Richard A. Groeneveld

Measuring skewness and kurtosis

The question of how to measure the degree of skewness of a continuous random variable is addressed. In van Zwet (1964) a method for ordering two distributions with regard to skewness is given. Here, using the concept of comparative skewness, we consider properties that a measure of skewness should satisfy. Several extensions of the Bowley measure of skewness taking values on (-1, 1) are discussed. How well these measures reflect ones intuitive idea of skewness is examined. These measures of skewness are extended to measures of kurtosis for symmetric distributions.

The nontruncated marginal of a truncated bivariate normal distribution

Inference is considered for the marginal distribution ofX, when (X, Y) has a truncated bivariate normal distribution. TheY variable is truncated, but only theX values are observed. The relationship of this distribution to Azzalinis “skew-normal” distribution is obtained. Method of moments and maximum likelihood estimation are compared for the three-parameter Azzalini distribution. Samples that are uniformative about the skewness of this distribution may occur, even for largen. Profile likelihood methods are employed to describe the uncertainty involved in parameter estimation. A sample of 87 Otis test scores is shown to be well-described by this model.

Measuring Skewness with Respect to the Mode

Abstract There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or medians of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained the convex or c-ordering of distributions that he introduced. This ordering has been widely accepted as appropriate for ordering two distributions in relation to skewness. More recently, measures based on the medians have been shown to honor the convex ordering. The measure of skewness (μ – M) / [sgrave] where μ, [sgrave], and M are, respectively, the expectation, standard deviation, and mode of the distribution was initially proposed by Karl Pearson. It unfortunately does not maintain the convex ordering. Here we introduce a measure based on the mode of a distribution that maintains the c-ordering. For many classes of right-skewed distributions, it is easily computed as a function of the shape parameter of the family and ...

The Mode, Median, and Mean Inequality

Abstract An elementary method of proof of the mode, median, and mean inequality is given for skewed, unimodal distributions of continuous random variables. A proof of the inequality for the gamma, F, and beta random variables is sketched.

A class of quantile measures for kurtosis

Abstract Several authors have concluded that the standardized fourth central moment of a symmetric distribution is not a good measure of the shape of a distribution. Here we consider the properties of a class of parameters of a distribution based on its percentiles, as alternative measures of kurtosis. It is shown that this scale and location invariant measure maintains the symmetric ordering of van Zwet. Influence functions are used to show how this measure reflects the kurtosis of a distribution. Results of a simulation study indicate that the power of the Shapiro-Wilk test, for a large number of symmetric distributions alternative to the normal distribution, is almost linear as a function of appropriate functionals in this class. This suggests the use of this functional as a kurtosis measure.

An Influence Function Approach to Describing the Skewness of a Distribution

Abstract Skewness, like kurtosis, is a qualitative property of a distribution. A comparison of several measures of skewness of univariate distributions is carried out. Hampels influence function is used to clarify the differences and similarities among these measures. A general concept of skewness as a location- and scale-free deformation of the probability mass of a symmetric distribution emerges. Positive skewness can be thought of as resulting from movement of mass at the right of the median from the center to the right tail of the distribution together with movement of mass at the left of the median from the left tail to the center of the distribution.

Some Properties of the Arcsine Distribution

Abstract Three characterizations of the arcsine distribution are presented. A symmetric random variable X has the arcsine distribution if and only if X 2 and (1 + X) /2 are identically distributed. Also, a symmetric random variable has an arcsine distribution if and only if X 2 and 1 – X 2 are identically distributed and X and 2X (1 – X 2) are identically distributed. A related result dealing with two i.i.d. random variables is also presented. The fact that the sine of a uniform random variable has the arcsine distribution is used to motivate these characterizations. One of the characterizations is used to quantify the heavy-tailed nature of the arcsine distribution.

Maximal Deviation between Sample and Population Means in Finite Populations

Abstract Bounds are presented for the maximal deviation in finite populations between sample and population means in units of (a) the population mean deviation, (b) the population range, and (c) the population mean, extending previous results for the population standard deviation. The efficiency of sampling is illustrated by various sample size calculations. The effect of symmetry on these bounds is considered. An application is made, in the symmetric case, to the expression for the bias of the ratio estimator.

Best bounds for order statistics and their expectations in range and mean units with applications

Best bounds for the order statistics are obtained in terras of the sample range and, for non-negative samples, in terms of the sample mean. Best bounds for the differences of two order statistics are found in this case also. Corresponding bounds on the expectation of order statistics and their differences are also found. Several applications of these bounds are considered

Ranking Teams in a League with Two Divisions of t Teams

Abstract A league with an equal number of teams in each of two divisions is considered. Each team plays every other one in its division g 1 times and every team in the other division g 2 times. Using the numbers of wins and losses in games between each pair of teams, several methods of ranking all of the teams are discussed and compared. A nonparametric ranking method based on the numbers of iterated wins and losses of each team is shown to have suitable theoretical properties. Analysis of National League baseball data (1973–1988) suggests that this method performs well in relation to ranking based on the classic Bradley–Terry parametric procedure. The nonparametric ranking method has the advantage of ease of computation and simplicity of explanation.