On the robustness and johnson's modification of the one sample T-Statistic

Cicchitelli (1989) conducted an extensive Monte Carlo study to investigate the robustness of the one sample T-statistic under non-normal parent populations. He considered a rich family of distributions, viz., the generalized λ-distribution which was introduced by Ramberg et al. (1979), as the family of parent populations. We shall address and reinforce his empirical findings by means of Edgeworth expansion of the T-statistic. As the skewness of the parent population affects the T-statistic more than the kurtosis, Johnson (1978) suggested a modification to the T-statistic to reduce the effect of skewness. We investigate the performance of this modified T-statistic under the same family of distributions as Cicchitelli considered by means of a Monte Carlo study and give some recommendations on its use.     

Distributional properties of simpson's index of diversity

Closed expressions for the first four moments of Simpson's index of diversity are derived using techniaues suggested by Haldane (1937). As the samole size increases the behavior of the skewness and kurtosis is studied for several Dopulations with varying degrees of diversity, If the populationproportions decrease accordinq to a geometric progression, graphs of β₁and β₂ indicate that convergence to normality in general is more rapid for populations which are less diverse.     

Bayesian analysis of the linear regression model with an edgeworth series prior distribution

The purpose of the present investigation 1s to observe the effect of departure from normahty of the prior distribution of regresslon parameters on the Bayman analysis of a h e a r regresslon model Assuming an Edgeworth serles prior distribution for the regresslon coefficients and gamma prior for the disturbances precision, the expressions for the posterlor distribution, posterlor mean and Bayes risk under a quadratic loss function are obtalned The results of a numerical evaluation are also analyzed     

Using the studentized range to assess kurtosis

Because it is easy to compute from three common statistics (minimum, maximum, standard deviation) the studentized range is a useful test for non-normality when the original data are unavailable. For samples from symmetric populations, the studentized range allows an assessment of kurtosis with Type I and II error rates similar to those obtained from the moment coefficients.     

Invariant tests for multivariate normality: a critical review

d -dimensional random vector X is some nondegenerate d-variate normal distribution, on the basis of i.i.d. copies X ₁, ..., X _x of X. Particular emphasis is given to progress that has been achieved during the last decade. Furthermore, we stress the typical diagnostic pitfall connected with purportedly ‘directed’ procedures, such as tests based on measures of multivariate skewness. Received: April 30, 2001; revised version: October 30, 2001     

ON THE FOURTH ROOT TRANSFORMATION OF CHI-SQUARE

We show that, within the family of power transformations of a Chisquare variable, the square and fourth roots minimize Pearson's index of kurtosis. Two new transtormations of the fourth root, a symmetrized-truncated version and its linear combination with the square root are also studied. The first transformation shows a considerable improvement over the fourth root while the second one turns out to be even more accurate than Hilferty-Wilson's cube root transformation.     

On the distribution of a new measure of heaviness of tails

Distributional properties are given for a statistic T*, which has previously been reported to have power properties as a test of normality as attractive as those of the sample kurtosis or perhaps slightly more attractive. Asymptotic results, the mean and variance under normality, the range of variation, and approximation of critical values for testing normality are obtained     

Some Improvements on Markov's Theorem with Extensions

ABSTRACT

Markov's theorem for an upper bound of the probability related to a nonnegative random variable has been improved using additional information in almost the nontrivial entire range of the variable. In the improvement, Cantelli's inequality is applied to the square root of the original variable, whose expectation is finite when that of the original variable is finite. The improvement has been extended to lower bounds and monotonic transformations of the original variable. The improvements are used in Chebyshev's inequality and its multivariate version.     

The null distribution of multivariate kurtosis

The paper introduces a x²-approximation to multivariate kurtosis b_2,punder normality. It requires calculating the third moment of b_2,pwhich is obtained. We compare the approximation with simulated percentage points and the normal approximation, and find it to be adequate for p=l and 2. For p=3, the simple average of this estimate and the normal approximation is found to be generally superior to either approximation on its own. For p=4, the normal approximation is best for non-extreme values of ∝