The one- and multi-sample problem for functional data with application to projective shape analysis

In this paper tests are derived for testing neighborhood hypotheses for the one- and multi-sample problem for functional data. Our methodology is used to generalize testing in projective shape analysis, which has traditionally involving data consisting of finite number of points, to the functional case. The one-sample test is applied to the problem of scene identification, in the context of the projective shape of a planar curve.     

Tests for independence in non-parametric heteroscedastic regression models

Consistent procedures are constructed for testing independence between the regressor and the error in non-parametric regression models. The tests are based on the Fourier formulation of independence, and utilize the joint and the marginal empirical characteristic functions of the regressor and of estimated residuals. The asymptotic null distribution as well as the behavior of the test statistic under alternatives is investigated. A simulation study compares bootstrap versions of the proposed tests to corresponding procedures utilizing the empirical distribution function.     

A weighted multivariate signed-rank test for cluster-correlated data

A weighted multivariate signed-rank test is introduced for an analysis of multivariate clustered data. Observations in different clusters may then get different weights. The test provides a robust and efficient alternative to normal theory based methods. Asymptotic theory is developed to find the approximate p-value as well as to calculate the limiting Pitman efficiency of the test. A conditionally distribution-free version of the test is also discussed. The finite-sample behavior of different versions of the test statistic is explored by simulations and the new test is compared to the unweighted and weighted versions of Hotelling’s T² test and the multivariate spatial sign test introduced in [D. Larocque, J. Nevalainen, H. Oja, A weighted multivariate sign test for cluster-correlated data, Biometrika 94 (2007) 267-283]. Finally, a real data example is used to illustrate the theory.     

Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors

We consider a panel data semiparametric partially linear regression model with an unknown parameter vector for the linear parametric component, an unknown nonparametric function for the nonlinear component, and a one-way error component structure which allows unequal error variances (referred to as heteroscedasticity). We develop procedures to detect heteroscedasticity and one-way error component structure, and propose a weighted semiparametric least squares estimator (WSLSE) of the parametric component in the presence of heteroscedasticity and/or one-way error component structure. This WSLSE is asymptotically more efficient than the usual semiparametric least squares estimator considered in the literature. The asymptotic properties of the WSLSE are derived. The nonparametric component of the model is estimated by the local polynomial method. Some simulations are conducted to demonstrate the finite sample performances of the proposed testing and estimation procedures. An example of application on a set of panel data of medical expenditures in Australia is also illustrated.     

Structural test in regression on functional variables

Many papers deal with structural testing procedures in multivariate regression. More recently, various estimators have been proposed for regression models involving functional explanatory variables. Thanks to these new estimators, we propose a theoretical framework for structural testing procedures adapted to functional regression. The procedures introduced in this paper are innovative and make the link between former works on functional regression and others on structural testing procedures in multivariate regression. We prove asymptotic properties of the level and the power of our procedures under general assumptions that cover a large scope of possible applications: tests for no effect, linearity, dimension reduction, …     

A new test for multivariate normality

We propose a new class of rotation invariant and consistent goodness-of-fit tests for multivariate distributions based on Euclidean distance between sample elements. The proposed test applies to any multivariate distribution with finite second moments. In this article we apply the new method for testing multivariate normality when parameters are estimated. The resulting test is affine invariant and consistent against all fixed alternatives. A comparative Monte Carlo study suggests that our test is a powerful competitor to existing tests, and is very sensitive against heavy tailed alternatives.     

Partially linear single-index beta regression model and score test

An important model in handling the multivariate data is the partially linear single-index regression model with a very flexible distribution—beta distribution, which is commonly used to model data restricted to some open intervals on the line. In this paper, the score test is extended to the partially linear single-index beta regression model. The penalized likelihood estimation based on P-spline is proposed. Based on the estimation, the score test statistics about varying dispersion parameter is given. Its asymptotical property is investigated. Both simulated examples are used to illustrate our proposed methods.     

