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

Abstract

To Professor Joseph L. Gastwirth whose creativity and fondness for indices have been matchlessly inspiring 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.

Inference by linearization for Zenga’s new inequality index: a comparison with the Gini index

“…Although literature on the Zenga index is not as plentiful as on the Gini index, it has drawn increasing attention in the scientific community. The literature includes some publications on the properties of the index (Maffenini and Polisicchio 2010), inference and applications (Polisicchio 2008;Greselin et al 2009Greselin et al , 2010 as well as subgroup decomposition (Radaelli 2008(Radaelli , 2010. The literature on the Zenga index and curve also focuses greatly on its advantages compared to the Gini index.…”

Asymptotic distribution of the quintile share ratio estimator

“…the survey papers Xu, 2004 and Langel and Tillé;2013). Indices, most notably the Zenga index, have recently been explored from various points of view; for example finite and asymptotic variance cases have been considered in Greselin et al (2009Greselin et al ( ), (2010, (2013) and (2014). Another trend, somewhat different from Zengaâ's but equally interesting, which is based on the Palma index, is considered in e.g.…”

Stability of -statistics from weakly dependent observations

“…In financial context L-statistics accommodate numerous indices of economic inequality as well as risk measures of actuarial science (see Davydov et al, 2007, Greselin et al, 2009). In particular, they constitute a natural class of estimators for spectral risk measures.…”