On the sufficient statistics for multivariate ARMA models: approximate approach

This paper is an investigation on the sufficient statistic for the parameters of the vector-valued (multivariate) ARMA models, when a finite sample is available. In the simplest case ARMA(1,1), by using the factorization theorem, we present a sufficient statistic whose dimension depends on the sample size and this dimension is even larger than the sample size. In this case and under some restrictions, we have solved this problem and have presented a sufficient statistic whose dimension does not depend on the sample size. In the general case, due to the complexity of the problem, we will use the modified versions of the likelihood function to find an approximate sufficient statistic in terms of the periodogram. The dimension of this sufficient statistic depends on the sample size; however, this dimension is much lower than the sample size.     

Kernel naive Bayes discrimination for high‐dimensional pattern recognition

Kernel discriminant analysis translates the original classification problem into feature space and solves the problem with dimension and sample size interchanged. In high‐dimension low sample size (HDLSS) settings, this reduces the ‘dimension’ to that of the sample size. For HDLSS two‐class problems we modify Mika's kernel Fisher discriminant function which – in general – remains ill‐posed even in a kernel setting; see Mika et al. (1999). We propose a kernel naive Bayes discriminant function and its smoothed version, using first‐ and second‐degree polynomial kernels. For fixed sample size and increasing dimension, we present asymptotic expressions for the kernel discriminant functions, discriminant directions and for the error probability of our kernel discriminant functions. The theoretical calculations are complemented by simulations which show the convergence of the estimators to the population quantities as the dimension grows. We illustrate the performance of the new discriminant rules, which are easy to implement, on real HDLSS data. For such data, our results clearly demonstrate the superior performance of the new discriminant rules, and especially their smoothed versions, over Mika's kernel Fisher version, and typically also over the commonly used naive Bayes discriminant rule.     

Testing identity of high-dimensional covariance matrix

Two new statistics are proposed for testing the identity of high-dimensional covariance matrix. Applying the large dimensional random matrix theory, we study the asymptotic distributions of our proposed statistics under the situation that the dimension p and the sample size n tend to infinity proportionally. The proposed tests can accommodate the situation that the data dimension is much larger than the sample size, and the situation that the population distribution is non-Gaussian. The numerical studies demonstrate that the proposed tests have good performance on the empirical powers for a wide range of dimensions and sample sizes.     

Selection of Variables in Multivariate Regression Models for Large Dimensions

The Akaike information criterion, AIC, and Mallows’ C _p statistic have been proposed for selecting a smaller number of regressors in the multivariate regression models with fully unknown covariance matrix. All of these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix is much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge-type estimators for the covariance matrix. In this article, we use a ridge-type estimators for the covariance matrix and obtain the modified AIC and modified C _p statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.     

A projection pursuit index for large <Emphasis Type="Italic">p</Emphasis> small <Emphasis Type="Italic">n</Emphasis> data

In high-dimensional data, one often seeks a few interesting low-dimensional projections which reveal important aspects of the data. Projection pursuit for classification finds projections that reveal differences between classes. Even though projection pursuit is used to bypass the curse of dimensionality, most indexes will not work well when there are a small number of observations relative to the number of variables, known as a large p (dimension) small n (sample size) problem. This paper discusses the relationship between the sample size and dimensionality on classification and proposes a new projection pursuit index that overcomes the problem of small sample size for exploratory classification.     

Testing linear hypotheses in high-dimensional regressions

For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤20. On the other hand, assuming that the data dimension p as well as the number q of regression variables are fixed while the sample size n grows, several asymptotic approximations are proposed in the literature for Wilk's Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension p and a large sample size n. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null hypothesis and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large p and large n context, but also for moderately large data dimensions such as p=30 or p=50. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in multivariate analysis of variance which is valid for high-dimensional data.     

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.     

