Multivariate Normal Populations Research Articles

Cost-Risk Analysis in Statistical Validating Simulation Models of Service Processes

In this paper, for validating computer simulation models of service processes, an uniformly most powerful invariant (UMPI) test is developed from the generalized maximum likelihood ratio (GMLR). This test can be considered as a result of a new approach to solving the Behrens-Fisher problem when covariance matrices of two multivariate normal populations (compared with respect to their means) are different and unknown. The test is based on invariant statistic whose distribution, under the null hypothesis, does not depend on the unknown (nuisance) parameters. The sample size and threshold of the UMPI test are determined from minimization of the weighted sum of the model builder's risk and the model user's risk. The proposed test could result in the saving of sample items, if the items of the sample are observed sequentially. In this paper we present the exact form of the proposed curtailed procedure and examine the expected sample size savings under the null hypothesis. The sample size savings can be bounded by a constant, which is independent of the sample size. Tables are given for the expected sample size savings and maximum sample size saving under the null hypothesis for a range of significance levels (α), dimensions (p) and sample sizes (n). The curtailed test considered in this paper represents improvement over the noncurtailed or standard fixed sample tests.

TWO-SAMPLE BAYESIAN TESTS USING INTRINSIC BAYES FACTORS FOR MULTIVARIATE NORMAL OBSERVATIONS

Unlike estimation problems, it is necessary to be careful to use noninformative priors in Bayesian model selection or testing problems. Since these priors are typically improper and are defined only up to arbitrary constants, the resulting Bayes factors are then not well defined. A recently proposed model selection criterion, the intrinsic Bayes factor (IBF), overcomes such problems by using a part of the sample as a training sample. In this article, the IBF criterion was used to conduct a multiple test of two independent multivariate normal populations, which is commonly used in longitudinal data analysis and repeated measurements experiments. Finally, a simulation study was performed to support theoretical results.

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. I. The Equal-Means Case

We consider the problem of discriminating, on the basis of random “training” samples, between two independent multivariate normal populations, Np(μ, Σ1) and Np(μ, Σ2), which have a common mean vector μ and distinct covariance matrices Σ1 and Σ2. Using the theory of Bessel functions of the second kind of matrix argument developed by Herz (1955, Ann. Math.61, 474–523), we derive stochastic representations for the exact distributions of the “plug-in” quadratic discriminant functions for classifying a newly obtained observation. These stochastic representations involve only chi-squared and F-distributions, hence we obtain an efficient method for simulating the discriminant functions and estimating the corresponding probabilities of misclassification. For some special values of p, Σ1 and Σ2 we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks (1933, Ann. Eugen.5, 1–55) in a biometric investigation of the physical characteristics of twins, and to data provided by Rencher (1995, “Methods of Multivariate Analysis,” Wiley, New York) in a study of the relationship between football helmet design and neck injuries. For each application we estimate the exact probabilities of misclassification, and in the case of Stocks' data we make extensive comparisons with previously published estimates.

CONFIDENCE INTERVALS FOR THE MAHALANOBIS DISTANCE

Mahalanobis distances appear, often in a disguised form, in many statistical problems dealing with comparing two multivariate normal populations. Assuming a common covariance matrix the overlapping coefficient (Bradley, E.L. Encyclopedia of Statistical Sciences; 1985), optimal error rates (Rao, P.S.R.S.; Dorvlo, A.S.S. Commun. Statist.—Simula. Computation 1985, 14, 774–790), and the generalized ROC criterion (Reiser, B.; Faraggi, D. Biometrics 1997, 53, 644–652) are all monotonic functions of the Mahalanobis distance. Approximate confidence intervals for all of these have appeared in the literature on an ad-hoc basis. In this paper, we provide a unified approach to obtaining an effectively exact confidence interval for the Mahalanobis distance and all the above measures.

