Computational Statistics & Data Analysis最新文献

Kolmogorov-Smirnov (KS) statistic is quite popular in many areas as the major performance evaluation criterion for binary classification due to its explicit business intension. Fang and Chen (2019) proposed a novel DMKS method that directly maximizes the KS statistic and compares favorably with the popular existing methods. However, DMKS did not consider the critical problem of variable selection since the special form of KS brings great challenge to establish the DMKS estimator's asymptotic distribution which is most likely to be nonstandard. This intractable issue is handled by introducing a surrogate loss function which leads to a $\sqrt{n}$-consistent estimator for the true parameter up to a multiplicative scalar. Then a nonconcave penalty function is combined to achieve the variable selection consistency and asymptotical normality with the oracle property. Results of empirical studies confirm the theoretical results and show advantages of the proposed SKS (Surrogated Kolmogorov-Smirnov) method compared to the original DMKS method without variable selection.

The interpretation of regression models with compositional vectors as response and/or explanatory variables has been approached from different perspectives. The initial approaches are performed in coordinate space subsequent to applying a log-ratio transformation to the compositional vectors. Given that these models exhibit non-linearity concerning classical operations within real space, an alternative approach has been proposed. This approach relies on infinitesimal increments or derivatives, interpreted within a simplex framework. Consequently, it offers interpretations of elasticities or semi-elasticities in the original space of shares which are independent of any log-ratio transformations. Some functions of these elasticities or semi-elasticities turn out to be constant throughout the sample observations, making them natural parameters for interpreting CoDa models. These parameters are linked to relative variations of pairwise share ratios of the response and/or of the explanatory variables. Approximations of share ratio variations are derived and linked to these natural parameters. A real dataset on the French presidential election is utilized to illustrate each type of interpretation in detail.

Screening experiments are useful for identifying a small number of truly important factors from a large number of potentially important factors. The Gauss-Dantzig Selector (GDS) is often the preferred analysis method for screening experiments. Just considering main-effects models can result in erroneous conclusions, but including interaction terms, even if restricted to two-factor interactions, increases the number of model terms dramatically and challenges the GDS analysis. A new analysis method, called Gauss-Dantzig Selector Aggregation over Random Models (GDS-ARM), which performs a GDS analysis on multiple models that include only some randomly selected interactions, is proposed. Results from these different analyses are then aggregated to identify the important factors. The proposed method is discussed, the appropriate choices for the tuning parameters are suggested, and the performance of the method is studied on real and simulated data.

Heterogeneity among patients commonly exists in clinical studies and leads to challenges in medical research. It is widely accepted that there exist various sub-types in the population and they are distinct from each other. The approach of identifying the sub-types and thus tailoring disease prevention and treatment is known as precision medicine. The mixture model is a classical statistical model to cluster the heterogeneous population into homogeneous sub-populations. However, for the highly heterogeneous population with multiple components, its parameter estimation and clustering results may be ambiguous due to the dependence of the EM algorithm on the initial values. For sub-typing purposes, the finite mixture of regression models with concomitant variables is considered and a novel statistical method is proposed to identify the main components with large proportions in the mixture sequentially. Compared to existing typical statistical inferences, the new method not only requires no pre-specification on the number of components for model fitting, but also provides more reliable parameter estimation and clustering results. Simulation studies demonstrated the superiority of the proposed method. Real data analysis on the drug response prediction illustrated its reliability in the parameter estimation and capability to identify the important subgroup.

The hypothesis testing problem of homogeneity of k $(\ge 2)$ inverse Gaussian means against ordered alternatives is studied when nuisance or scale-like parameters are unknown and unequal. The maximum likelihood estimators (MLEs) of means and scale-like parameters are obtained when means satisfy some simple order restrictions and scale-like parameters are unknown and unequal. An iterative algorithm is proposed for finding these estimators. It has been proved that under a specific condition, the proposed algorithm converges to the true MLEs uniquely. A likelihood ratio test and two simultaneous tests are proposed. Further, an algorithm for finding the MLEs of parameters is given when means are equal but unknown. Using the estimators, the likelihood ratio test is developed for testing against ordered alternative means. Using the asymptotic distribution, the asymptotic likelihood ratio test is proposed. However, for small samples, it does not perform well. Hence, a parametric bootstrap likelihood ratio test (PB LRT) is proposed. Therefore, the asymptotic validity of the bootstrap procedure has been shown. Using the Box-type approximation method, test statistics are developed for the two-sample problem of equality of means when scale-like parameters are heterogeneous. Using these, two PB-based heuristic tests are proposed. Asymptotic null distributions are derived and PB accuracy is also developed. Two asymptotic tests are also proposed using the asymptotic null distributions. To get the critical points and test statistics of the three PB tests and two asymptotic tests, an ‘R’ package is developed and shared on GitHub. Applications of the tests are illustrated using real data.