Distance-based outlier detection for high dimension,low sample size data

Despite the popularity of high dimension, low sample size data analysis, there has not been enough attention to the sample integrity issue, in particular, a possibility of outliers in the data. A new outlier detection procedure for data with much larger dimensionality than the sample size is presented. The proposed method is motivated by asymptotic properties of high-dimensional distance measures. Empirical studies suggest that high-dimensional outlier detection is more likely to suffer from a swamping effect rather than a masking effect, thus yields more false positives than false negatives. We compare the proposed approaches with existing methods using simulated data from various population settings. A real data example is presented with a consideration on the implication of found outliers.     

Feature screening for case‐cohort studies with failure time outcome

Case‐cohort design has been demonstrated to be an economical and efficient approach in large cohort studies when the measurement of some covariates on all individuals is expensive. Various methods have been proposed for case‐cohort data when the dimension of covariates is smaller than sample size. However, limited work has been done for high‐dimensional case‐cohort data which are frequently collected in large epidemiological studies. In this paper, we propose a variable screening method for ultrahigh‐dimensional case‐cohort data under the framework of proportional model, which allows the covariate dimension increases with sample size at exponential rate. Our procedure enjoys the sure screening property and the ranking consistency under some mild regularity conditions. We further extend this method to an iterative version to handle the scenarios where some covariates are jointly important but are marginally unrelated or weakly correlated to the response. The finite sample performance of the proposed procedure is evaluated via both simulation studies and an application to a real data from the breast cancer study.     

Structure-preserving mappings of uniform order statistics

There are several well-known mappings which transform the first r common order statistics in a sample of size n from a standard uniform distribution to a full vector of dimension r of order statistics in a sample of size r from a uniform distribution. Continuing the results reported in a previous paper by the authors, it is shown that transformations of these types do not lead to order statistics from an i.i.d. sample of random variables, in general, when being applied to order statistics from non-uniform distributions. By accepting the loss of one dimension, a structure-preserving transformation exists for power function distributions.     

A two-sample test for mean functions with increasing number of projections

We propose a test for equality of two means when data are functions and obtain the asymptotic properties of the test statistic as data dimension increases with the sample size. We also derive the asymptotic power of the test under some local alternatives and show that the test statistic is root-n consistent. A simulation study is conducted to evaluate the performance of the test numerically and to compare the proposed test with other existing four popular tests.     

Binary discrimination methods for high-dimensional data with a geometric representation

Four binary discrimination methods are studied in the context of high-dimension, low sample size data with an asymptotic geometric representation, when the dimension increases while the sample sizes of the classes are fixed. We show that the methods support vector machine, mean difference, distance-weighted discrimination, and maximal data piling have the same asymptotic behavior as the dimension increases. We study the consistent, inconsistent, and strongly inconsistent cases in terms of angles between the normal vectors of the separating hyperplanes of the methods and the optimal direction for classification. A simulation study is done to assess the theoretical results.     

Goodness‐of‐fit testing based on a weighted bootstrap: A fast large‐sample alternative to the parametric bootstrap

The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness‐of‐fit testing. The simplest way to carry out such goodness‐of‐fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data increase. An alternative resampling technique based on a fast weighted bootstrap is proposed in this paper, and is studied both theoretically and empirically. The outcome of this work is a generic and computationally efficient multiplier goodness‐of‐fit procedure that can be used as a large‐sample alternative to the parametric bootstrap. In order to approximately determine how large the sample size needs to be for the parametric and weighted bootstraps to have roughly equivalent powers, extensive Monte Carlo experiments are carried out in dimension one, two and three, and for models containing up to nine parameters. The computational gains resulting from the use of the proposed multiplier goodness‐of‐fit procedure are illustrated on trivariate financial data. A by‐product of this work is a fast large‐sample goodness‐of‐fit procedure for the bivariate and trivariate t distribution whose degrees of freedom are fixed. The Canadian Journal of Statistics 40: 480–500; 2012 © 2012 Statistical Society of Canada     

Robust centroid based classification with minimum error rates for high dimension,low sample size data

A new method of statistical classification (discrimination) is proposed. The method is most effective for high dimension, low sample size data. It uses a robust mean difference as the direction vector and locates the classification boundary by minimizing the error rates. Asymptotic results for assessment and comparison to several popular methods are obtained by using a type of asymptotics of finite sample size and infinite dimensions. The value of the proposed approach is demonstrated by simulations. Real data examples are used to illustrate the performance of different classification methods.     

