Multivariate Normal Populations Research Articles

Predicting Microbial Protein Synthesis in Cattle: Evaluation of Extant Equations and Steps Needed to Improve Accuracy and Precision of Future Equations.

Predictions of microbial crude protein (MCP) synthesis for beef cattle generally rely on empirical regression equations, with intakes of energy and protein as key variables. Using a database from published literature, we developed new equations based on the intake of organic matter (OM) and intakes or concentrations of crude protein (CP) and neutral detergent fiber (NDF). We compared these new equations to several extant equations based on intakes of total digestible nutrients (TDN) and CP. Regression fit statistics were evaluated using both resampling and sampling from a simulated multivariate normal population. Newly developed equations yielded similar fit statistics to extant equations, but the root mean square error of prediction averaged 155 g (28.7% of the mean MCP of 540.7 g/d) across all equations, indicating considerable variation in predictions. A simple approach of calculating MCP as 10% of the TDN intake yielded MCP estimates and fit statistics that were similar to more complicated equations. Adding a classification code to account for unique dietary characteristics did not have significant effects. Because MCP synthesis is measured indirectly, most often using surgically altered animals, literature estimates are relatively few and highly variable. A random sample of individual studies from our literature database indicated a standard deviation for MCP synthesis that averaged 19.1% of the observed mean, likely contributing to imprecision in the MCP predictions. Research to develop additional MCP estimates across various diets and production situations is needed, with a focus on developing consistent and reliable methodologies for MCP measurements. The use of new meta-omics tools might improve the accuracy and precision of MCP predictions, but further research will be needed to assess the utility of such tools.

Inference about a Common Mean Vector from Several Independent Multinormal Populations with Unequal and Unknown Dispersion Matrices

This paper addresses the problem of making inferences about a common mean vector from several independent multivariate normal populations with unknown and unequal dispersion matrices. We propose an unbiased estimator of the common mean vector, along with its asymptotic estimated variance, which can be used to test hypotheses and construct confidence ellipsoids, both of which are valid for large samples. Additionally, we discuss an approximate method based on generalized p-values. The paper also presents exact test procedures and methods for constructing exact confidence sets for the common mean vector, with a comparison of the local power of these exact tests. The performance of the proposed methods is demonstrated through a simulation study and an application to data from the Current Population Survey (CPS) Annual Social and Economic (ASEC) Supplement 2021 conducted by the U.S. Census Bureau for the Bureau of Labor Statistics.

Exact sampling distribution of the general case sample correlation matrix

The distribution of the sample correlation matrix when the sample comes from a real multivariate normal population and when the population covariance matrix is diagonal or when the population variables are independently distributed is available in the literature. In the general case, that is, when the covariance matrix in the normal population is a general real positive definite matrix, the distribution of the sample correlation matrix is not available in a compact computable form in the literature. Some scattered results on some aspects of the general case are available, but the representations are not easily tractable. This article deals with a covariance structure coming from a real matrix-variate gamma when the scale parameter matrix is a general real positive definite matrix. General procedures for deriving asymptotic chi-square and asymptotic normal for some general moment structures are also presented in this article. The density in some particular cases are given explicitly. Graphs are also provided in the particular cases.

High-dimensional hypothesis testing for allometric extension model

Allometric extension model is a multivariate mixture model, in which two population distributions have a common first principal component vector of their covariance matrices and the direction of the difference of their mean vectors coincides with the first principal component vector. This paper studies hypothesis testing for the allometric extension model in a high-dimensional setting when the multivariate normal population distributions are assumed. A test statistic is constructed and its asymptotic normality is shown under the null hypothesis. The consistency of the test is discussed under the alternative hypothesis. Several numerical results on the size and power of the test are also presented.

An Edgeworth-type expansion for the distribution of a likelihood-based discriminant function

ABSTRACT The exact distribution of a classification function is often complicated to allow for easy numerical calculations of misclassification errors. The use of expansions is one way of dealing with this difficulty. In this paper, approximate probabilities of misclassification of the maximum likelihood-based discriminant function are established via an Edgeworth-type expansion based on the standard normal distribution for discriminating between two multivariate normal populations.

Consecutive Bayes factor for the mean vector

In this article, we propose an explicit closed-form fully objective Bayes factor for one-sample hypotheses testing of the mean vector of multivariate normal population. The proposed approach can be regarded as a Bayesian version of the Hotelling’s T 2 test and has various appealing properties in practical applications. It relies on data only through the T 2-statistic and can easily be calculated. The proposed Bayes factor is applicable for the multivariate paired test as well as the univariate case. In this article, we also introduce a simple idea of consecutive minimal training samples which leads to a fully objective Bayes factor. Simulated and real data examples are provided for illustrative purposes.

