

For high-dimensional data, it is a tedious task to determine anomalies such as outliers. We present a novel outlier detection method for high-dimensional contingency tables. We use the class of decomposable graphical models to model the relationship among the variables of interest, which can be depicted by an undirected graph called the interaction graph. Given an interaction graph, we derive a closed-form expression of the likelihood ratio test (LRT) statistic and an exact distribution for efficient simulation of the test statistic. An observation is declared an outlier if it deviates significantly from the approximated distribution of the test statistic under the null hypothesis. We demonstrate the use of the LRT outlier detection framework on genetic data modeled by Chow–Liu trees.     

A resampling‐based test for two crossing survival curves

The area between two survival curves is an intuitive test statistic for the classical two‐sample testing problem. We propose a bootstrap version of it for assessing the overall homogeneity of these curves. Our approach allows ties in the data as well as independent right censoring, which may differ between the groups. The asymptotic distribution of the test statistic as well as of its bootstrap counterpart are derived under the null hypothesis, and their consistency is proven for general alternatives. We demonstrate the finite sample superiority of the proposed test over some existing methods in a simulation study and illustrate its application by a real‐data example.     

Nonlinear regression models with general distortion measurement errors

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.     

An F-type test for detecting departure from monotonicity in a functional linear model

When studying associations between a functional covariate and scalar response using a functional linear model (FLM), scientific knowledge may indicate possible monotonicity of the unknown parameter curve. In this context, we propose an F-type test of monotonicity, based on a full versus reduced nested model structure, where the reduced model with monotonically constrained parameter curve is nested within an unconstrained FLM. For estimation under the unconstrained FLM, we consider two approaches: penalised least-squares and linear mixed model effects estimation. We use a smooth then monotonise approach to estimate the reduced model, within the null space of monotone parameter curves. A bootstrap procedure is used to simulate the null distribution of the test statistic. We present a simulation study of the power of the proposed test, and illustrate the test using data from a head and neck cancer study.     

Two‐Sample Bootstrap Hypothesis Tests for Three‐Dimensional Labelled Landmark Data

Abstract. We investigate resampling methodologies for testing the null hypothesis that two samples of labelled landmark data in three dimensions come from populations with a common mean reflection shape or mean reflection size‐and‐shape. The investigation includes comparisons between (i) two different test statistics that are functions of the projection onto tangent space of the data, namely the James statistic and an empirical likelihood statistic; (ii) bootstrap and permutation procedures; and (iii) three methods for resampling under the null hypothesis, namely translating in tangent space, resampling using weights determined by empirical likelihood and using a novel method to transform the original sample entirely within refection shape space. We present results of extensive numerical simulations, on which basis we recommend a bootstrap test procedure that we expect will work well in practise. We demonstrate the procedure using a data set of human faces, to test whether humans in different age groups have a common mean face shape.     

Goodness-of-Fit for Longitudinal Count Data with Overdispersion

In this article, we develop a method for checking the estimation equations, which is for joint estimation of the regression parameters and the overdispersion parameters, based on one dimension projected covariate. This method is different from the general testing methods in that our proposed method can be applied to high-dimensional response while the classical testing methods can not be extended to high dimension problem simply to construct a powerful test. Furthermore, the properties of the test statistics are investigated and Nonparametric Monte Carlo Test (NMCT) is suggested to determine the critical values of the test statistics under null hypothesis.     

Testing diagonality of high-dimensional covariance matrix under non-normality

Under non-normality, this article is concerned with testing diagonality of high-dimensional covariance matrix, which is more practical than testing sphericity and identity in high-dimensional setting. The existing testing procedure for diagonality is not robust against either the data dimension or the data distribution, producing tests with distorted type I error rates much larger than nominal levels. This is mainly due to bias from estimating some functions of high-dimensional covariance matrix under non-normality. Compared to the sphericity and identity hypotheses, the asymptotic property of the diagonality hypothesis would be more involved and we should be more careful to deal with bias. We develop a correction that makes the existing test statistic robust against both the data dimension and the data distribution. We show that the proposed test statistic is asymptotically normal without the normality assumption and without specifying an explicit relationship between the dimension p and the sample size n. Simulations show that it has good size and power for a wide range of settings.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

An improved test of equality of mean directions for the Langevin‐von Mises‐Fisher distribution

A multi‐sample test for equality of mean directions is developed for populations having Langevin‐von Mises‐Fisher distributions with a common unknown concentration. The proposed test statistic is a monotone transformation of the likelihood ratio. The high‐concentration asymptotic null distribution of the test statistic is derived. In contrast to previously suggested high‐concentration tests, the high‐concentration asymptotic approximation to the null distribution of the proposed test statistic is also valid for large sample sizes with any fixed nonzero concentration parameter. Simulations of size and power show that the proposed test outperforms competing tests. An example with three‐dimensional data from an anthropological study illustrates the practical application of the testing procedure.     

Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities

Summary.  We propose a general bootstrap procedure to approximate the null distribution of non-parametric frequency domain tests about the spectral density matrix of a multivariate time series. Under a set of easy-to-verify conditions, we establish asymptotic validity of the bootstrap procedure proposed. We apply a version of this procedure together with a new statistic to test the hypothesis that the spectral densities of not necessarily independent time series are equal. The test statistic proposed is based on an L ₂-distance between the non-parametrically estimated individual spectral densities and an overall, 'pooled' spectral density, the latter being obtained by using the whole set of m time series considered. The effects of the dependence between the time series on the power behaviour of the test are investigated. Some simulations are presented and a real life data example is discussed.     

