Modified Wald statistics for generalized linear models

Summary: Wald statistics in generalized linear models are asymptotically ² distributed. The asymptotic chi–squared law of the corresponding quadratic form shows disadvantages with respect to the approximation of the finite–sample distribution. It is shown by means of a comprehensive simulation study that improvements can be achieved by applying simple finite–sample size approximations to the distribution of the quadratic form in generalized linear models. These approximations are based on a ² distribution with an estimated degree of freedom that generalizes an approach by Patnaik and Pearson. Simulation studies confirm that nominal level is maintained with higher accuracy compared to the Wald statistics.     

On special linear identities for order statistics

Linear identities for the distribution functions of order statistics from an iid sample are defined. It is shown that such identities are true for all distributions or to some discrete distributions taking a finite number of values.     

Optimal estimation in functional linear regression for sparse noise‐contaminated data

In this article, we propose a novel approach to fit a functional linear regression in which both the response and the predictor are functions. We consider the case where the response and the predictor processes are both sparsely sampled at random time points and are contaminated with random errors. In addition, the random times are allowed to be different for the measurements of the predictor and the response functions. The aforementioned situation often occurs in longitudinal data settings. To estimate the covariance and the cross‐covariance functions, we use a regularization method over a reproducing kernel Hilbert space. The estimate of the cross‐covariance function is used to obtain estimates of the regression coefficient function and of the functional singular components. We derive the convergence rates of the proposed cross‐covariance, the regression coefficient, and the singular component function estimators. Furthermore, we show that, under some regularity conditions, the estimator of the coefficient function has a minimax optimal rate. We conduct a simulation study and demonstrate merits of the proposed method by comparing it to some other existing methods in the literature. We illustrate the method by an example of an application to a real‐world air quality dataset. The Canadian Journal of Statistics 47: 524–559; 2019 © 2019 Statistical Society of Canada     

Optimal designs for generalized linear models

Summary This paper solves some D-optimal design problems for certain Generalized Linear Models where the mean depends on two parameters and two explanatory variables. In all of the cases considered the support point of the optimal designs are found to be independent of the unknown parameters. While in some cases the optimal design measures are given by two points with equal weights, in others the support is given by three point with weights depending on the unknown parameters, hence the designs are locally optimal in general. Empirical results on the efficiency of the locally optimal designs are also given. Some of the designs found can also be used for planning D-optimal experiments for the normal linear model, where the mean must be positive. This research was carried out in part at University College, London as an M.Sc. project. Thanks are due to Prof. I. Ford (University of Glasgow) and Prof. A. Giovagnoli (University of Perugia) for their valuable suggestions and critical observations.     

Optimal allocation for linear combinations of linear predictors in generalized linear models

The problem of the allocation of experimental units to experimental groups is studied within the context of generalized linear models. Optimal designs for the estimation of linear combinations of linear predictors are characterized, using concepts from the theory of optimal design. If there is only one linear combination of interest, then the D-optimal allocation is equivalent to the well-known Neyman allocation of subsamples in stratified sampling. However, if the number of linear combinations equals the number of design points, or experimental groups, then the equal replication of all design points is D-optimal. For cases in between, there are no easily accessible general solutions to the problem, although some particular cases are solved, including: i estimation of the n- 1 possible comparisons with a control group in an n-point, one-factor design; and ii estimation of 2 one or two of the four natural parameters of a 2 factorial design. The A-optimal allocations are determined in general.     

Optimal testing for equicorrelated linear regression models

Recently, Knautz and Trenkler (1993) considered Christensen’s (1987) equicorrelated linear regression model as an example to show that S² and are independent even though the disturbances are equicorrelated. This paper addresses the issue of testing for the equicorrelation coefficient in the linear regression model based on survey data. It computes exact and approximate critical values using Point optimal and F-test statistics, respectively. An empirical comparison of these critical values at five percent nominal level are presented to demonstrate the performance of the new tests.     

Inhomogeneous higher-order summary statistics for point processes on linear networks

Statistics and Computing - As a workaround for the lack of transitive transformations on linear network structures, which are required to consider different notions of distributional invariance,...     