Time warping function provides a mathematical representation to measure phase variability in functional data. Recent studies have developed various approaches to estimate optimal warping between functions. However, a principled, linear, generative representation on time warping functions is still under-explored. This is highly challenging because the warping functions are non-linear in the conventional $L^2$ space. To address this problem, a new linear warping space is defined and a stochastic process representation is proposed to characterize time warping functions. The key is to define an inner-product structure on the time warping space, followed by a transformation which maps the warping functions into a sub-space of the $L^2$ space. With certain constraints on the warping functions, this transformation is an isometric isomorphism. In the transformed space, the $L^2$ basis in the Hilbert space is adopted for representation, which can be easily utilized to generate time warping functions by using different types of stochastic process. The effectiveness of this representation is demonstrated through its use as a new penalty in the penalized function registration, accompanied by an efficient gradient method to minimize the cost function. The new penalized method is illustrated through simulations that properly characterize nonuniform and correlated constraints in the time domain. Furthermore, this representation is utilized to develop a boxplot for warping functions, which can estimate templates and identify warping outliers. Finally, this representation is applied to a Covid-19 dataset to construct boxplots and identify states with outlying growth patterns.

For multivariate nonparametric regression, doubly penalized ANOVA modeling (DPAM) has recently been proposed, using hierarchical total variations (HTVs) and empirical norms as penalties on the component functions such as main effects and multi-way interactions in a functional ANOVA decomposition of the underlying regression function. The two penalties play complementary roles: the HTV penalty promotes sparsity in the selection of basis functions within each component function, whereas the empirical-norm penalty promotes sparsity in the selection of component functions. To facilitate large-scale training of DPAM using backfitting or block minimization, two suitable primal-dual algorithms are developed, including both batch and stochastic versions, for updating each component function in single-block optimization. Existing applications of primal-dual algorithms are intractable for DPAM with both HTV and empirical-norm penalties. The validity and advantage of the stochastic primal-dual algorithms are demonstrated through extensive numerical experiments, compared with their batch versions and a previous active-set algorithm, in large-scale scenarios.

Mixed membership models are an extension of finite mixture models, where each observation can partially belong to more than one mixture component. A probabilistic framework for mixed membership models of high-dimensional continuous data is proposed with a focus on scalability and interpretability. The novel probabilistic representation of mixed membership is based on convex combinations of dependent multivariate Gaussian random vectors. In this setting, scalability is ensured through approximations of a tensor covariance structure through multivariate eigen-approximations with adaptive regularization imposed through shrinkage priors. Conditional weak posterior consistency is established on an unconstrained model, allowing for a simple posterior sampling scheme while keeping many of the desired theoretical properties of our model. The model is motivated by two biomedical case studies: a case study on functional brain imaging of children with autism spectrum disorder (ASD) and a case study on gene expression data from breast cancer tissue. These applications highlight how the typical assumption made in cluster analysis, that each observation comes from one homogeneous subgroup, may often be restrictive in several applications, leading to unnatural interpretations of data features.

To address the challenges in estimating parameters of the widely applied Student-Lévy process, the study introduces two distinct methods: a likelihood-based approach and a data-driven approach. A two-step quasi-likelihood-based method is initially proposed, countering the non-closed nature of the Student-Lévy process's distribution function under convolution. This method utilizes the limiting properties observed in high-frequency data, offering estimations via a quasi-likelihood function characterized by asymptotic normality. Additionally, a novel neural-network-based parameter estimation technique is advanced, independent of high-frequency observation assumptions. Utilizing a CNN-LSTM framework, this method effectively processes sparse, local jump-related data, extracts deep features, and maps these to the parameter space using a fully connected neural network. This innovative approach ensures minimal assumption reliance, end-to-end processing, and high scalability, marking a significant advancement in parameter estimation techniques. The efficacy of both methods is substantiated through comprehensive numerical experiments, demonstrating their robust performance in diverse scenarios.

Generalized linear models (GLMs) form one of the most popular classes of models in statistics. The gamma variant is used, for instance, in actuarial science for the modelling of claim amounts in insurance. A flaw of GLMs is that they are not robust against outliers (i.e., against erroneous or extreme data points). A difference in trends in the bulk of the data and the outliers thus yields skewed inference and predictions. To address this problem, robust methods have been introduced. The most commonly applied robust method is frequentist and consists in an estimator which is derived from a modification of the derivative of the log-likelihood. The objective is to propose an alternative approach which is modelling-based and thus fundamentally different. Such an approach allows for an understanding and interpretation of the modelling, and it can be applied for both frequentist and Bayesian statistical analyses. The proposed approach possesses appealing theoretical and empirical properties.