Journal of Multivariate Analysis最新文献

Estimation of the mean and covariance parameters for functional data is a critical task, with local linear smoothing being a popular choice. In recent years, many scientific domains are producing multivariate functional data for which $p$, the number of curves per subject, is often much larger than the sample size $n$. In this setting of high-dimensional functional data, much of developed methodology relies on preliminary estimates of the unknown mean functions and the auto- and cross-covariance functions. This paper investigates the convergence rates of local linear estimators in terms of the maximal error across components and pairs of components for mean and covariance functions, respectively, in both $L^2$ and uniform metrics. The local linear estimators utilize a generic weighting scheme that can adjust for differing numbers of discrete observations $N_{ij}$ across curves $j$ and subjects $i$, where the $N_{ij}$ vary with $n$. Particular attention is given to the equal weight per observation (OBS) and equal weight per subject (SUBJ) weighting schemes. The theoretical results utilize novel applications of concentration inequalities for functional data and demonstrate that, similar to univariate functional data, the order of the $N_{ij}$ relative to $p$ and $n$ divides high-dimensional functional data into three regimes (sparse, dense, and ultra-dense), with the high-dimensional parametric convergence rate of ${\left(log\left(p\right)/n\right)}^{1/2}$ being attainable in the latter two.

In this paper, we consider the problem of dependent censoring models with a positive probability that the times of failure are equal. In this context, we propose to consider the Marshall–Olkin type model and studied some properties of the associated survival copula in its application to censored data. We also introduce estimators for the marginal distributions and the joint survival probabilities under different schemes and show their asymptotic normality under appropriate conditions. Finally, we evaluate the finite-sample performance of our approach relying on a small simulation study with synthetic data real data applications.

Testing the equality of the covariance matrices of two high-dimensional samples is a fundamental inference problem in statistics. Several tests have been proposed but they are either too liberal or too conservative when the required assumptions are not satisfied which attests that they are not always applicable in real data analysis. To overcome this difficulty, a normal-reference test is proposed and studied in this paper. It is shown that under some regularity conditions and the null hypothesis, the proposed test statistic and a chi-squared-type mixture have the same limiting distribution. It is then justified to approximate the null distribution of the proposed test statistic using that of the chi-squared-type mixture. The distribution of the chi-squared-type mixture can be well approximated using a three-cumulant matched chi-squared-approximation with its approximation parameters consistently estimated from the data. The asymptotic power of the proposed test under a local alternative is also established. Simulation studies and a real data example demonstrate that the proposed test works well in general scenarios and outperforms the existing competitors substantially in terms of size control.

We study power approximation formulas for peak detection using Gaussian random field theory. The approximation, based on the expected number of local maxima above the threshold $u$, $E\left[M_u\right]$, is proved to work well under three asymptotic scenarios: small domain, large threshold, and sharp signal. An adjusted version of $E\left[M_u\right]$ is also proposed to improve accuracy when the expected number of local maxima $E\left[M_{-\infty }\right]$ exceeds 1. Cheng and Schwartzman (2018) developed explicit formulas for $E\left[M_u\right]$ of smooth isotropic Gaussian random fields with zero mean. In this paper, these formulas are extended to allow for rotational symmetric mean functions, making them applicable not only for power calculations but also for other areas of application that involve non-centered Gaussian random fields. We also apply our formulas to 2D and 3D simulated datasets, and the 3D data is induced by a group analysis of fMRI data from the Human Connectome Project to measure performance in a realistic setting.

In this study, we address the selection of both fixed and random effects in partial linear mixed effects models. By combining B-spline and QR decomposition techniques, we propose a double-penalized likelihood procedure for both estimating and selecting these effects. Furthermore, we introduce an orthogonality-based method to estimate the non-parametric component, ensuring that the fixed and random effects are separated without any mutual interference. The asymptotic properties of the resulting estimators are investigated under mild conditions. Simulation studies are conducted to evaluate the finite sample performance of the proposed method. Finally, we demonstrate the practical applicability of our methodology by analyzing a real data.

The article proposes and justifies an optimal rank-based portmanteau test of multivariate elliptical strict white noise against multivariate serial dependence. It is based on new stochastic hyperplane-based ranks that are simpler and easier to compute than other usable hyperplane-based competitors and still share with them many good properties such as their distribution-free nature, affine invariance, efficiency, robustness and weak moment assumptions. The finite-sample performance of the portmanteau test is illustrated empirically in a small Monte Carlo simulation study.

Recent research and substantive studies have shown growing interest in expectile regression (ER) procedures. Similar to quantile regression, ER with respect to different expectile levels can provide a comprehensive picture of the conditional distribution of a response variable given predictors. This study proposes three composite-type ER estimators to improve estimation accuracy. The proposed ER estimators include the composite estimator, which minimizes the composite expectile objective function across expectiles; the weighted expectile average estimator, which takes the weighted average of expectile-specific estimators; and the weighted composite estimator, which minimizes the weighted composite expectile objective function across expectiles. Under certain regularity conditions, we derive the convergence rate of the slope function, obtain the mean squared prediction error, and establish the asymptotic normality of the slope vector. Simulations are conducted to assess the empirical performances of various estimators. An application to the analysis of capital bike share data is presented. The numerical evidence endorses our theoretical results and confirm the superiority of the composite-type ER estimators to the conventional least squares and single ER estimators.

We obtain exponential inequalities for regularized Hotelling’s $T_n^2$ statistics, that take into account the potential high dimensional aspects of the problem. We explore the finite sample properties of the tail of these statistics by deriving exponential bounds for symmetric distributions and also for general distributions under weak moment assumptions (we never assume exponential moments). For this, we use a penalized estimator of the covariance matrix and propose an optimal choice for the penalty coefficient.

Sir Ronald Aylmer Fisher opened many new areas in Multivariate Analysis, and the one which we will consider is discriminant analysis. Several papers by Fisher and others followed from his seminal paper in 1936 where he coined the name discrimination function. Historically, his four papers on discriminant analysis during 1936–1940 connect to the contemporaneous pioneering work of Hotelling and Mahalanobis. We revisit the famous iris data which Fisher used in his 1936 paper and in particular, test the hypothesis of multivariate normality for the data which he assumed. Fisher constructed his genetic discriminant motivated by this application and we provide a deeper insight into this construction; however, this construction has not been well understood as far as we know. We also indicate how the subject has developed along with the computer revolution, noting newer methods to carry out discriminant analysis, such as kernel classifiers, classification trees, support vector machines, neural networks, and deep learning. Overall, with computational power, the whole subject of Multivariate Analysis has changed its emphasis but the impact of this Fisher’s pioneering work continues as an integral part of supervised learning in Artificial Intelligence (AI).

In this paper, we introduce a new class of models for spatial data obtained from max-convolution processes based on indicator kernels with random shape. We show that these models have appealing dependence properties including tail dependence at short distances and independence at long distances. We further consider max-convolutions between such processes and processes with tail independence, in order to separately control the bulk and tail dependence behaviors, and to increase flexibility of the model at longer distances, in particular, to capture intermediate tail dependence. We show how parameters can be estimated using a weighted pairwise likelihood approach, and we conduct an extensive simulation study to show that the proposed inference approach is feasible in relatively high dimensions and it yields accurate parameter estimates in most cases. We apply the proposed methodology to analyze daily temperature maxima measured at 100 monitoring stations in the state of Oklahoma, US. Our results indicate that our proposed model provides a good fit to the data, and that it captures both the bulk and the tail dependence structures accurately.