LIKELIHOOD RATIO TEST OF EQUALITY OF SEVERAL VARIANCES IN THE INTRACLASS CORRELATION MODEL

The likelihood ratio test of the hypothesis of equality of several variances is deduced for a sample of independent observations from a multivariate normal population whose covariance matrix has intraclass correlation. The resulting test, related to Bartlett's test, is derived when all population correlations are zero. The analysis is related to a test in the mixed linear model. A numerical illustration is provided.

Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions

A completely nonparametric approach to population bioequivalence in crossover trials has been suggested by Munk and Czado (1999). It is based on the Mallows (1972) metric as a nonparametric distance measure which allows the comparison between the entire distribution functions of test and reference formulations. It was shown that a separation between carry-over and period effects is not possible in the nonparametric setting. However when carry-over effects can be excluded, treatment effects can be assessed when period effects are or not. Munk and Czado (1999) proved bootstrap limit laws of the corresponding test statistics because estimation of the limiting variance of the test statistic is very cumbersome. The purpose of this paper is to investigate the small sample behavior of various bootstrap methods and to compare it with the asymptotic test obtained by estimation of the limiting variance. The percentile (PC) and bias correct- ed and accelerated (BCA) bootstrap were compared for multivariate normal and nonnormal populations. From the simulation results presented, the BCA bootstrap is found to be less conservative and provides higher power compared to the PC bootstrap, especially when skewed multivariate populations are present.

Simultaneous Estimation of Several Intraclass Correlation Coefficients

Based on shrinkage and preliminary test rules, various estimators are proposed for estimation of several intraclass correlation coefficients when independent samples are drawn from multivariate normal populations. It is demonstrated that the James-Stein type estimators are asymptotically superior to the usual estimators. Furthermore, it is also indicated through asymptotic results that none of the preliminary test and shrinkage estimators dominate each other, though they perform relatively well as compared to the classical estimator. The relative dominance picture of the estimators is presented. A Monte Carlo study is performed to appraise the properties of the proposed estimators for small samples.

Discriminant Analysis When a Block of Observations is Missing

We consider the problem of classifying a p× 1 observation into one of two multivariate normal populations when the training samples contain a block of missing observations. A new classification procedure is proposed which is a linear combination of two discriminant functions, one based on the complete samples and the other on the incomplete samples. The new discriminant function is easy to use. We consider the estimation of error rate of the linear combination classification procedure by using the leave-one-out estimation and bootstrap estimation. A Monte Carlo study is conducted to evaluate the error rate and the estimation of it. A numerical example is given tof illustrate its use.

Estimating the Mahalanobis distance from mixed continuous and discrete data.

We present a method for estimating the Mahalanobis distance between two multivariate normal populations when a subset of the measurements is observed as ordered categorical responses. Asymptotic properties of the proposed estimator are developed. Two examples are discussed.

Error Bounds for Asymptotic Approximations of the Linear Discriminant Function When the Sample Sizes and Dimensionality are Large

Theoretical accuracies are studied for asymtotic approximations of the expected probabilities of misclassification (EPMC) when the linear discriminant function is used to classify an observation as coming from one of two multivariate normal populations with a common covariance matrix. The asymptotic approximations considered are the ones under the situation where both the sample sizes and the demensionality are large. We give explicit error bounds for asymptotic approximations of EPMC, based on a general approximation result. We also discuss with a method of obtaining asymptotic expansions for EPMC and their error bounds.

A Simulation Study of a Partitioning Procedure for Signal Detection with an Application in Medical Imaging

