Australian & New Zealand Journal of Statistics最新文献

In this paper, we propose a novel robust principal component analysis (PCA) for high-dimensional data in the presence of various heterogeneities, in particular strong tailing and outliers. A transformation motivated by the characteristic function is constructed to improve the robustness of the classical PCA. The suggested method has the distinct advantage of dealing with heavy-tail-distributed data, whose covariances may be non-existent (positively infinite, for instance), in addition to the usual outliers. The proposed approach is also a case of kernel principal component analysis (KPCA) and employs the robust and non-linear properties via a bounded and non-linear kernel function. The merits of the new method are illustrated by some statistical properties, including the upper bound of the excess error and the behaviour of the large eigenvalues under a spiked covariance model. Additionally, using a variety of simulations, we demonstrate the benefits of our approach over the classical PCA. Finally, using data on protein expression in mice of various genotypes in a biological study, we apply the novel robust PCA to categorise the mice and find that our approach is more effective at identifying abnormal mice than the classical PCA.

Frequentist and Bayesian methods differ in many aspects but share some basic optimal properties. In real-life prediction problems, situations exist in which a model based on one of the above paradigms is preferable depending on some subjective criteria. Nonparametric classification and regression techniques, such as decision trees and neural networks, have both frequentist (classification and regression trees (CARTs) and artificial neural networks) as well as Bayesian counterparts (Bayesian CART and Bayesian neural networks) to learning from data. In this paper, we present two hybrid models combining the Bayesian and frequentist versions of CART and neural networks, which we call the Bayesian neural tree (BNT) models. BNT models can simultaneously perform feature selection and prediction, are highly flexible, and generalise well in settings with limited training observations. We study the statistical consistency of the proposed approaches and derive the optimal value of a vital model parameter. The excellent performance of the newly proposed BNT models is shown using simulation studies. We also provide some illustrative examples using a wide variety of standard regression datasets from a public available machine learning repository to show the superiority of the proposed models in comparison to popularly used Bayesian CART and Bayesian neural network models.

In stating the results of their research, scientists usually want to publish narrow confidence intervals because they give precise estimates of the effects of interest. In many cases, the researcher would want to use the narrowest interval that maintains the desired coverage probability. In this manuscript, we propose a new method of finding confidence intervals that are often narrower than traditional confidence intervals for any individual parameter in a linear model if the errors are from a skewed distribution or from a long-tailed symmetric distribution. If the errors are normally distributed, we show that the width of the proposed normal scores confidence interval will not be much greater than the width of the traditional interval. If the dataset includes predictor variables that are uncorrelated or moderately correlated then the confidence intervals will maintain their coverage probability. However, if the covariates are highly correlated, then the coverage probability of the proposed confidence interval may be slightly lower than the nominal value. The procedure is not computationally intensive and an R program is available to determine the normal scores 95% confidence interval. Whenever the covariates are not highly correlated, the normal scores confidence interval is recommended for the analysis of datasets having 50 or more observations.

Principal Component Analysis (PCA) is a popular multivariate analytic tool which can be used for dimension reduction without losing much information. Data vectors containing a large number of features arriving sequentially may be correlated with each other. An effective algorithm for such situations is online PCA. Existing Online PCA research works revolve around proposing efficient scalable updating algorithms focusing on compression loss only. They do not take into account the size of the dataset at which further arrival of data vectors can be terminated and dimension reduction can be applied. It is well known that the dataset size contributes to reducing the compression loss – the smaller the dataset size, the larger the compression loss while larger the dataset size, the lesser the compression loss. However, the reduction in compression loss by increasing dataset size will increase the total data collection cost. In this paper, we move beyond the scalability and updation problems related to Online PCA and focus on optimising a cost-compression loss which considers the compression loss and data collection cost. We minimise the corresponding risk using a two-stage PCA algorithm. The resulting two-stage algorithm is a fast and an efficient alternative to Online PCA and is shown to exhibit attractive convergence properties with no assumption on specific data distributions. Experimental studies demonstrate similar results and further illustrations are provided using real data. As an extension, a multi-stage PCA algorithm is discussed as well. Given the time complexity, the two-stage PCA algorithm is emphasised over the multi-stage PCA algorithm for online data.

The discrete minification model based on the modified negative binomial operator, as an extension to the continuous minification model, can be used to describe an extreme value after few increasing values. To make this model more practical and flexible, a new minification integer-valued autoregressive process driven by explanatory variables is proposed. Ergodicity of the new process is discussed. The estimators of the unknown parameters are obtained via the conditional least squares and conditional maximum likelihood methods, and the asymptotic properties are also established. A testing procedure for checking existence of the explanatory variables is developed. Some Monte Carlo simulations are given to illustrate the finite-sample performances of the estimators under specification and misspecification and the test, respectively. A real example is applied to illustrate the performance of our model.

In recent years, small area estimation has played an important role in statistics as it deals with the problem of obtaining reliable estimates for parameters of interest in areas with small or even zero sample sizes corresponding to population sizes. Nested error linear regression models are often used in small area estimation assuming that the covariates are measured without error and also the relationship between covariates and response variable is linear. Small area models have also been extended to the case in which a linear relationship may not hold, using penalised spline (P-spline) regression, but assuming that the covariates are measured without error. Recently, a nested error regression model using a P-spline regression model, for the fixed part of the model, has been studied assuming the presence of measurement error in covariate, in the Bayesian framework. In this paper, we propose a frequentist approach to study a semi-parametric nested error regression model using P-splines with a covariate measured with error. In particular, the pseudo-empirical best predictors of small area means and their corresponding mean squared prediction error estimates are studied. Performance of the proposed approach is evaluated through a simulation and also by a real data application. We propose a frequentist approach to study a semi-parametric nested error regression model using P-splines with a covariate measured with error.