Moments of the likelihood-based discriminant function

The likelihood approach used in this paper leads to quadratic discriminant functions. Classification into one of two known multivariate normal populations with a known and unknown covariance matrix are separately considered, where the two cases depend on the sample size and an unknown squared Mahalanobis distance. Their exact distributions are complicated to obtain. Therefore, moments for the likelihood based discriminant functions are established to express the basic characteristics of respective distribution.

Asymptotic approximation of misclassification probabilities in linear discriminant analysis with repeated measurements

We propose asymptotic approximations for the probabilities of misclassification in linear discriminant analysis when the group means follow a growth curve structure. The discriminant function can classify a new observation vector of p repeated measurements into one of several multivariate normal populations with equal covariance matrix. We derive certain relations of the statistics under consideration in order to obtain asymptotic approximation of misclassification errors for the two group case. Finally, we perform Monte Carlo simulations to evaluate the reliability of the proposed results.

Robust generalised quadratic discriminant analysis

Quadratic discriminant analysis (QDA) is a widely used statistical tool to classify observations from different multivariate Normal populations. The generalized quadratic discriminant analysis (GQDA) classification rule/classifier, which generalizes the QDA and the minimum Mahalanobis distance (MMD) classifiers to discriminate between populations with underlying elliptically symmetric distributions competes quite favorably with the QDA classifier when it is optimal and performs much better when QDA fails under non-Normal underlying distributions with heavy tail, e.g. Cauchy distribution. However, the classification rule in GQDA is still based on the sample mean vector and the sample dispersion matrix of a training set, which are extremely non-robust under data contamination. In real world, however, it is quite common to face data which are highly vulnerable to outliers and so the lack of robustness of the classical estimators of the mean vector and the dispersion matrix reduces the efficiency of the GQDA classifier significantly, increasing the misclassification errors. The present paper investigates the performance of the GQDA classifier when the classical estimators of the mean vector and the dispersion matrix used therein are replaced by various robust counterparts. Applications to various real data sets as well as simulation studies reveal far better performance of the proposed robust versions of the GQDA classifier. A comparative study has been made to advocate the appropriate choice of the robust estimators to be used in a specific situation.

Test for equality of generalized variance in high-dimensional and large sample settings

Generalized variance (GV), proposed by Wilks [8], is a one-dimensional measure of multidimensional scatter. It plays an important role in both theoretical and applied research on analyzing big data. This article examines the problem of testing equality of generalized variances of k multivariate normal populations in high-dimensional and large sample settings. The conventional likelihood-ratio test statistic reveals a serious bias as dimensions increase. We present a new test statistic that eliminates this bias, and propose an asymptotic approximation-based test. The likelihood-ratio test statistic can be interpreted as an estimator of criteria related to Jensen’s inequality. Our test statistic is obtained by appropriately estimating this criteria in high-dimensional and large sample settings. In addition, our proposed test is valid not only in high dimensional settings but also in large sample settings. We also obtain the asymptotic non-null distribution of the proposed test in high-dimensional and large sample settings. Finally, we study the finite sample and dimension behavior of this test through Monte Carlo simulations.

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

In a 2-step monotone missing dataset drawn from a multivariate normal population, T2-type test statistic (similar to Hotelling’s T2 test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T2 test and LR test are equivalent, however T2-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T2-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.

A k-stage procedure for estimating the mean vector of a multivariate normal population

In this article, we have estimated the mean vector of a multivariate normal population by using a k-stage sequential estimation procedure. Point estimation as well as confidence region estimation is done. Second-order approximations are obtained in both the cases. In case of minimum risk point estimation of , negative regret is achieved.

Second-order approximations for a multivariate analog of Behrens-Fisher problem through three-stage procedure

In this article, we have proposed a three-stage procedure for the estimation of the difference of the means of two multivariate normal populations having unknown and unequal variances. Point as well as confidence region estimation is done for the same. Here, we have used the concept of classical Behrens-Fisher problem. Second-order approximations are obtained in both the cases, i.e., point estimation and confidence region estimation.

Testing equality of standardized generalized variances of k multivariate normal populations with arbitrary dimensions

For a p-variate normal distribution with covariance matrix $$ {\varvec{\Sigma }}$$, the standardized generalized variance (SGV) is defined as the positive pth root of $$ |{\varvec{\Sigma }}| $$ and used as a measure of variability. Testing equality of the SGVs, for comparing the variability of multivariate normal distributions with different dimensions, is still regarded as matter of interest. The most classical test for this problem is the likelihood ratio test (LRT). In this article, testing equality of the SGVs of k multivariate normal distributions with possibly unequal dimensions is studied. To test this hypothesis, two approximations for the null distribution of the LRT statistic are proposed based on the well known Welch–Satterthwaite and Bartlett adjustment distribution approximation methods. Furthermore, the high-dimensional behavior of these approximated distributions is also investigated. Through a wide simulation study: first, the performance of the proposed tests with the classical LRT is compared in terms of type I error, power, and alpha adjusted equivalents; second, the robustness of the procedures with respect to departures from normality assumption is evaluated. Finally, the proposed methods are illustrated with two real data examples.

