Some R graphics for bivariate distributions

There is no package in R to plot bivariate distributions for discrete variables or variables given by classes. Therefore, with the help of the already implemented R routine persp R functions will be proposed for 3-D plots of the bivariate distribution of discrete variables, the so-called stereogram that generalizes the well-known histogram for cross-classified data and the approximative bivariate distribution function for cross-classified data. --Bivariate distribution function,bivariate frequency density,stereogram,approximate bivariate distribution function

On Idempotent Estimators of Location

Idempotence is a well-known property of functionals of location. It means that the value of the functional at a singular distribution must be identically to the mass point of this distribution. First, we explain the role of idempotence in the known axiomizations of location functionals. Then we derive the distribution of idempotent and sufficient statistics. In the special cases of parametric families of location we get the so-called power-n-distributions. Power-n-distributions again are distributions with a parameter of location and can be derived from every location family for which the density is constrained. Additionally we show that the completeness of the populations family insures the completeness of the family of power-n-distributions. And at last, we give a further, now very easy proof that the normal distribution is the only one for which a idempotent, sufficient and unbiased estimator attains the Cramer-Rao-lower bound. --

Van Zwet ordering and the Ferreira-Steel family of skewed distributions

There are several procedures to construct a skewed distribution. One of these procedures is based on a symmetric distribution that will be distorted by a skewed distribution defined on (0; 1). This proposal stems from Arellano-Valle et al. and was refined by Ferreira and Steel. Up to now, it is an open question whether the famous skewness ordering of van Zwet will be preserved for this proposal. There is a general condition under which the van Zwets skewness ordering will be preserved by the Ferreira-Steel family. But this condition is not easy to verify for the most families of distribution. Therefore, for the skewness mechanism we choose a special beta distribution with only one parameter. Then, we get three results. First, the skewness ordering will be preserved starting for symmetric distributions that are leptokurtic like the logistic distribution. Larger parameter values give distributions that are more skewed to the right. Second, the same skewness mechanism can generate distributions that are more skewed to left if the support of the underlying symmetric distribution is compact. Third, for underlying symmetric distributions on R with platykurtic behavior the van Zwet ordering of skewness will be preserved. This restricts a little bit the benefit of the Ferreira-Steel family. --skewness,skewness to the right,skewness ordering,measure of skewness

Van Zwet ordering for Fechner asymmetry

There are several procedures to construct a skewed distribution. One of these procedures splits the value of a parameter of scale for the two halfs of a symmetric distribution. Fechner proposed this procedure in his famous book Kollektivmaßlehre (1897), p. 295ff.. A similar proposal comes from Fernandez et al. (1995). We consider the very general approach from Arellano-Valle et al. (2005) of splitting a scale parameter and show that this technique of generating skewed distributions incorporates a well-defined parameter of skewness. It is well-defined in the sense that the parameter of skewness is compatible with the orderingSkewness,skewness to the right,skewness ordering,measure of skewness

Some critical remarks on Zhang's gamma test for independence

Zhang (2008) defines the quotient correlation coefficient to test for dependence and tail dependence of bivariate random samples. He shows that asymptotically the test statistics are gamma distributed. Therefore, he called the corresponding test gamma test. We want to investigate the speed of convergence by a simulation study. Zhang discusses a rank-based version of this gamma test that depends on random numbers drawn from a standard Frechet distribution. We propose an alternative that does not depend on random numbers. We compare the size and the power of this alternative with the well-known t-test, the van der Waerden and the Spearman rank test. Zhang proposes his gamma test also for situations where the dependence is neither strictly increasing nor strictly decreasing. In contrast to this, we show that the quotient correlation coefficient can only measure monotone patterns of dependence. --test on dependence,rank correlation test,Spearman's p,copula,Lehmann ordering

On J.M. Keynes' The principal averages and the laws of error which lead to them: refinement and generalisation

