ROBUST M-ESTIMATION OF THE INTRACLASS CORRELATION COEFFICIENT

Robust M-estimators of intraclass correlation coefficient, location and scale parameters are defined for familial data. It is shown that these estimators are strongly consistent. Also the asymptotic distributions of these estimators are derived when the underlying distribution is elliptically and permutationally symmetric.     

A ROBUST TEST FOR MULTIPLE COMPARISONS OF CORRELATION COEFFICIENTS1

A limiting expression is derived for the tail of the distribution of the maximum of a set of product moment correlation coefficients. The technique used is quite general and may be applied to non-normal observations as well as to rank correlation coefficients. The result obtained for the latter leads to a test procedure for multiple comparisons of these non-parametric measures of dependence.     

THE WEIGHTED MEDIAN AND MULTIPLE REGRESSION1

This paper reviews the iterative use of the weighted median to estimate the parameter vector in the classical linear model when the fitting criterion is (i) least absolute deviation sum (LAD); and (ii) the Cauchy criterion. The implications of the Cauchy criterion, little developed hitherto, are compared and contrasted with results for the better-known LAD procedure. Since the weighted median is essentially an estimation technique for the simplest regression model, its use in these contexts illustrates the central role in statistical theory that is played by regression analysis, a focal area of the work of E. J. Williams (1959).     

MULTIPLE COMPARISONS AND SELECTION1

The selection of the “best” of k populations and subsequent prediction for this population versus “the rest” is compared with the Newman-Keuls multiple comparison procedure for “separating” populations.     

GOODNESS-OF-FIT TESTS FOR FISHER'S DISTRIBUTION ON THE SPHERE

A set of three goodness-of-fit procedures is proposed to investigate the adequacy of fit of Fisher's distribution on the sphere as a model for a given sample of spherical data. The procedures are all based on standard tests using the empirical distribution function.     

A NEW APPROXIMATION FOR FISHER'S Z

As the sample size increases, the coefficient of skewness of the Fisher's transformation z= tanh^-1r, of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the distribution of standardized z can be approximated more accurately in terms of the t distribution with matching kurtosis than by the unit normal distribution. This t distribution can, in turn be subjected to Wallace's approximation resulting in a new normal approximation for the Fisher's z transform. This approximation, which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1974). Fisher (1921) suggested approximating distribution of the variance stabilizing transform z=(1/2) log ((1 +r)/(1r)) of the correlation coefficient r by the normal distribution with mean = (1/2) log ((1 + p)/(lp)) and variance =l/(n3). This approximation is generally recognized as being remarkably accurate when ||Gr| is moderate but not so accurate when ||Gr| is large, even when n is not small (David (1938)). Among various alternatives to Fisher's approximation, the normalizing transformation due to Ruben (1966) and a t approximation due to Kraemer (1973), are interesting on the grounds of novelty, accuracy and/or aesthetics. If r?= r/√ (1r²) and r?|Gr = |Gr/√(1|Gr²), then Ruben (1966) showed that (1) g_n (r,|Gr) ={(2n5)/2}^1/2r?r{(2n3)/2}^1/2r?|GR, {1 + (1/2)(r?r²+r?|Gr²)}^1/2 is approximately unit normal. Kraemer (1973) suggests approximating (2) t_n (r, |Gr) = (r|GR¹) √ (n2), √(11r²) √(1|Gr²) by a Student's t variable with (n2) degrees of freedom, where after considering various valid choices for |Gr¹ she recommends taking |Gr¹= |Gr*, the median of r given n and |Gr.     

SERIAL CORRELATION AND SPECTRAL MOMENT STATISTICS1

Distributions of ratios of weighted independent chi-square variables are found in closed form. Such density functions are of interest in the theory of serial correlation and spectral moment estimation.     

A SADDLEPOINT APPROXIMATION TO THE DISTRIBUTION FUNCTION OF THE ANDERSON-DARLING TEST STATISTIC

The Anderson-Darling goodness-of-fit test has a highly skewed and non-standard limit distribution. Various attempts have been made to tabulate the associated critical points, using both theoretical approximations and simulation methods. We show that a standard saddlepoint approximation performs well in both tails of the distribution. It is markedly superior to other theoretical approximations in the lower tail of the distribution.     

A BETA-OPTIMAL TEST OF THE EQUICORRELATION COEFFICIENT

This paper considers the problem of testing for nonzero values of the equicorrelation coefficient of a standard symmetric multivariate normal distribution. Recently, SenGupta (1987) proposed a locally best test. We construct a beta-optimal test and present selected one and five percent critical values. An empirical power comparison of SenGupta's test with two versions of the beta-optimal test and the power envelope shows the relative strengths of the three tests. It also allows us to assess and confirm Efron's (1975) rule of when to question the use of a locally best test, at least for this testing problem. On the basis of these results, we argue that the two beta-optimal tests can be considered as approximately uniformly most powerful tests, at least at the five percent significance level.     

