Taylor's law of fluctuation scaling for semivariances and higher moments of heavy-tailed data.

We generalize Taylor's law for the variance of light-tailed distributions to many sample statistics of heavy-tailed distributions with tail index in (0, 1), which have infinite mean. We show that, as the sample size increases, the sample upper and lower semivariances, the sample higher moments, the skewness, and the kurtosis of a random sample from such a law increase asymptotically in direct proportion to a power of the sample mean. Specifically, the lower sample semivariance asymptotically scales in proportion to the sample mean raised to the power 2, while the upper sample semivariance asymptotically scales in proportion to the sample mean raised to the power [Formula: see text] The local upper sample semivariance (counting only observations that exceed the sample mean) asymptotically scales in proportion to the sample mean raised to the power [Formula: see text] These and additional scaling laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the Omega index, the upside potential ratio, and the Farinelli-Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution.

More powerful goodness-of-fit tests for uniform stochastic ordering.

The ordinal dominance curve (ODC) is a useful graphical tool to compare two population distributions. These distributions are said to satisfy uniform stochastic ordering (USO) if the ODC for them is star-shaped. A goodness-of-fit test for USO was recently proposed when both distributions are unknown.

Testing for positive quadrant dependence.

We develop an empirical likelihood approach to test independence of two univariate random variables and versus the alternative that and are strictly positive quadrant dependent (PQD). Establishing this type of ordering between and is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague (2003, ), we create a distribution-free test statistic that integrates a localized empirical likelihood ratio test statistic with respect to the empirical joint distribution of and .

NONPARAMETRIC GOODNESS-OF-FIT TESTS FOR UNIFORM STOCHASTIC ORDERING.

We propose distance-based goodness-of-fit (GOF) tests for uniform stochastic ordering with two continuous distributions and , both of which are unknown. Our tests are motivated by the fact that when and are uniformly stochastically ordered, the ordinal dominance curve = is star-shaped. We derive asymptotic distributions and prove that our testing procedure has a unique least favorable configuration of and for ∈ [1,∞].