Normal Mean Vector Research Articles

Parametric test based on the bootstrapping approach for the MANOVA under a Behrens-Fisher problem

This article presents a comparison of multivariate normal mean vectors under covariance positive definite matrices. We introduce an improved parametric bootstrap (IPB) approach for addressing the multivariate Behrens-Fisher problem, specifically focusing on cases with unequal covariance matrices. Additionally, we evaluate the performance of the IPB test by comparing it with three existing tests: the parametric bootstrap (PB) test, the generalized variable (GV) test, and the Johansen test. Through Monte Carlo simulation, our results demonstrate that both the IPB test and the PB test exhibit superior control over Type I error rates compared to the GV and Johansen tests. Notably, the IPB test outperforms the PB test in terms of controlling Type I error rates. Consequently, our study concludes that the IPB test represents a preferred statistical method for testing the equality of mean vectors in the multivariate Behrens-Fisher problem.

Uniformly more powerful tests for a subset of the components of a Normal Mean Vector

A class of tests that are uniformly more powerful than the likelihood ratio test is derived for testing the hypothesis about the means of a subset of the components of a multivariate normal distribution with unknown covariance matrix, when the means of the other subset of the components are known.

Machine learning and the James–Stein estimator

It is now 62 years since the publication of James and Stein’s seminal article on the estimation of a multivariate normal mean vector. The paper made a spectacular first impression on the statistical community through its demonstration of inadmissability of the maximum likelihood estimator. It continues to be influential, but not for the initial reasons. Empirical Bayes shrinkage estimation, now a major topic, found its early justification in the James–Stein formula. Less obvious downstream topics include Tweedie’s formula and Benjamini and Hochberg’s false discovery rate algorithm. This is a short and mainly non-technical review of the James–Stein rule and its effects on the machine learning era of statistical innovation.

An exact test for equality of two normal mean vectors with monotone missing data

The problem of testing equality of two normal mean vectors with incomplete data when the covariance matrices are equal is considered. For data matrices with monotone missing pattern, an exact test is proposed as an alternative one to the traditional likelihood ration test. Numerical power comparisons show that the powers of the proposed test and the likelihood ration test are comparable. However, the proposed test is an exact one. It is easy to use and useful to identify the component that caused the rejection of null hypothesis. It is illustrated using an example.

A Two-Sample Test of High Dimensional Means Based on Posterior Bayes Factor

In classical statistics, the primary test statistic is the likelihood ratio. However, for high dimensional data, the likelihood ratio test is no longer effective and sometimes does not work altogether. By replacing the maximum likelihood with the integral of the likelihood, the Bayes factor is obtained. The posterior Bayes factor is the ratio of the integrals of the likelihood function with respect to the posterior. In this paper, we investigate the performance of the posterior Bayes factor in high dimensional hypothesis testing through the problem of testing the equality of two multivariate normal mean vectors. The asymptotic normality of the linear function of the logarithm of the posterior Bayes factor is established. Then we construct a test with an asymptotically nominal significance level. The asymptotic power of the test is also derived. Simulation results and an application example are presented, which show good performance of the test. Hence, taking the posterior Bayes factor as a statistic in high dimensional hypothesis testing is a reasonable methodology.

Multivariate Behrens-Fisher problem using means of auxiliary variables

The authors considered the problem of testing equality of two multivariate normal mean vectors when the covariance matrices are unknown and arbitrary. Given auxiliary variables with known means, the authors proposed a pivotal quantity which is similar to the Hotelling T 2 statistic and obtained a satisfying approximation to its distribution. The authors also outlined hypothesis testing and confidence estimation based on the approximate distribution. The merits of the test were studied using Monte Carlo simulation. Monte Carlo studies indicated that the test is very satisfactory even for moderately small samples. At last, the authors illustrated the proposed methods by an example.

