The Mean, Median, Mode Inequality and Skewness for a Class of Densities

Recent work on the mean, median, mode inequality and skewness for unimodal continuous distributions is discussed with reference to the properties of totally positive kernels. These properties also give simple conditions which are sufficient for the inequality and which include the characterization of the Pearson family.     

THE RELATIONSHIPS BETWEEN SKEWNESS AND KURTOSIS

Theoretical considerations of kurtosis, whether of partial orderings of distributions with respect to kurtosis or of measures of kurtosis, have tended to focus only on symmetric distributions. With reference to historical points and recent work on skewness and kurtosis, this paper defines anti-skewness and uses it as a tool to discuss the concept of kurtosis in asymmetric univariate distributions. The discussion indicates that while kurtosis is best considered as a property of symmetrised versions of distributions, symmetrisation does not simply remove skewness. Skewness, anti-skewness and kurtosis are all inter-related aspects of shape. The Tukey g and h family and the Johnson Su family are considered as examples.     

SOME REMARKS ON THE INFORMATION GEOMETRY OF THE GAMMA DISTRIBUTION

ABSTRACT

Bounds for the Fisher information metric associated with the Gamma statistical model are found in terms of Poincaré type metric. This results in the determination of bounds for the Rao distance, that is the Riemannian distance induced by the information metric, between Gamma distributions, and of bounds for the Gaussian curvature of the Gamma model. The bounds seem to be sharp, where the lower and upper Rao distance bounds are Poincaré distances with Gaussian curvatures ?1/4 and ?1/2, respectively. In addition, the sign of the Gaussian curvature of the Gamma model is shown to be negative which means, in particular, that the geometry of the model is hyperbolic.     

ON THE GENERALIZED RAO-RUBIN CONDITION and SOME VARIANTS

This paper gives a characterization of some members of the compound Poisson family of distributions based on the generalized Rao-Rubin condition. By considering some variants of this condition and using power series arguments, characterizations of the Poisson distribution are also obtained.     

AFFINE‐EQUIVARIANT SPATIAL MEDIAN AND ITS USE IN THE MULTIVARIATE MULTI‐SAMPLE LOCATION PROBLEM

The classical spatial median is not affine‐equivariant, which often turns out to be an unfavourable property. In this paper, the asymptotic properties of an affine‐equivariant modification of the spatial median are investigated. It is shown that under some weak regularity conditions, the modified spatial median computed by means of the sample norming matrix is asymptotically equivalent to the one computed by means of the population norming matrix, which yields its asymptotic normality. A consistent estimate of the asymptotic covariance matrix of the modified spatial median is also presented. These results are implemented in a scheme, where the sample norm is determined by means of the sample Dümbgen scatter matrix. The results are utilized in the construction of affine‐invariant test statistics for testing the multi‐sample hypothesis of equality of location parameters. The performance of the proposed tests is demonstrated through a simulation study.     

UPPER BOUNDS FOR THE HARMONIC MEAN, WITH AN APPLICATION TO EXPERIMENTAL DESIGN

For positive-valued random variables, the paper provides a sequence of upper bounds for the harmonic mean, the ith of these bounds being exact if and only if the random variable is essentially i-valued. Sufficient conditions for the convergence of the bounds to the harmonic mean are given. The bounds have a number of applications, particularly in experimental design where they may be used to check how close a given design is to A-optimality     

ON THE CHANGE POINT OF THE MEAN RESIDUAL LIFE OF SERIES AND PARALLEL SYSTEMS

This paper considers the mean residual life in series and parallel systems with independent and identically distributed components and obtains relationships between the change points of the mean residual life of systems and that of their components. Compared with the change point for single components, should it exists, the change point for a series system occurs later. For a parallel system, however, the change point is located before that for the components, if it exists at all. Moreover, for both types of systems, the distance between the change points of the mean residual life for systems and for components increases with the number of components. These results are helpful in the determination of optimal burn-in time and related decision making in reliability analysis.     

ON THE INDEPENDENCE OF THE SAMPLE MEAN AND TRANSLATION-INVARIANT STATISTICS FOR MATRIX NORMAL DISTRIBUTIONS

We present an explicit characterization of the joint dependency structure of an n×p matrix normal random matrix such that the p-dimensional sample mean vector is independent of all translation invariant statistics.     

THE EXAMINATION AND ANALYSIS OF RESIDUALS FOR SOME BIASED ESTIMATORS IN LINEAR REGRESSION

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares (OLS) estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean-squared error.     

ON STUDENTIZING AND BLOCKING METHODS FOR IMPLEMENTING THE BOOTSTRAP WITH DEPENDENT DATA

Gōtze & Kūnsch (1990) announced that a certain version of the bootstrap percentile-t method, and the blocking method, can be used to improve on the normal approximation to the distribution of a Studentized statistic computed from dependent data. This paper shows that this result depends fundamentally on the method of Studentization. Indeed, if the percentile-t method is implemented naively, for dependent data, then it does not improve by an order of magnitude on the much simpler normal approximation despite all the computational effort that is required to implement it. On the other hand, if the variance estimator used for the percentile-t bootstrap is adjusted appropriately, then percentile-t can improve substantially on the normal approximation.     

ESTIMATION OF THE CONCENTRATION PARAMETERS OF THE FISHER MATRIX DISTRIBUTION ON 50(3) AND THE BINGHAM DISTRIBUTION ON Sq, q≥ 2

Two procedures are considered for estimating the concentration parameters of the Fisher matrix distribution for rotations or orientations in three dimensions. The first is maximum likelihood. The use of a convenient 1-dimensional integral representation of the normalising constant, which greatly simplifies the computation, is suggested. The second approach exploits the equivalence of the Fisher distribution for rotations in three dimensions, and the Bingham distribution for axes in four dimensions. We describe a pseudo likelihood procedure which works for the Bingham distribution in any dimension. This alternative approach does not require numerical integration. Results on the asymptotic efficiency of the pseudo likelihood estimator relative to the maximum likelihood estimator are given, and the two estimators are compared in the analysis of a well-known vectorcardiography dataset.