We study the performance of a screening procedure R of CHEN, MELVIN, and WICKS (1999) when it is adopted prior to the signal detection procedure R kkr proposed independently by KELLY (1986) and KHATRI and RAO (1987). Through simulation results, we show that the probability of detection for R kkr is significantly improved when procedure R is first used to screen out nonhomogeneous data. Procedure R is a selection procedure which compares k (≥1) experimental populations with a control population and eliminates the dissimilar experimental populations. An experimental population with covariance matrix Σ is said to be similar to the control population with covariance matrix Σ 0 if Σ 0 Σ -1 is close to the identity matrix in certain meaning of closeness to be defined later in the paper. As mentioned in CHEN et al. (1999), procedure R can be used as a screening process prior to any traditional signal processing detection algorithm which requires the assumption of the same covariance matrices for the experimental populations as for the control population. A commonly used such detection algorithm is the one proposed independently by KELLY (1986) and KHATRI and RAO (1987). In this paper, we first simulate data from k + I complex multivariate normal populations which all have zero mean vectors and have different covariance matrices. One of the populations represents the control population and the remaining populations are the experimental populations. Then we apply procedure R to the simulated experimental populations to screen out the dissimilar populations. Finally, R kkr is used to detect a target using respectively the unscreened and the screened data. We present simulation results on the powers of the test R kkr when it is applied to the unscreened data and the screened data. The results illustrate that, under the nonhomogeneous environment where covariance matrices of the experimental populations are different from the covariance matrix of the control population, we can always improve the power of the test R kkr by employing the procedure R. We also present an example illustrating the potential application of our study in medical imaging.

Comparison of Approximation Methods for Computing Tolerance Factors for a Multivariate Normal Population

In this article, we compare several approximation methods for computing the tolerance factors of a multivariate normal population. These approximate methods are evaluated by comparing the Monte Carlo estimates of the coverage probabilities with those of the specified ones. Numerical studies indicate that, in general, the tolerance factors based on an approximate method given by John, which is commonly used in the literature, are inaccurate when the number of variates is greater than or equal to 2, but another approximation (which is also due to John) with slight modification gives better results. Using John's idea, we also suggest some new approximations, which give satisfactory results. Two of the new approximations emerge as satisfactory candidates for practical use. Our fairly extensive numerical results provide guidelines regarding the choice of the tolerance factor for practical applications.

Comparison of Approximation Methods for Computing Tolerance Factors for a Multivariate Normal Population

In this article, we compare several approximation methods for computing the tolerance factors of a multivariate normal population. These approximate methods are evaluated by comparing the Monte Carlo estimates of the coverage probabilities with those of the specified ones. Numerical studies indicate that, in general, the tolerance factors based on an approximate method given by John, which is commonly used in the literature, are inaccurate when the number of variates is greater than or equal to 2, but another approximation (which is also due to John) with slight modification gives better results. Using John's idea, we also suggest some new approximations, which give satisfactory results. Two of the new approximations emerge as satisfactory candidates for practical use. Our fairly extensive numerical results provide guidelines regarding the choice of the tolerance factor for practical applications.

Some computational issues in cluster analysis with no a priori metric

Recent interest in data mining and knowledge discovery underscores the need for methods by which patterns can be discovered in data without any prior knowledge of their existence. In this paper, we explore computational methods of finding clusters of multivariate data points when there is no metric given a priori. We are given a sample, X , of n points in R p that come from g distinct multivariate normal populations with unknown parameters each of which contributes in excess of p points. Based on the assumption that we are given the number of groups, g, and a computational budget of T seconds of computer time, the paper reviews choices for cluster finding that have been described in the literature and introduces a new method that is a structured combination of two of them. We investigate these algorithms on some real data sets and describe simulation experiments. A principal conclusion is strong support for the contention that a two-stage algorithm based on a combinatorial search followed by the EM algorithm is the best way to find clusters.

Distributions of Hotelling's T 2 and Multiple and Partial Correlation Coefficients for the Mixture of Two Multivariate Gaussian Populations

For the mixture of two multivariate normal populations Srivastava and Awan (1982), and Srivastava (1983) derive the densities of Hotelling's-T 2, and the multiple correlation coefficient. They do not express their results in the standard forms available for a single multivariate normal population. In this paper we derive these distributions in the standard forms and also study the distribution of partial correlation coefficient. The results depend on the fact that the latent vectors of an identity matrix and of a projection idempotent matrix are the same.

