Stochastic dominance relations for generalised parametric distributions obtained through composition
- PDF / 442,451 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 42 Downloads / 220 Views
Stochastic dominance relations for generalised parametric distributions obtained through composition Tommaso Lando1,2
· Lucio Bertoli-Barsotti1
Received: 15 May 2019 / Accepted: 21 July 2020 © The Author(s) 2020
Abstract Investigating stochastic dominance within flexible multi-parametric families of distributions is often complicated, owing to the high number of parameters or non-closed functional forms. To simplify the problem, we use the T–X method, making it possible to obtain generalised models through the composition of cumulative distributions and quantile functions. We derive conditions for the second-order stochastic dominance and for the increasing convex order within multi-parametric families in two steps, namely: (i) breaking them down via the T–X approach and (ii) checking dominance conditions of the (more) manageable distributions composing the model. We apply our method to some special distributions and focus on the beta-generated family, which enables the comparisons of order statistics of i.i.d. samples from (possibly) different random variables. Keywords Stochastic dominance · T–X family · Generalised distributions · Generalised beta · Order statistics Mathematics Subject Classification 60E15 · 62XX · 60E05
1 Introduction Stochastic orders are primary tools in ranking probability distributions based on some preference relations [36]. Owing to several applications of ordering theory in areas such as economics, econometrics, and finance, the identification of stochastic orders within flexible generalised families of distributions (i.e. suitable to approximate a wide range of phenomena) is a relevant issue. The literature on stochastic orders contains several studies aimed at deriving sufficient conditions for the second-order stochastic dominance (SSD) and the closely related Lorenz order (LO) for some basic parametric families. Many such conditions can be derived by what we denote as the single-crossing property. This is a fundamental result ascribable to [20] that
B
Tommaso Lando [email protected]
1
University of Bergamo, Bergamo, Italy
2
VŠB-TU Ostrava, Ostrava, Czech Republic
123
T. Lando, L. Bertoli-Barsotti
has also been presented elsewhere [17,33]. This property provides a sufficient condition for SSD when distributions cross once, which turns out to be especially simple to verify for basic models having few parameters. When the single-crossing condition is not directly verifiable, we rely on alternative conditions that must be verified with densities [11,33,35]. Most generalised models are generally not easily tractable owing to the large number of parameters or non-closed functional forms. Thus, deriving dominance conditions may be an issue with the methods currently available. For instance, some authors derived LO conditions within (and between) some generalised size distributions with remarkable computational effort [12,21,22,34,40–43]. Regarding generalised parametric models, the so-called T–X family, recently introduced by [5], provides an interesting modern method for
Data Loading...
