A simple moment-ordering condition is shown to be necessary for stochastic dominance. Closely related results on generalizations of the geometric and harmonic means are also provided. An ordering of the moment-generating functions is shown to be necessary and sufficient for stochastic dominance. The results have a straightforward and useful interpretation in terms of constant relative and absolute risk aversion utility functions. These results are used to provide necessary and sufficient conditions for optimality of distributions on an important class of utility functions.

THIS PAPER PROVIDES the large sample properties of two inequality indices, Atkinson's (1970) index and the generalized entropy index (Cowell and Kuga (1981), Shorrocks (1984)). The large sample properties of the equally-distributed-equivalent (e.d.e.) income measures associated with these indices are also derived. The asymptotic distributions of these inequality and welfare measures allow the construction of confidence intervals and tests for differences in the measures; these confidence intervals and tests are asymptotically distribution-free. Given the increasing availability of large micro-data sets on incomes, these results are directly applicable to empirical work on income distributions. The sampling distributions of some inequality indices, such as the Gini coefficient and Pietra index, are known.' But the ethical bases of these indices have been subject to criticism, and some researchers may prefer the Atkinson or generalized entropy indices on ethical grounds. These researchers are confronted with a choice between using an ethically unattractive index with a known sampling distribution, or using an ethically preferable index whose sampling properties are unknown. The increasing use of the Atkinson and generalized entropy indices in empirical analyses of income distributions suggests many are choosing the second option. It then becomes important to understand the sampling properties of these indices, so that statistical inference procedures may be applied. The analysis in this paper complements the recent research on sampling properties and inference for Lorenz curves. For example, Beach and Davidson (1983), and Gastwirth and Gail (1985) propose tests for equality of Lorenz curves, and Beach and Richmond (1985) provide joint confidence intervals for the Lorenz ordinate vector. Bishop, Chakraborti, and Thistle (1987, 1988a, 1988b) and Bishop, Formby, and Thistle (1989) provide a number of extensions. These papers utilize results from the theory of linear functions of order statistics.2 However, this approach is not directly applicable to the Atkinson and generalized entropy indices; these indices are not linear functions of order statistics. The Atkinson and generalized entropy indices are functions of fractional and negative moments of the distribution, and their large sample properties follow from the properties of the sample moments.

We examine CEOs' risk of termination, its determinants and its effect on firm value. Using survival analysis, we find that the risk of termination increases for about thirteen years before decreasing slightly with CEO tenure; 82% of CEOs have tenure of less than thirteen years. We also find that tenure increases with performance and compensation and decreases with monitoring by the board. Changes in the risk of termination do not have a significant effect on firm value. Taken as a whole, our results are consistent with the view that corporate governance functions reasonably well for the vast majority of firms.

Luck, skill and labor markets all have empirical support as determinants of managerial compensation. We examine the relative importance of pay for luck, managerial skill and labor market opportunities in determining compensation. We measure luck as the predictable component of firm performance, measure skill using managerial fixed effects and measure labor market opportunities as the compensation of executives at matched firms. Our results imply that managerial skill is the most important determinant of managers' compensation, followed by firm size and labor market opportunities, and that luck is not an important determinant of managerial compensation.

John A. Bishop, John P. Formby, Paul D. Thistle, Convergence and Divergence of Regional Income Distributions and Welfare, The Review of Economics and Statistics, Vol. 76, No. 2 (May, 1994), pp. 228-235

This paper investigates interstate variation in income inequality. By avoiding inequality indices and focusing directly on the Lorenz curve, the authors provide a more general explanation of the differences in inequality. They find that mean family income, the standard deviation of years of schooling, per capita educational expenditure, and property income are robust predictors of inequality. Of particular interest is their finding that, ceteris paribus, higher per capita education expenditures tend to be associated with states that have income inequality which is greater than the U.S. average.

In a recent study, Tinic and West (1986) empirically reexamine the risk-return relationship posited by the traditional mean-variance CAPM. They find a positive nonlinear relationship between risk and return, except during January when the market rewards bearing nonsystematic risk. This study examines the hypothesis that nonnormality of return distributions may account for some of these anomalous results. We compare Shalit and Yitzhaki's (1984) mean-extended Gini CAPM—an equilibrium asset pricing relation that is independent of the form of the underlying asset distribution—with the traditional CAPM. Our results indicate that the nonlinear risk-return relationship and the size and January effects are robust to nonnormality of return distributions.

John A. Bishop, S. Chakraborti, Paul D. Thistle, Asymptotically Distribution-Free Statistical Inference for Generalized Lorenz Curves, The Review of Economics and Statistics, Vol. 71, No. 4 (Nov., 1989), pp. 725-727