Generalized cochran-mantel-haenszel test statistics for correlated categorical data

Three new test statistics are introduced for correlated categorical data in stratified R×C tables. They are similar in form to the standard generalized Cochran-Mantel-Haenszel statistics but modified to handle correlated outcomes. Two of these statistics are asymptotically valid in both many-strata (sparse data) and large-strata limiting models. The third one is designed specifically for the many-strata case but is valid even with a small number of strata. This latter statistic is also appropriate when strata are assumed to be random.     

Bayesian kernel projections for classification of high dimensional data

A Bayesian multi-category kernel classification method is proposed. The algorithm performs the classification of the projections of the data to the principal axes of the feature space. The advantage of this approach is that the regression coefficients are identifiable and sparse, leading to large computational savings and improved classification performance. The degree of sparsity is regulated in a novel framework based on Bayesian decision theory. The Gibbs sampler is implemented to find the posterior distributions of the parameters, thus probability distributions of prediction can be obtained for new data points, which gives a more complete picture of classification. The algorithm is aimed at high dimensional data sets where the dimension of measurements exceeds the number of observations. The applications considered in this paper are microarray, image processing and near-infrared spectroscopy data.     

Edgeworth expansion for signed linear rank statistics with regression constants

An Edgeworth expansion with remainder o(N^?1) is obtained for signed linear rank statistics under suitable assumptions. The theorem is proved for a wide class of score generating functions including the Chi-quantile function by adapting van Zwet's methodand Does's conditioning arguments.     

Best linear unbiased interpolation of order statistics

We derive the best linear unbiased interpolation for the missing order statistics of a random sample using the well-known projection theorem. The proposed interpolation method only needs the first two moments on both sides of a missing order statistic. A simulation study is performed to compare the proposed method with a few interpolation methods for exponential and Lévy distributions.     

Multivariate linear regression for heterogeneous data

The problem of multivariate regression modelling in the presence of heterogeneous data is dealt to address the relevant issue of the influence of such heterogeneity in assessing the linear relations between responses and explanatory variables. In spite of its popularity, clusterwise regression is not designed to identify the linear relationships within ‘homogeneous’ clusters exhibiting internal cohesion and external separation. A within-clusterwise regression is introduced to achieve this aim and, since the possible presence of a linear relation ‘between’ clusters should be also taken into account, a general regression model is introduced to account for both the between-cluster and the within-cluster regression variation. Some decompositions of the variance of the responses accounted for are also given, the least-squares estimation of the parameters is derived, together with an appropriate coordinate descent algorithms and the performance of the proposed methodology is evaluated in different datasets.     

Asymptotic distributions of some test statistics for a linear functional relationship

We consider the testing problems of the structural parameters for the multivariate linear functional relationship model. We treat the likelihood ratio test statistics and the test statistics based on the asymptotic distributions of the maximum likelihood estimators. We derive their asymptotic distributions under each null hypothesis respectively. A simulation study is made to evaluate how we can trust our asymptotic results when the sample size is rather small.     

Edgeworth expansions for signed linear rank statistics under near location alternatives

Edgeworth expansions with the uniform remainder of order o(N⁻¹) are established for signed linear rank statistics with regression constants under near location alternatives. The results are obtained both with exact scores and with approximate scores, normalized with natural parameters as well as with simple constants.     

On the distribution of summary statistics for missing data

Under an assumption that missing values occur randomly in a matrix, formulae are developed for the expected value and variance of six statistics that summarize the number and location of the missing values. For a seventh statistic, a regression model based on simulated data yields an estimate of the expected value. The results can be used in the development of methods to control the Type I error and approximate power and sample size for multilevel and longitudinal studies with missing data.     

Regression model for surrogate data in high dimensional statistics

Abstract

This paper deals with the problem of estimating the regression of a surrogated scalar response variable given a functional random one. We construct an estimator of the regression operator by using, in addition to the available (true) response data, a surrogate data. We then establish some asymptotic properties of the constructed estimator in terms of the almost-complete and the quadratic mean convergences. Notice that the obtained results generalize a part of the results obtained in the finite dimensional framework. Finally, an illustration on the applicability of our results on both simulated data and a real dataset was realized. We have thus shown the superiority of our estimator on classical estimators when we are lacking complete data.