RECURSIVE ESTIMATION OF THE GENERAL LINEAR MODEL WITH DEPENDENT ERRORS AND MULTIPLE ADDITIONAL OBSERVATIONS1

The techniques for recursive estimation of the general linear model with dependent errors and known second order properties, is generalised to allow for simultaneous addition of an arbitrary number of additional observations. Computational formulae for recursive updating of parameter estimates are derived, together with a sequence of univariate recursive residuals for testing the constancy of the regression relation over time.     

ON THE DISTRIBUTION OF THE REGRESSION COEFFICIENT MATRIX OF A NORMAL DISTRIBUTION

In this paper we generalize a result of Kshirsagar's (1960, pp. 83–84) on the distribution of the regression coefficient matrix for a multivariate normal population.     

DECOMPOSITIONS OF THE QUEUE LENGTH DISTRIBUTIONS IN THE MAP/G/1 QUEUE UNDER MULTIPLE AND SINGLE VACATIONS WITH N-POLICY

In this paper, we establish an explicit form of matrix decompositions for the queue length distributions of the MAP/G/1 queues under multiple and single vacations with N-policy. We show that the vector generating function Y (z) of the queue length at an arbitrary time and X (z) at departures are decomposed into Y (z) = p _idle (z)ζ _Y (z) and X (z) = p _idle (z)ζ _X (z) where p _idle (z) is the vector generating function of the queue length at an arbitrary epoch at which the server is not in service, and ζ _Y (z) and ζ _X (z) are unidentified matrix generating functions.     

TESTING THE LARGEST OF A SET OF CORRELATION COEFFICIENTS

A previous paper which studied the distribution of the smallest distance between N independent random points on the surface of a sphere is generalised to higher dimensions in order to study the distribution of the largest sample correlation coefficient between a set of independent normally distributed variables. Inclusion-exclusion arguments provide accurate bounds for the tail of this distribution, and by another argument more exact bounds are also found, one of which is an improvement on the result in the previous paper. Bounds are also found for the power of the test against the alternative hypothesis that one only of the population correlation coefficients is non-zero. The test is also shown to be the likelihood ratio test against the latter alternative.     

PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE

This paper investigates the roles of partial correlation and conditional correlation as measures of the conditional independence of two random variables. It first establishes a sufficient condition for the coincidence of the partial correlation with the conditional correlation. The condition is satisfied not only for multivariate normal but also for elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial and Dirichlet distributions. Such families of distributions are characterized by a semigroup property as a parametric family of distributions. A necessary and sufficient condition for the coincidence of the partial covariance with the conditional covariance is also derived. However, a known family of multivariate distributions which satisfies this condition cannot be found, except for the multivariate normal. The paper also shows that conditional independence has no close ties with zero partial correlation except in the case of the multivariate normal distribution; it has rather close ties to the zero conditional correlation. It shows that the equivalence between zero conditional covariance and conditional independence for normal variables is retained by any monotone transformation of each variable. The results suggest that care must be taken when using such correlations as measures of conditional independence unless the joint distribution is known to be normal. Otherwise a new concept of conditional independence may need to be introduced in place of conditional independence through zero conditional correlation or other statistics.     

AN ANALYTIC FORM FOR THE CUMULATIVE DISTRIBUTION FUNCTION OF THE BELL-DOKSUM STATISTIC

A closed form analytic expression for the Bell-Doksum Statistic is developed. The use of the generalized hypergeometric function to evaluate the expression is demonstrated, and a table of critical values for 2 n 25 is presented.     

RAO'S STATISTIC FOR THE ANALYSIS OF UNIFORM ASSOCIATION IN CROSS-CLASSIFICATIONS

In this paper we introduce a new measure for the analysis of association in cross-classifications having ordered categories. Association is measured in terms of the odd-ratios in 2 × 2 subtables formed from adjacent rows and adjacent columns. We focus our attention in the uniform association model. Our measure is based in the family of divergences introduced by Burbea and Rao [1] Burbea, J. and Rao, C. R. 1982a. On the convexity of some divergence measures based on entropy functions. IEEE Transactions on Information Theory, 28: 489–495. [Crossref], [Web of Science ®] , [Google Scholar]. Some well-known sets of data are reanalyzed and a simulation study is presented to analyze the behavior of the new families of test statistics introduced in this paper.