Comparing Conditional Quantile Curves

Abstract. We consider the problem of testing the equality of J quantile curves from independent samples. A test statistic based on an L²‐distance between non‐crossing non‐parametric estimates of the quantile curves from the individual samples is proposed. Asymptotic normality of this statistic is established under the null hypothesis, local and fixed alternatives, and the finite sample properties of a bootstrap‐based version of this test statistic are investigated by means of a simulation study.     

The power of the circular cone test: A noncentral chi-bar-squared distribution

Pincus (1975) derived the null distribution of the likelihood-ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood-ratio test statistic has a chi-bar-squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood-ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi-bar-squared” distribution. To our knowledge, this is the first order-restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear-model setting. That is, the distribution of the likelihood-ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.     

Wavelet goodness-of-fit test for dependent data

We introduce a new goodness-of-fit test which can be applied to hypothesis testing about the marginal distribution of dependent data. We derive a new test for the equivalent hypothesis in the space of wavelet coefficients. Such properties of the wavelet transform as orthogonality, localisation and sparsity make the hypothesis testing in wavelet domain easier than in the domain of distribution functions. We propose to test the null hypothesis separately at each wavelet decomposition level to overcome the problem of bi-dimensionality of wavelet indices and to be able to find the frequency where the empirical distribution function differs from the null in case the null hypothesis is rejected. We suggest a test statistic and state its asymptotic distribution under the null and under some of the alternative hypotheses.     

On empirical likelihood for semiparametric two-sample density ratio models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a dual form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ$\phi$-divergences and duality techniques. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and approximation of the power function of the test is deduced.     

Hypothesis testing for two-group incomplete-data growth curve problems

Using simulation techniques, the null distribution properties of seven hypothesis testing procedures and a comparison of their powers are investigated for incomplete-data small-sample growth curve situations. The testing procedures are a combination of two growth curve models (the Potthoff and Roy model for complete data and Kleinbaum's extention to incomplete data) and three estimation techniques (two involving means of existing observations and the other using the EM algorithm) plus an analysis of a subset of complete data. All of the seven tests use the Kleinbaum Wald statistic, but different tests use different information. The hypotheses of identical and parallel growth curves are tested under the assumptions of multivariate normality and a linear polynomial mean growth curve for each of two groups. Good approximate null distributions are found for all procedures and one procedure is identified as empirically most powerful for the situations investigated.     

Bootstrapping in a high dimensional but very low-sample size problem

This article is concerned with testing multiple hypotheses, one for each of a large number of small data sets. Such data are sometimes referred to as high-dimensional, low-sample size data. Our model assumes that each observation within a randomly selected small data set follows a mixture of C shifted and rescaled versions of an arbitrary density f. A novel kernel density estimation scheme, in conjunction with clustering methods, is applied to estimate f. Bayes information criterion and a new criterion weighted mean of within-cluster variances are used to estimate C, which is the number of mixture components or clusters. These results are applied to the multiple testing problem. The null sampling distribution of each test statistic is determined by f, and hence a bootstrap procedure that resamples from an estimate of f is used to approximate this null distribution.     

Empirical likelihood test for equality of two distributions using distance of characteristic functions

In this paper, we investigate the problem of testing for the equality of two distributions. We employ a two-sample Jackknife Empirical Likelihood (JEL) approach to construct a test statistic whose limiting distribution is Chi-square distribution with degree of freedom 1, no matter what the data dimension (fixed) is. A variety of synthetic data experiments demonstrate that our JEL test statistic performs very well, with a very neat asymptotic distribution under the null hypothesis. Furthermore, we apply the test procedure to a real dataset to obtain competitive results.     

A simple test for comparing regression curves versus one-sided alternatives

In this article we present a simple procedure to test for the null hypothesis of equality of two regression curves versus one-sided alternatives in a general nonparametric and heteroscedastic setup. The test is based on the comparison of the sample averages of the estimated residuals in each regression model under the null hypothesis. The test statistic has asymptotic normal distribution and can detect any local alternative of rate n^-1/2. Some simulations and an application to a data set are included.     

Omnibus test of normality using the Q statistic

A new statistical procedure for testing normality is proposed. The Q statistic is derived as the ratio of two linear combinations of the ordered random observations. The coefficients of the linear combinations are utilizing the expected values of the order statistics from the standard normal distribution. This test is omnibus to detect the deviations from normality that result from either skewness or kurtosis. The statistic is independent of the origin and the scale under the null hypothesis of normality, and the null distribution of Q can be very well approximated by the Cornish-Fisher expansion. The powers for various alternative distributions were compared with several other test statistics by simulations.     

Testing the linearity of negative binomial regression models

The negative binomial (NB) is frequently used to model overdispersed Poisson count data. To study the effect of a continuous covariate of interest in an NB model, a flexible procedure is used to model the covariate effect by fixed-knot cubic basis-splines or B-splines with a second-order difference penalty on the adjacent B-spline coefficients to avoid undersmoothing. A penalized likelihood is used to estimate parameters of the model. A penalized likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the continuous covariate effect. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. The smoothing parameter value is determined by setting a specified value equal to the asymptotic expectation of the test statistic under the null hypothesis. The power performance of the proposed test is studied with simulation experiments.