Wald statistic for iiomogeneity of multiple parameters

The paper proposes Wald statistic for testing homogeneity of parameters of k populations and gives the asymptotic expansion of the distribution function under a null hypothesis and of power function of it under contiguous alternatives. It shows that Wald statistic is the same as Nagao's statistic (1973) for testing homogeneity of covariance matrices of multivariate normal populations.

Polynomial estimation of eigenvalues

In estimating the eigenvalues of the covariance matrix of a multivariate normal population, the usual estimates are the eigenvalues of the sample covariance matrix. It is well known that these estimates are biased. This paper investigates obtaining improved eigenvalue estimates through improved estimates of the characteristic polynomial, which is a function of the sample eigenvalues. A numerical study investigates the improvements evaluated under both a square error and an entropy loss function.

Three Types of Gamma‐Ray Bursts

A multivariate analysis of gamma-ray burst (GRB) bulk properties is presented to discriminate between distinct classes of GRBs. Several variables representing burst duration, Nuence, and spectral hardness are considered. Two multivariate clustering procedures are used on a sample of 797 bursts from the Third BATSE Catalog, a nonparametric average linkage hierarchical agglomerative clustering pro- cedure validated with WilksI * and other multivariate analysis of variance tests and a parametric maximum likelihood modelEbased clustering procedure assuming multinormal populations calculated with the Expectation-Maximization algorithm and validated with the Bayesian Information Criterion. The two methods yield very similar results. The BATSE GRB population consists of three classes with the following duration/Nuence/spectrum bulk properties: class I with long/bright/soft bursts, class II with short/faint/hard bursts, and class III with intermediate/intermediate/soft bursts. One outlier due to spu- rious data is also present. Classes I and II correspond to those reported by Kouveliotou et al., but class III is clearly de-ned here for the -rst time. Subject headings: gamma rays: bursts E methods: data analysis E methods: statistical

Calculating the mvue of the fraction nonconforming for the bivariate normal

The minimum variance unbiased estimator of the proportion lying outside an m-dimensional rectangle for multivariate normal populations was derived by Baillie (1987a, b). The estimator is a natural extension of a univariate estimator widely used in acceptance sampling. Computation of the multivariate estimator is nontrivial; one must integrate a multivariate density over the intersection of an m-dimensional ellipsoid and an m-dimensional rectangle. We propose an algorithm for the bivariate case which involves a one-dimensional numerical integration and calls to routines for either an incomplete beta function or a Student's t cumulative distribution function

Inferential Statistical Method for Analysis of Nonsinusoidal Hybrid Time Series with Unequidistant Observations

Most variables of interest in laboratory medicine show predictable changes with several frequencies in the span of time investigated. The waveform of such nonsinusoidal rhythms can be well described by the use of multiple components rhythmometry, a method that allows fitting a linear model with several cosine functions. The method, originally described for analysis of longitudinal time series, is here extended to allow analysis of hybrid data (time series sampled from a group of subjects, each represented by an individual series). Given k individual series, we can fit the same linear model with m different frequencies (harmonics or not from one fundamental period) to each series. This fit will provide estimations for 2m + 1 parameters, namely, the amplitude and acrophase of each component, as well as the rhythm-adjusted mean. Assuming that the set of parameters obtained for each individual is a random sample from a multivariate normal population, the corresponding population parameter estimates can be based on the means of estimates obtained from individuals in the sample. Their confidence intervals depend on the variability among individual parameter estimates. The vari-ance-covariance matrix can then be estimated on the basis of the sample covariances. Confidence intervals for the rhythm-adjusted mean, as well as for the amplitude-acrophase pair, of each component can then be computed using the estimated covariance matrix. The p-values for testing the zero-amplitude assumption for each component, as well as for the global model, can finally be derived using those confidence intervals and the t and F distributions. The method, validated by a simulation study and illustrated by an example of modeling the circadian variation of heart rate, represents a new step in the development of statistical procedures in chronobiology.