Statistical inference of partially linear regression models with heteroscedastic errors

The authors study a heteroscedastic partially linear regression model and develop an inferential procedure for it. This includes a test of heteroscedasticity, a two-step estimator of the heteroscedastic variance function, semiparametric generalized least-squares estimators of the parametric and nonparametric components of the model, and a bootstrap goodness of fit test to see whether the nonparametric component can be parametrized.     

Analysis of two-sample truncated data using generalized logistic models

Parallel to Cox's [JRSS B34 (1972) 187-230] proportional hazards model, generalized logistic models have been discussed by Anderson [Bull. Int. Statist. Inst. 48 (1979) 35-53] and others. The essential assumption is that the two densities ratio has a known parametric form. A nice property of this model is that it naturally relates to the logistic regression model for categorical data. In astronomic, demographic, epidemiological, and other studies the variable of interest is often truncated by an associated variable. This paper studies generalized logistic models for the two-sample truncated data problem, where the two lifetime densities ratio is assumed to have the form exp{α+φ(x;β)}. Here φ is a known function of x and β, and the baseline density is unspecified. We develop a semiparametric maximum likelihood method for the case where the two samples have a common truncation distribution. It is shown that inferences for β do not depend the nonparametric components. We also derive an iterative algorithm to maximize the semiparametric likelihood for the general case where different truncation distributions are allowed. We further discuss how to check goodness of fit of the generalized logistic model. The developed methods are illustrated and evaluated using both simulated and real data.     

Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis

Motivated by the likelihood functions of several incomplete categorical data, this article introduces a new family of distributions, grouped Dirichlet distributions (GDD), which includes the classical Dirichlet distribution (DD) as a special case. First, we develop distribution theory for the GDD in its own right. Second, we use this expanded family as a new tool for statistical analysis of incomplete categorical data. Starting with a GDD with two partitions, we derive its stochastic representation that provides a simple procedure for simulation. Other properties such as mixed moments, mode, marginal and conditional distributions are also derived. The general GDD with more than two partitions is considered in a parallel manner. Three data sets from a case-control study, a leprosy survey, and a neurological study are used to illustrate how the GDD can be used as a new tool for analyzing incomplete categorical data. Our approach based on GDD has at least two advantages over the commonly used approach based on the DD in both frequentist and conjugate Bayesian inference: (a) in some cases, both the maximum likelihood and Bayes estimates have closed-form expressions in the new approach, but not so when they are based on the commonly-used approach; and (b) even if a closed-form solution is not available, the EM and data augmentation algorithms in the new approach converge much faster than in the commonly-used approach.     

Analysis of correlated binary data under partially linear single-index logistic models

Clustered data arise commonly in practice and it is often of interest to estimate the mean response parameters as well as the association parameters. However, most research has been directed to address the mean response parameters with the association parameters relegated to a nuisance role. There is relatively little work concerning both the marginal and association structures, especially in the semiparametric framework. In this paper, our interest centers on the inference of both the marginal and association parameters. We develop a semiparametric method for clustered binary data and establish the theoretical results. The proposed methodology is investigated through various numerical studies.     

Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors

We develop optimal rank-based procedures for testing affine-invariant linear hypotheses on the parameters of a multivariate general linear model with elliptical VARMA errors. We propose a class of optimal procedures that are based either on residual (pseudo-)Mahalanobis signs and ranks, or on absolute interdirections and lift-interdirection ranks, i.e., on hyperplane-based signs and ranks. The Mahalanobis versions of these procedures are strictly affine-invariant, while the hyperplane-based ones are asymptotically affine-invariant. Both versions generalize the univariate signed rank procedures proposed by Hallin and Puri (J. Multivar. Anal. 50 (1994) 175), and are locally asymptotically most stringent under correctly specified radial densities. Their AREs with respect to Gaussian procedures are shown to be convex linear combinations of the AREs obtained in Hallin and Paindaveine (Ann. Statist. 30 (2002) 1103; Bernoulli 8 (2002) 787) for the pure location and purely serial models, respectively. The resulting test statistics are provided under closed form for several important particular cases, including multivariate Durbin-Watson tests, VARMA order identification tests, etc. The key technical result is a multivariate asymptotic linearity result proved in Hallin and Paindaveine (Asymptotic linearity of serial and nonserial multivariate signed rank statistics, submitted).     