Keynes (1911) derived general forms of probability density functions for which the “most probable value” is given by the arithmetic mean, the geometric mean, the harmonic mean, or the median. His approach was based on indirect (i.e., posterior) distributions and used a constant prior distribution for the parameter of interest. It was therefore equivalent to maximum likelihood (ML) estimation, the technique later introduced by Fisher (1912). Keynes' results suffer from the fact that he did not discuss the supports of the distributions, the sets of possible parameter values, and the normalising constants required to make sure that the derived functions are indeed densities. Taking these aspects into account, we show that several of the distributions proposed by Keynes reduce to well-known ones, like the exponential, the Pareto, and a special case of the generalised inverse Gaussian distribution. Keynes' approach based on the arithmetic, the geometric, and the harmonic mean can be generalised to the class of quasi-arithmetic means. This generalisation allows us to derive further results. For example, assuming that the ML estimator of the parameter of interest is the exponential mean of the observations leads to the most general form of an exponential family with location parameter introduced by Dynkin (1961) and Ferguson (1962, 1963). --ML estimator,criterion function,median,quasi-arithmetic mean,exponential family

Kurtosis ordering of the generalized secant hyperbolic distribution: a technical note

Two major generalizations of the hyperbolic secant distribution have been proposed in the statistical literature which both introduce an additional parameter that governs the kurtosis of the generalized distribution. The generalized hyperbolic secant (GHS) distribution was introduced by Harkness and Harkness (1968) who considered the p-th convolution of a hyperbolic secant distribution. Another generalization, the so-called generalized secant hyperbolic (GSH) distribution was recently suggested by Vaughan 2002). In contrast to the GHS distribution, the cumulative and inverse cumulative distribution function of the GSH distribution are available in closedform expressions. We use this property to proof that the additional shape parameter of the GSH distribution is actually a kurtosis parameter in the sense of van Zwet (1964). --kurtosis ordering,hyperbolic secant distribution

Kurtosis transformation and kurtosis ordering

Leptokurtic distributions can be generated by applying certain non-linear transformations to a standard normal random variable. Within this work we derive general conditions for these transformations which guarantee that the generated distributions are ordered with respect to the partial ordering of van Zwet for symmetric distributions and the partial ordering of MacGillivray for arbitrary distributions. In addition, we propose a general power transformation which nests the H-, J- and K-transformations which have already been proposed in the literature. Within this class of power transformations the above mentioned condition can be easily verified and the power can be interpreted as parameter of leptokurtosis. --Power kurtosis transformation,leptokurtosis,kurtosis ordering

Some results on weak and strong tail dependence coefficients for means of copulas

Copulas represent the dependence structure of multivariate distributions in a natural way. In order to generate new copulas from given ones, several proposals found its way into statistical literature. One simple approach is to consider convex-combinations (i.e. weighted arithmetic means) of two or more copulas. Similarly, one might consider weighted geometric means. Consider, for instance, the Spearman copula, defined as the geometric mean of the maximum and the independence copula. In general, it is not known whether weighted geometric means of copulas produce copulas, again. However, applying a recent result of Liebscher (2006), we show that every weighted geometric mean of extreme-value copulas produces again an extreme-value copula. The second contribution of this paper is to calculate extremal dependence measures (e.g. weak and strong tail dependence coe±cients) for (weighted) geometric and arithmetic means of two copulas. --Tail Dependence,Extreme-value copulas,arithmetic and geometric mean

Constructing symmetric generalized FGM copulas by means of certain univariate distributions

In this paper we focus on symmetric generalized Fairlie-Gumbel-Morgenstern (or symmetric Sarmanov) copulas which are characterized by means of so-called generator functions. In particular, we introduce a class of generator functions which is based on univariate distributions with certain properties. Some of the generator functions from the literature are recovered. Moreover two new generators are suggested, implying two new copulas. Finally, the opposite way around, it is exemplarily shown how to calculate the univariate distribution which belongs to a given copula generator function. --