L-functions, processes, and statistics in measuring economic inequality and actuarial risks

Abstract

$L$-statistics play prominent roles in various research areas and applications, including development of robust statistical methods, measuring economic inequality and insurance risks. In many applications the score functions of $L$-statistics depend on parameters (e.g., distortion parameter in insurance, risk aversion parameter in econometrics), which turn the $L$-statistics into functions that we call $L$-functions. A simple example of an $L$-function is the Lorenz curve. Ratios of $L$-functions play equally important roles, with the Zenga curve being a prominent example. To illustrate real life uses of these functions/curves, we analyze a data set from the Bank of Italy year 2006 sample survey on household budgets. Naturally, empirical counterparts of the population $L$-functions need to be employed and, importantly, adjusted and modified in order to meaningfully capture situations well beyond those based on simple random sampling designs. In the processes of our investigations, we also introduce the $L$-process on which statistical inferential results about the population $L$-function hinges. Hence, we provide notes and references facilitating ways for deriving asymptotic properties of the $L$-process.