Nonparametric methods for unbalanced multivariate data and many factor levels

We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large a, small n_i case). Quasi gratis, some new parametric multivariate tests suitable for the large a asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.     

Moving estimates test with time varying bandwidth

In this paper, we consider the problem of testing for parameter changes in time series models based on a moving estimates (ME) test. It is widely accepted that detecting some changes, for instance, those caused by temporary parameter shifts by the existing cusum test is difficult. A MV test with a fixed bandwidth has been developed to circumvent the defect, but the test still does not perform well under certain conditions. Motivated by this, we propose a MV test with a time varying bandwidth to outperform the original test. In order to illustrate our findings, we have provided simulation results.     

Quantile regression for longitudinal data

The penalized least squares interpretation of the classical random effects estimator suggests a possible way forward for quantile regression models with a large number of “fixed effects”. The introduction of a large number of individual fixed effects can significantly inflate the variability of estimates of other covariate effects. Regularization, or shrinkage of these individual effects toward a common value can help to modify this inflation effect. A general approach to estimating quantile regression models for longitudinal data is proposed employing ?₁ regularization methods. Sparse linear algebra and interior point methods for solving large linear programs are essential computational tools.     

Robust estimation in generalized semiparametric mixed models for longitudinal data

In this paper, we consider robust generalized estimating equations for the analysis of semiparametric generalized partial linear mixed models (GPLMMs) for longitudinal data. We approximate the non-parametric function in the GPLMM by a regression spline, and make use of bounded scores and leverage-based weights in the estimating equation to achieve robustness against outliers and influential data points, respectively. Under some regularity conditions, the asymptotic properties of the robust estimators are investigated. To avoid the computational problems involving high-dimensional integrals in our estimators, we adopt a robust Monte Carlo Newton-Raphson (RMCNR) algorithm for fitting GPLMMs. Small simulations are carried out to study the behavior of the robust estimates in the presence of outliers, and these estimates are also compared to their corresponding non-robust estimates. The proposed robust method is illustrated in the analysis of two real data sets.     

A generalized Hollander-Proschan type test for NBUE alternatives

In this note we develop a family of test statistics for testing exponentiality against NBUE alternatives. The asymptotic distribution of the test statistics is derived. The test statistics are shown to be asymptotically normal and consistent. This family of test statistics includes the test proposed by Hollander and Proschan (1975) as a special case. Efficiency studies have also been done.     

Nonparametric lack-of-fit tests for parametric mean-regression models with censored data

We developed two kernel smoothing based tests of a parametric mean-regression model against a nonparametric alternative when the response variable is right-censored. The new test statistics are inspired by the synthetic data and the weighted least squares approaches for estimating the parameters of a (non)linear regression model under censoring. The asymptotic critical values of our tests are given by the quantiles of the standard normal law. The tests are consistent against fixed alternatives, local Pitman alternatives and uniformly over alternatives in Hölder classes of functions of known regularity.     

Complexity-penalized estimation of minimum volume sets for dependent data

A minimum volume (MV) set, at level α, is a set having minimum volume among all those sets containing at least α probability mass. MV sets provide a natural notion of the ‘central mass’ of a distribution and, as such, have recently become popular as a tool for the detection of anomalies in multivariate data. Motivated by the fact that anomaly detection problems frequently arise in settings with temporally indexed measurements, we propose here a new method for the estimation of MV sets from dependent data. Our method is based on the concept of complexity-penalized estimation, extending recent work of Scott and Nowak for the case of independent and identically distributed measurements, and has both desirable theoretical properties and a practical implementation. Of particular note is the fact that, for a large class of stochastic processes, choice of an appropriate complexity penalty reduces to the selection of a single tuning parameter, which represents the data dependency of the underlying stochastic process. While in reality the dependence structure is unknown, we offer a data-dependent method for selecting this parameter, based on subsampling principles. Our work is motivated by and illustrated through an application to the detection of anomalous traffic levels in Internet traffic time series.