Wavelet Shrinkage Generalized Bayes Estimation for Multivariate Normal Distribution Mean Vectors with unknown Covariance Matrix under Balanced-LINEX Loss

In this paper, the generalized Bayes estimator of mean vector parameter for multivariate normal distribution with Unknown mean vector and covariance matrix is considered. This estimation is performed under the balanced-LINEX error loss function. The generalized Bayes estimator by using wavelet transformation is investigated. We also prove admissibility and minimaxity of shrinkage estimator and we present the simulation study and real data set for test validity of new estimator.

Estimation of Simultaneous Signals Using Absolute Inner Product with Applications to Integrative Genomics

Integrating the summary statistics from genome-wide association study (\textsc{gwas}) and expression quantitative trait loci (e\textsc{qtl}) data provides a powerful way of identifying the genes whose expression levels are potentially associated with complex diseases. A parameter called $T$-score that quantifies the genetic overlap between a gene and the disease phenotype based on the summary statistics is introduced based on the mean values of two Gaussian sequences. Specifically, given two independent samples $\mathbf{x}_n\sim N(\theta, \Sigma_1)$ and $\mathbf{y}_n\sim N(\mu, \Sigma_2)$, the $T$-score is defined as $\sum_{i=1}^n |\theta_i\mu_i|$, a non-smooth functional, which characterizes the amount of shared signals between two absolute normal mean vectors $|\theta|$ and $|\mu|$. Using approximation theory, estimators are constructed and shown to be minimax rate-optimal and adaptive over various parameter spaces. Simulation studies demonstrate the superiority of the proposed estimators over existing methods. The method is applied to an integrative analysis of heart failure genomics datasets and we identify several genes and biological pathways that are potentially causal to human heart failure.

A Regression Modeling Approach to Structured Shrinkage Estimation

Problems involving the simultaneous estimation of multiple parameters arise in many areas of theoretical and applied statistics. A canonical example is the estimation of a vector of normal means. Frequently, structural information about relationships between the parameters of interest is available. For example, in a gene expression denoising problem, genes with similar functions may have similar expression levels. Despite its importance, structural information has not been well-studied in the simultaneous estimation literature, perhaps in part because it poses challenges to the usual geometric or empirical Bayes shrinkage estimation paradigms. This article proposes that some of these challenges can be resolved by adopting an alternate paradigm, based on regression modeling. This approach can naturally incorporate structural information and also motivates new shrinkage estimation and inference procedures. As an illustration, this regression paradigm is used to develop a class of estimators with asymptotic risk optimality properties that perform well in simulations and in denoising gene expression data from a single cell RNA-sequencing experiment.

Simultaneous estimation of normal means with side information

The integrative analysis of multiple datasets is an important strategy in data analysis. It is increasingly popular in genomics, which enjoys a wealth of publicly available datasets that can be compared, contrasted, and combined in order to extract novel scientific insights. This paper studies a stylized example of data integration for a classical statistical problem: leveraging side information to estimate a vector of normal means. This task is formulated as a compound decision problem, an oracle integrative decision rule is derived, and a data-driven estimate of this rule based on minimizing an unbiased estimate of its risk is proposed. The data-driven rule is shown to asymptotically achieve the minimum possible risk among all separable decision rules, and it can outperform existing methods in numerical properties. The proposed procedure leads naturally to an integrative high-dimensional classification procedure, which is illustrated by combining data from two independent gene expression profiling studies.

Admissible estimators of a multivariate normal mean vector when the scale is unknown

Summary We study admissibility of a subclass of generalized Bayes estimators of a multivariate normal vector in the case where the variance is unknown, under scaled quadratic loss. Minimaxity is established for some of these estimators.

On the conditional test of order restricted multivariate normal mean vectors: simulation study and application

Multivariate isotonic regression theory plays an important role in the field of testing statistical hypotheses under order restriction for several vector valued parameters and this problem may arise in several situations. Sasabuchi et al. derived a likelihood ratio type test for homogeneity of order restricted mean vectors under a multivariate normality assumption with unknown and common covariance matrices; however, its null distribution depends on the unknown covariance matrix. In order to overcome this difficulty, in the present paper we propose a test that conditions on the sufficient statistic for the covariance matrix. We remove the dependence of the null distribution of test statistic proposed by Sasabuchi et al. on the unknown value of covariance matrix and present a condition on a sufficient statistic for covariance matrix under the null hypothesis. A method based on Markov Chain Monte Carlo sampling is used to calculate the value of the test. We evaluate the operating characteristics of the proposed test through simulation studies. Our extensive simulation studies demonstrate that the power of this new test is generally higher than the test proposed by Sasabuchi et al. and also by Sasabuchi. A real data example is presented.