A test for the complete independence of high-dimensional random vectors

ABSTRACT

This paper discusses the problem of testing the complete independence of random variables when the dimension of observations can be much larger than the sample size. It is reported that two typical tests based on, respectively, the biggest off-diagonal entry and the largest eigenvalue of the sample correlation matrix lose their control of type I error in such high-dimensional scenarios, and exhibit distinct behaviours in type II error under different types of alternative hypothesis. Given these facts, we propose a permutation test procedure by synthesizing these two extreme statistics. Simulation results show that for finite dimension and sample size the proposed test outperforms the existing methods in various cases.     

ON ESTIMATION OF THE POPULATION SPECTRAL DISTRIBUTION FROM A HIGH‐DIMENSIONAL SAMPLE COVARIANCE MATRIX

Sample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.     

A model selection criterion for discriminant analysis of high-dimensional data with fewer observations

This paper is concerned with the problem of selecting variables in two-group discriminant analysis for high-dimensional data with fewer observations than the dimension. We consider a selection criterion based on approximately unbiased for AIC type of risk. When the dimension is large compared to the sample size, AIC type of risk cannot be defined. We propose AIC by replacing maximum likelihood estimator with ridge-type estimator. This idea follows Srivastava and Kubokawa (2008). It has been further extended by Yamamura et al. (2010). Simulation revealed that the proposed AIC performs well.     

Clustering and classification problems in genetics through U-statistics

ABSTRACT

Genetic data are frequently categorical and have complex dependence structures that are not always well understood. For this reason, clustering and classification based on genetic data, while highly relevant, are challenging statistical problems. Here we consider a versatile U-statistics-based approach for non-parametric clustering that allows for an unconventional way of solving these problems. In this paper we propose a statistical test to assess group homogeneity taking into account multiple testing issues and a clustering algorithm based on dissimilarities within and between groups that highly speeds up the homogeneity test. We also propose a test to verify classification significance of a sample in one of two groups. We present Monte Carlo simulations that evaluate size and power of the proposed tests under different scenarios. Finally, the methodology is applied to three different genetic data sets: global human genetic diversity, breast tumour gene expression and Dengue virus serotypes. These applications showcase this statistical framework's ability to answer diverse biological questions in the high dimension low sample size scenario while adapting to the specificities of the different datatypes.     

L'analyse de la variation saisonniére quand l'amplitude et la taille sont faibles

Studies of seasonal variation are valuable in biomedical research because they can help to discover the etiology of diseases that are not well understood. Generally in these studies the data have certain characteristics that require specialized tests and methods for the statistical analysis. But the effectiveness of these specialized tests is variable, especially according to the seasonal variation, the dimension of the amplitude in the seasonal variation, and the sample size. The purpose of this paper is to present a test and methods appropriate for the analysis and modeling of data whose seasonal variation has small amplitude and whose sample size is small. This test can detect different kinds of seasonal variation. The results from a simulation study show that the test performs very well. The application of these methods is illustrated by two examples.     

Random sliced inverse regression

Sliced Inverse Regression (SIR; 1991) is a dimension reduction method for reducing the dimension of the predictors without losing regression information. The implementation of SIR requires inverting the covariance matrix of the predictors—which has hindered its use to analyze high-dimensional data where the number of predictors exceed the sample size. We propose random sliced inverse regression (rSIR) by applying SIR to many bootstrap samples, each using a subset of randomly selected candidate predictors. The final rSIR estimate is obtained by aggregating these estimates. A simple variable selection procedure is also proposed using these bootstrap estimates. The performance of the proposed estimates is studied via extensive simulation. Application to a dataset concerning myocardial perfusion diagnosis from cardiac Single Proton Emission Computed Tomography (SPECT) images is presented.