Unbiasedness of Invariant Tests for Manova and Other Multivariate Problems

Perlman, Michael D.; Olkin, Ingram

doi:10.1214/aos/1176345204

Cited by 60 publications

(22 citation statements)

References 29 publications

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“…Furthermore, it is possible to construct a wide class of tests based on the eigenvalues {Am} of HE-I. Perlman and Olkin (1980) showed that any test with an acceptance region g ( ~, ... , AM ) ::;; c, where g is nondecreasing in each argument, is unbiased. They also supply monotonicity results of the power functions of such tests.…”

Section: Distribution Since the F(it-m-i) Distribution Limits The Xmentioning

confidence: 99%

Testing for Multivariate Autocorrelation

Holgersson

2004

Journal of Applied Statistics

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This paper concerns the problem of assessing autocorrelation of multivariate (i.e. system wise) models. It is well known that systemwise diagnostic tests for autocorrelation often suffers from poor small sample properties in the sense that the true size overstates the nominal size gravely. The failure of keeping control of the size usually stems from the fact that the critical values (used to decide the rejection area) originate from the slowly converging asymptotic null distribution.Another drawback of existing tests is that the power may be rather low if the deviation from the null is not symmetrical over the marginal models. In this paper we consider four quite different test techniques for autocorrelation. These are (i) Pillai's trace, (ii) Roy's largest root, (iii) the maximum F-statistic and (iv) the maximum (-test. We show how to obtain control of the size ofthe tests, and then examine the true (small sample) size and power properties by means of Monte Carlo simulations.

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Section: Distribution Since the F(it-m-i) Distribution Limits The Xmentioning

confidence: 99%

Testing for Multivariate Autocorrelation

Holgersson

2004

Journal of Applied Statistics

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“…Example 4 illustrates the use offamily weights to investigate the effects of sibship size on the degree of sibling similarity of total finger ridge;count ofthe two hands for children in 129 families. This is done by forming all Ki(Ki -1) possible pairs of siblings (Ki = number ofchildren in family i) for each family and assigning weights 1 (giving all sibpairs equal weight), 1/(Ki -1) (giving each child equal weight), or 1/Kj(K, -1) (giving each family equal weight) to each sibpair in the calculation of the association matrices (15).…”

Section: Examples From Real Datamentioning

confidence: 99%

“…Only the upper triangular portion of the association matrix is presented because the matrices are symmetric. 4. Representative association matrices with respect to all monotone functions P' for total finger ridge count of 129 Indian families Comparing the resultant association matrices reveals that corresponding elements are greater when larger families are more heavily weighted (i.e., sibpair > individual > family), suggesting that there is greater similarity between siblings for total ridge count in larger families [see Karlin et aL (13) Families with Asian-born parents perhaps maintain more disciplined (homogeneous) life-styles and cultural traditions, whereas European/American-born parents may follow more heterogeneous life-styles.…”

Section: Examples From Real Datamentioning

confidence: 99%

“…The concept of strong association with respect to families of functions I define the association array (an array of measures of dependence) for the variables {X,Y} to consist of the ensemble of correlations for an array oftransformed variables {+(X), +(Y)}, where 4 and qi belong to a natural class offunctions 4 E i and qi ',ES (9 and S may be the same) motivated by the problem.…”

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Association arrays in assessing forms of dependencies between bivariate random variables

Karlin¹

1983

Proc. Natl. Acad. Sci. U.S.A.

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The bivariate distribution of pairs of random variables (X,Y) is said to be associated with respect to the classes of functions 9; and ifthe product-moment correlation r[4(X), qf(Y)] 20 for all 4 E 9 and i E %. In the case in which both 9; = g = 9;* consist of all increasing functions, then the bivariate distribution of(X,Y) is said to be positive quadrant dependent. To apply the concept to data, I examine the correlations for classes of extremal functions that span by positive combinations the totality of functions 4 E Z and 4i E '9 to investigate whether the pair of random variables (X,Y) are associated with respect to i and g and to assess the relative degree (or strength) ofassociation when comparing two sets of random variables (X,Y) and (Z,W).paring degrees of relative deviation from a central point (e.g., deviations from the mean or median) by taking 9 and 'S to consist of all functions increasing away from a central point. In this specification, the association array corresponding to i or ' ; reduces to studying the pattern ofcorrelations ofthe pairs ofvariables O(IX -mxl) and qt(IY -myl) for all similarly monotone functions 4 and 4i defined for positive arguments where mx and my are suitable central points of the random variables X and Y, respectively (see example 6, Section 4). Henceforth, we designate i;* to be thefamily of all monotone increasingfunctions.Definition 1: The random variables X and Y are said to be associated with respect to the classes 9Z and '9 if The standard approach to measuring the degree ofdependence between two random variables X and Y involves the computation of a single statistic for the sample, resulting in some estimated measure of "overall" dependence for the distribution of(X,Y). The Pearson correlation and nonparametric correlation constructions such as Spearman's correlation coefficient, Kendall's ar and many other "monotonicity coefficients" are all of this type. However, the relationship among variables of real data is often stochastically nonlinear and usually cannot be properly summarized by a deterministic functional form having identified residual random components. Further, these statistics may be misleading when the form ofdependence varies over the range of the distribution.In this report, I present an approach to assessing the level and form of dependence for multivariate observations that provides a fine tuning in evaluating relationships ofpairs ofrandom variables by transforming the data in natural manifold ways and then computing the associated correlations whose totality reflects on the nature of dependence between the array of transformed variables.1. The concept of strong association with respect to families of functions I define the association array (an array of measures of dependence) for the variables {X,Y} to consist of the ensemble of correlations for an array oftransformed variables {+(X), +(Y)}, where 4 and qi belong to a natural class offunctions 4 E i and qi ',ES (9 and S may be the same) motivated by the problem.In many of the example...

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“…Also in Chapter 2 the Schur concave functions are characterized and the so-called Convolution Theorem of Marshall and Olkin (1974) is established. (Perlman and Olkin (1980)), and the second shows that the coordinates of a multivariate normal random vector are associated iff the elements of the covariance are all non-negative (Pitt (1982)). …”

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confidence: 99%