Density Prediction and the Stein Phenomenon

The Stein phenomenon is a path-breaking discovery in mathematical statistics in the last century. A large number of researchers followed Stein’s footsteps and developed a wide variety of minimax shrinkage point estimators of a multivariate normal mean vector, each dominating the sample mean. More recently, the problem resurfaced, but this time with minimax shrinkage predictive density estimation, illustrating once again the Stein phenomenon In this review paper, we discuss parallel developments for normal and Poisson distributions under the Kullback-Leibler and more general divergence losses.

Improved loss estimation for a normal mean matrix

We investigate improved loss estimation in the matrix mean estimation problem. Specifically, for estimators of a normal mean matrix, we consider estimation of the Frobenius loss. Based on the singular values of the observation, we develop loss estimators that dominate the unbiased loss estimator for a broad class of matrix mean estimators including the Efron–Morris estimator. This is an extension of the results of Johnstone (1988) for a normal mean vector. We also provide improved estimators of loss for reduced-rank estimators. Numerical results show the effectiveness of the proposed loss estimators.

Testing the equality of several multivariate normal mean vectors under heteroscedasticity: A fiducial approach and an approximate test

ABSTRACTWe consider the problem of testing the equality of several multivariate normal mean vectors under heteroscedasticity. We first construct a fiducial confidence region (FCR) for the differences between normal mean vectors and we then propose a fiducial test for comparing mean vectors by inverting the FCR. We also propose a simple approximate test that is based on a modification of the χ2 approximation. This simple test avoids the complications of simulation-based inference methods. We show that the proposed fiducial test has correct type one error rate asymptotically. We compare the proposed fiducial and approximate tests with the parametric bootstrap test in terms of controlling the type one error rate via an extensive simulation study. Our simulation results show that the proposed fiducial and approximate tests control the type one error rate, while there are cases that the parametric bootstrap test is out of control. We also discuss the power performance of the tests. Finally, we illustrate with a real example how our proposed methods are applicable in analyzing repeated measure designs including a single grouping variable.

Bootstrap approach to test the homogeneity of order restricted mean vectors when the covariance matrices are unknown

ABSTRACTTesting homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are unknown, arbitrary positive definite and unequal are considered. This problem of testing has been studied to some extent, for example, by Kulatunga and Sasabuchi (1984) when the covariance matrices are known and also Sasabuchi et al. (2003) and Sasabuchi (2007) when the covariance matrices are unknown but common. In this paper, a test statistic is proposed and because of the main advantage of the bootstrap test is that it avoids the derivation of the complex null distribution analytically, a bootstrap test statistic is derived and since the proposed test statistic is location invariance the bootstrap p-value defined logical and some steps are presented to estimate it. Our numerical studies via Monte Carlo simulation show that the proposed bootstrap test can correctly control the type I error rates. The power of the test for some of the p-dimensional normal distributions is computed by Monte Carlo simulation. Also, the null distribution of test statistic is estimated using kernel density. Finally, the bootstrap test is illustrated using a real data.

Tests for Normal Mean Vectors with Monotone Incomplete Data

SYNOPTIC ABSTRACTIn this article, we consider the distributions of the Hotelling’s T2-type test statistics for mean vectors when each dataset has a monotone missing data pattern. We give the approximate upper percentiles of the simplified Hotelling’s T2-type statistics in the case of data with k-step monotone missing data patterns. We also consider multivariate multiple comparisons for mean vectors with k-step monotone missing data. Approximate simultaneous confidence intervals for pairwise comparisons among mean vectors and comparisons with a control are obtained using Bonferroni’s approximation procedure. Finally, the accuracy and asymptotic behavior of the approximations are investigated by Monte Carlo simulation.

