Australian & New Zealand Journal of Statistics最新文献

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.

In nonlinear growth models, we considered the parameter estimation under subspace information for low-dimensional and high-dimensional data. We proposed novel estimators based on pretest and shrinkage strategies to improve the estimation efficiency and to establish asymptotic properties. We used simulation studies and a real data example to confirm the theoretical results. We also applied two well-known penalised methods—least absolute shrinkage and selection operator (LASSO) and adaptive LASSO (aLASSO)—for the dimensional reduction of the predictor variables. The results demonstrated that the pretest and shrinkage estimation strategies performed well in parameter estimations when the subspace information was incorrect for both low- and high-dimensional regimes.

Analysing multivariate and high_dimensional multi_sample data is essential in many scientific fields. One of the most crucial and popular topics in modern nonparametric statistics is multi_sample comparison problems for such multivariate and high_dimensional data. The Kruskal_Wallis test is widely used in the multi_sample problem. For multivariate or high_dimensional data, it is imperative to specify how to determine the ranks of individual vector_valued observations in terms of various distance metrics. Alternatively, one can combine the concept of principal component scores or independent component scores with the Kruskal_Wallis test. A simple but powerful Kruskal_Wallis test based on the principal component scores is discussed in this paper for the multivariate and high_dimensional data. Another type of Kruskal_Wallis test based on latent sources of independent component analysis is constructed as a competitor. These tests are suitable for testing the difference in the location vector, scale matrix or both and can be used with equal and unequal sample sizes. These tests_ power performances are thoroughly compared with traditional distance_based Kruskal_Wallis tests for multivariate data using simulation based on Monte Carlo for various population distributions. We include an illustration of the proposed tests using real data. The paper concludes with some remarks and directions for future research.