Some Applications of Order Statistics to Multivariate Analysis II

ABSTRACT

Harter (1979) summarized applications of order statistics to multivariate analysis up through 1949. The present paper covers the period 1950–1959. References in the two papers were selected from the first and second volumes, respectively, of the author's chronological annotated bibliography on order statistics [Harter (1978, 1983)]. Tintner (1950a) established formal relations between four special types of multivariate analysis: (1) canonical correlation, (2) principal components, (3) weighted regression, and (4) discriminant analysis, all of which depend on ordered roots of determinantal equations. During the decade 1950–1959, numerous authors contributed to distribution theory and/or computational methods for ordered roots and their applications to multivariate analysis. Test criteria for (i) multivariate analysis of variance, (ii) comparison of variance–covariance matrices, and (iii) multiple independence of groups of variates when the parent population is multivariate normal were usually derived from the likelihood ratio principle until S. N. Roy (1953) formulated the union–intersection principles on which Roy & Bose (1953) based their simultaneous test and confidence procedure. Roy & Bargmann (1958) used an alternative procedure, called the step–down procedure, in deriving a test for problem (iii), and J. Roy (1958) applied the step–down procedure to problem (i) and (ii), Various authors developed and applied distribution theory for several multivariate distributions. Advances were also made on multivariate tolerance regions [Fraser & Wormleighton (1951), Fraser (1951, 1953), Fraser & Guttman (1956), Kemperman (1956), and Somerville (1958)], a criterion for rejection of multivariate outliers [Kudô (1957)], and linear estimators, from censored samples, of parameters of multivariate normal populations [Watterson (1958, 1959)]. Textbooks on multivariate analysis were published by Kendall (1957) and Anderson (1958), as well as a monograph by Roy (1957) and a book of tables by Pillai (1957).     

Modified kolmogorov-smirnov tests of goodness of fit

The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D⁺ _n D^? _n and D_n are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C⁺ _n, C^? _n and C_n are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E⁺ _n, E^? _n and E_n corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.     

Modified Asymptotic Formulas for Critical Values of the Kolmogorov Test Statistic

A brief discussion is given of the Kolmogorov test of goodness of fit. Modified asymptotic formulas for critical values of the test statistic, much more accurate for small-to-moderate sample sizes than the usual asymptotic formulas, are given.     

Towards flexible organizations: A multidimensional design

The author considers one way by which the readiness, willingness, and ability of organizations to change themselves can be increased. Specifically, he considers how organizations can be designed to be more flexible and, therefore, more capable of being changed and changing themselves. Flexibility does not guarantee adaptability but it is essential for it.     

Affluence,Feelings of Stress,and Well-being

Data from the Gallup World Poll highlighted the differential relations between perceived stress, well-being, and wealth at the individual- versus nation-level. At the nation level, stress was a distinct concept from negative affect (NA). It correlated positively with well-being (positive affect, life satisfaction, and domain satisfaction) and wealth (as measured by income, gross domestic product, and modern conveniences). In contrast, NA correlated inversely with well-being and income. Although similar to NA at the individual level, stress showed weaker negative relations with well-being than NA did. In sum, nation-level stress and NA were related in the opposite direction to wealth (and poverty), well-being, and life expectancy. Furthermore, the concept of stress differed at the individual and nation levels. For the former, stress appeared to be purely a negative marker of affective well-being (albeit weaker than other discrete negative emotions); for the latter, it appeared to reflect lifestyle differences that were strongly associated with wealth, and with affective and cognitive well-being to a smaller degree.     

Marketing segmentation research—An application to bank services

The research reported here analyses data relating to bank services obtained by a stratified random sampling technique using stepwise linear regression and canonical correlation for the purpose of illustrating the application of segmentation principles to marketing research.     

Nonuniqueness of least absolute values regression

The question of nonuniqueness of the least absolute values (L₁) regression is discussed, and examples are given of situations where the L₁-regression is unique and where it has 2, 3 and 4 limiting positions. A method is proposed for finding a compromise regression line when the L₁ regression is not unique. Suggestions are made for further research.