마할라노비스 거리를 이용한 증기보일러 튜브의 고장탐지방법

화력발전소의 설비들은 매우 높은 온도와 압력의 환경에서 운전되므로, 설비고장은 상당한 인적 물적 손실로 이어진다. 그러므로 발전설비의 비정상정인 동작 상태를 사전에 확인할 수 있는 고장탐지 시스템이 필수적이다. 본 연구에서는, 화력발전소 증기보일러의 고장탐지를 위해서 마할라노비스 거리(Mahalanobis distance, MD)를 이용하였다. MD 기반의 고장탐지방법에서는, 비정상샘플은 정상샘플들로부터 멀리 떨어져 있다고 가정한다. 정상상태로 동작중인 대상시스템으로부터 수집된 다변량 샘플을 이용하여 평균벡터와 공분산행렬을 계산하고, MD값의 문턱값을 설정한다. 검증단계에서는, 평균벡터와 검증샘플들 간의 MD를 구한 후, 계산된 MD 값이 미리 설정된 문턱값보다 높으면 알람신호가 발생하게 된다. MD 기반의 고장탐지방법의 성능을 검증하기 위해서, 200MW 유연탄 화력발전소의 증기보일러 튜브누설로 인해서 발전정지 된 사례를 사용하였다. 실험결과는 MD 기반의 고장탐지기법이 발전정지가 발생하기 이전의 이상징후를 성공적으로 탐지할 수 있음을 보여준다. Since thermal power plant (TPP) equipment is operated under very high pressure and temperature, failures of the equipment give rise to severe losses of life and property. To prevent the losses, fault detection method is, therefore, absolutely necessary to identify abnormal operating conditions of the equipment in advance. In this paper, we present Mahalanobis distance (MD) based fault detection method for steam boiler tube in TPP. In the MD-based method, it is supposed that abnormal data samples are far away from normal samples. Using multivariate samples collected from normal target system, mean vector and covariance matrix are calculated and threshold value of MD is decided. In a test phase, after calculating the MDs between the mean vector and test samples, alarm signals occur if the MDs exceed the predefined threshold. To demonstrate the performance, a failure case due to boiler tube leakage in 200MW TPP is employed. The experimental results show that the presented method can perform early detection of boiler tube leakage successfully.

A generalized likelihood ratio test for normal mean when [formula omitted] is greater than [formula omitted

The problem of testing the population mean vector of high-dimensional multivariate data is considered. Inspired by Roy’s union–intersection test, a generalized high-dimensional likelihood ratio test for the normal population mean vector is proposed. The p-value for the test is obtained by using randomization method, which does not rely on assumptions about the structure of the covariance matrix. An interpretation of the new statistic is given, which does not rely on the normality assumption. Hence the proposed test is also available for non-normal multivariate population. Simulation studies show that the new test offers higher power than other two competing tests when the variables are dependent and performs particularly well for non-normal multivariate population.

Detection of structural changes through principal component analysis and multivariate statistical inference

This article introduces a new methodology for the detection of structural changes using a statistical data-driven modeling approach by means of a distributed piezoelectric active sensor network at different actuation phases. The three main features that characterize the proposed methodology are (a) the nature of the data used in the test since vectors of principal component analysis projections are used instead of the entire measured response of the structure or the coefficients of an AutoRegressive model, (b) the number of data used since the test is based on two random samples instead of some characteristic indicators, and (c) the samples come from a multidimensional variable and therefore a test for the plausibility of a value for a normal population mean vector is performed. The framework of multivariate statistical inference is used with the objective of the classification of structures in healthy or damaged. The novel scheme for damage detection presented in this article —based on multivariate inference over the principal component analysis projections of the raw data—is applied, validated, and tested on a small aluminum plate. The results show that the presented methodology is able to accurately detect damages, that is, for each actuation phase, a unique and reliable damage detection indicator is obtained no matter the number of sensors and/or actuators. It is worth noting that a major contribution of this article is that there exists an entire range of significance levels where the multivariate statistical inference is able to offer a correct decision although all of the univariate tests make a wrong decision.