Asymptotic Properties for the Eigenvalue and Eigenvector of a Covariance Matrix under Two-Step Monotone Incomplete Multivariate Normal Data

SYNOPTIC ABSTRACTWe derive the asymptotic properties for the eigenvalue and eigenvector of a covariance matrix in the context of two-step monotone incomplete data drawn from , which is a multivariate normal population with mean and covariance matrix . Our data consist of n observations of all p + q variables and an additional N – n observations of the first p variables; all observations are mutually independent. We use a maximum likelihood estimator (MLE) and an unbiased estimator (UBE) for a covariance matrix . Furthermore, we correct for bias of the eigenvalue and eigenvector by evaluating asymptotic expectations. Finally, we investigate the accuracy of our results using numerical simulations.

A Test on the Multivariate Behrens–Fisher Problem in High–Dimensional Data by Block Covariance Estimation

In this paper, we proposed a new testing statistic for testing the equality of mean vectors from two multivariate normal populations when the covariance matrices are unknown and unequal in high–dimensional data. A new test is proposed based on the idea of keeping more information from the sample covariance matrices as much as possible. A proposed test is invariant under scalar transformations and location shifts. We showed that the asymptotic distribution of proposed statistic is standard normal distribution when number of random variables approach infinity. We also compared the performance of the proposed test with other three existing tests by the simulation study. The simulation results showed that the attained significance level of proposed test close to setting nominal significance level satisfactorily. The attained power of proposed test outperforms as the other comparative tests under form of covariance matrices considered which can be arranged to block diagonal matrix structure. The attained power becomes more powerful when the dimension increases for a given sample size or vice versa, or relationship level between random variables in each sample increases. Finally, the proposed test is also illustrated with an analysis of DNA microarray data.

A general testing for order restriction on mean vectors of multivariate normal populations

Multivariate isotonic regression theory plays a key role in the field of testing statistical hypotheses under order restriction for vector valued parameters. This kind of statistical hypothesis 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 the present paper, we are interested in a general testing for order restriction of mean vectors against all possible alternatives based on a random sample from several dimensional normal populations when the unknown covariance matrices are common. In fact, this problem of testing is an extension of Bazyari and Chinipardaz's (2012) problem. We propose an approximate test statistic by likelihood ratio method based on orthogonal projections on the closed convex cones, study its upper tail probability under the null hypothesis and estimate its critical values for different significance levels by using Monte Carlo simulation. The problem of testing and obtained results is illustrated with a real example where this inference problem arises to evaluate the effect of Vinylidene fluoride on liver damage.

On the asymptotic distribution of T2-type statistic with two-step monotone missing data

In this article, we consider the asymptotic distribution of Hotelling’s T2-type test statistic when a two-step monotone missing data set is drawn from a multivariate normal population under a large-sample asymptotic framework. In particular, asymptotic expansions for the distribution and upper percentiles are derived using a perturbation method up to the terms of order n−1, where n = N-2 and N denotes the total sample size. Furthermore, making use of Fujikoshi’s transformations, we also have Bartlett-type corrections of the test statistic considered in this article. Finally, we investigate the performance of the proposed approximation to the upper percentiles and Bartletttype correction for the test statistic by conducting Monte Carlo simulations for some selected parameters.

Bayes minimax competitors of preliminary test estimators in k sample problems

In this paper, we consider the estimation of a mean vector of a multivariate normal population where the mean vector is suspected to be nearly equal to mean vectors of $k-1$ other populations. As an alternative to the preliminary test estimator based on the test statistic for testing hypothesis of equal means, we derive empirical and hierarchical Bayes estimators which shrink the sample mean vector toward a pooled mean estimator given under the hypothesis. The minimaxity of those Bayesian estimators are shown, and their performances are investigated by simulation.

Confidence intervals for product of powers of the generalized variances of k multivariate normal populations

ABSTRACTIn this paper, some confidence intervals (CIs) for the product of powers of the generalized variances of k multivariate normal populations with possibly different dimensions are proposed. The performance of these CIs in terms of the coverage probabilities and average lengths were evaluated via a Monte Carlo simulation study. The results were found to be satisfactory. To demonstrate utility of the proposed CIs, applications on three real data sets were provided.