Multiple outlier detection in multivariate data using projection pursuit techniques

Several methods of construction of generalised neighbour designs of equal and unequal block sizes are presented alongwith examples. Some methods for constructing neighbour designs of Rees are also given. The outline of analysis of generalised neighbour design has also been suggested.

Families of neighbour designs and their analysis

For any analysis-of-variance model with balanced data involving both fixed and random effects, uniformly most powerful unbiased (UMPU) and uniformly most powerful invariant unbiased (UMPIU) tests are derived for the significance of a fixed effect or a variance component, under the assumption of normality of random effects. The tests coincide with the usual F tests. Robustness of the UMPIU test against suitable deviations from normality is established.

Optimum Tests for Fixed Effects and Variance Components in Balanced Models

Environmental data are usually multivariate, with the variables conforming to some correlation structure. Occasionally, measurements which do not conform in structure or magnitude may occur in one or more variables. It is important (1) to characterize these discordancies in terms of the disturbed variables and the direction and magnitude of the anomalous error and (2) to associate each discordant observation with a specific cause of measurement error in order to prevent further mismeasurement. We describe a procedure for identifying suspected causes of discordant observations in otherwise multinormal data sets. Variables are assigned to groups, each of which is associated with a specific cause of measurement error. Discordant observations are identified with the generalized distance test or the multivariate kurtosis test. Suspected causes of measurement error are identified by repeating the tests with one of the groups of variables omitted in each analysis. The procedures are evaluated with simulated data sets having a correlation structure similar to that of a large environmental data set.

Finding suspected causes of measurement error in multivariate environmental data

In this note certain optimum and robustness properties of usual F test are derived, for balanced random and mixed effects nested models The underlying distribution assumed is elliptically symmetric WIJSMAN'S Representation theorem is applied as a tool Attention is confined to the testing of random effects only

On robustness of tests for random effects in balanced nested designs

: An extension of Ferguson's univariate normal results for detection of outliers with variance slippage is made to the multivariate elliptically symmetric case with dispersion slippage. The locally optimum test statistic derived possesses all three robustness properties: optimality, null and nonnull. As a technical tool, Wijsman's representation theorem is used. Additional keywords: Locally best invariant, Maximal invariant, and Variance slippage. (Author)

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Detection of Multivariate Outliers with Dispersion Slippage in Elliptically Symmetric Distributions

Consider the canonical-form MANOVA setup with X: n × p = (+ E, Xi ni × p, i = 1, 2, 3, Mi: ni × p, i = 1, 2, n1 + n2 + n3) p, where E is a normally distributed error matrix with mean zero and dispersion In (> 0 (positive definite). Assume (in contrast with the usual case) that M1i is normal with mean zero and dispersion In1) and M22 is either fixed or random normal with mean zero and different dispersion matrix In2 (being unknown. It is also assumed that M1E, and M2 (if random) are all independent. For testing H0) = 0 versus H1: (> 0, it is shown that when either n2 = 0 or M2 is fixed if n2 > 0, the trace test of Pillai (1955) is uniformly most powerful invariant (UMPI) if min(n1, p)= 1 and locally best invariant (LBI) if min(n1p) > 1 underthe action of the full linear group Gl (p). When p > 1, the LBI test is also derived under a somewhat smaller group GT(p) of p × p lower triangular matrices with positive diagonal elements. However, such results do not hold if n2 > 0 and M2 is random. The null, nonnull, and optimality robustness of Pillai's trace test under Gl(p) for suitable deviations from normality is pointed out.



Considerons la forme canonique du modele d'analyse de variance multivariee ou X: n × p = (E, Xi × ni × p, i = 1, 2, 3, Mi: ni × p, i = 1, 2, n1 + n2 + n3) p, ou E est une matrice de termes d'erreur de loi multinormale avec moyenne nulle et dispersion In (> 0 (definie positive). Contrairement au cas classique, supposons que M1 admette une loi multinormale de moyenne nulle et dispersion In1) que M2 soit, ou bien fixe, ou bien de loi multinormale avec moyenne nulle et dispersion In2 (et) etant inconnues. Supposons egalement que M1M2 (si aleatoire) et E soient mutuellement independantes. Nous desirons confronter les hypotheses H0: (= 0 et H1) 0. Nous montrons que, lorsque n2 = 0 ou lorsque M2 est fixe si n2 > 0, le test de la trace de Pillai (1955) est le test invariant uniformement le plus puissant (IUPP) si min (n1 p)= 1 et localement le plus puissant (ILPP) si min (n1p) > 1 sous ľaction du groupe lineaire complet Gl(p). Lorsque p > 1, nous construisons egalement le test ILPP mais sous l'action d'un groupe plus restreint, a savoir, le groupe Gr(p) des matrices triangulaires inferieures p × p ne comportant que des elements positifs sur la diagonale. Toutefois, de tels resultats ne tiennent plus lorsque n2 > 0 et M2 est aleatoire. Enfin, nous faisons etat de la robustesse du test de la trace de Pillai sous Gl(p) quant a ses lois de probabilite sous les deux hypotheses et quant a son optimalite lorsque la loi qui regit les observations s'ecarte de la normale.

Optimum invariant tests for random manova models

: In one-way random effects unbalanced model the locally best invariant test for the equality of the treatment effects is derived Surprisingly, this is different from the widely used familiar F-test In the balanced case, however the two tests coincide and represent the uniformly most powerful invariant tests, For two-way random effects and mixed effects balanced models, the uniformly most powerful invariant test for the equality of the treatment effects is derived both with and without interaction, and shown to be equivalent to the usual F-tests under fixed effects models The optimum invariant tests derived here are shown not to depend on the assumption of normality Different aspects of null, nonnull and optimality robustness of these tests (Kariya and Sinha, Annals of Statistics, 1985) are studied In the unbalanced two-way models however unlike in the fixed effects model providing a UMPI test, both random and mixed effects models present a difficulty which is pointed out Keywords: Multivariate analysis; Analysis of variance

Robust Optimum Invariant Tests in One-Way Unbalanced and Two-Way Balanced Models.

: This document considers left orthogonally invariant distribution, locally best invariants, MANOVA problems, maximal invariants, mixed effects models, random effects, robustness, spherically symmetric distribution, uniformly most powerful invariants, and Wijsman's representation theorm.

Robust Optimum Invariant Tests for Random MANOVA Models.

: This paper investigates robust optimum invariant tests of some covariance structurs that naturally arise in the context of robustness study in linear models. To describe this concept, let (Y, X beta, sigma sq I) be the assumed (probably incorrect) model while (Y, X beta, sigma sqV) be the correct model, resulting in the specification error in the dispersion matrix. Then it is well known that the BLUEs of all estimable linear parametric function A beta remain the same under both the models if and only if the following condition holds on the structure of V: LX'VZ = 07) where z denotes a matrix of maximal rank satisfying the condition Z'X = 0. Our object is to test the null hypothese that V possesses the structure based on samples on Y under the model (Y, X beta, sigma sq V) for a fixed design matrix X. This hypothesis is of considerable interest as its acceptance greatly simplifies determination of BLUEs of estimable linear parametric functions.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA174659&Location=U2&doc=GetTRDoc.pdf

Rita Das

Papers

Detection of Multivariate Outliers with Dispersion Slippage in Elliptically Symmetric Distributions

Optimum invariant tests for random manova models

Robust Optimum Invariant Tests in One-Way Unbalanced and Two-Way Balanced Models.

Robust Optimum Invariant Tests for Random MANOVA Models.

Robust Optimium Invariant Tests of Covariance Structures Useful in Linear Models.