A stress–strength model with dependent variables to measure household financial fragility

Parametric and nonparametric inference for the reliability of copula‐based stress‐strength models

“…The entries 2 and S are scalars. The elements (10) of A represent the Hessian and the elements (11) of B represent the outer product (square) of gradients (OPG). Hessians and OPGs interact whenever the expectation of derivatives and expectation of product of estimating functions do not coincide.…”

“…4,5 Dependence was initially modeled by common bivariate distributions, ranging from the bivariate normal 6 to the bivariate gamma 7 and the bivariate LogNormal, 8 among others. Recent methodological advances in this direction include the use of flexible dependence structures provided by copula-based models 9,10 with applications in econometrics 11 and engineering. [12][13][14] Copula-based stress-strength (CSS) models allow for the choice of the marginal distributions of Y 1 and Y 2 to be in the same family or not and the use of the many forms of dependence available through copulas.…”

“…[12][13][14] Copula-based stress-strength (CSS) models allow for the choice of the marginal distributions of Y 1 and Y 2 to be in the same family or not and the use of the many forms of dependence available through copulas. 15 Domma and Giordano 9,11 have used marginal distributions in the Burr system (notably the Dagum distribution) with the Farlie-Gumbel-Morgenstern (FGM) and Frank copulas to obtain a measure of financial fragility (income < expenditure) in the Italian economy. With the FGM copula, a closed-form expression for the fragility (1 − R) was obtained.…”

Assessing How the Dependence Structure Affects the Reliability Parameter of the Strength-Stress Model

“…Within the Bayesian framework, reference [7] studied the estimation of the reliability parameter assuming for the stress and strength variables asymmetric distributions obtained by skewing scale mixtures of normals; the margins are linked by the Gaussian copula. In [8] a stress-strength model is investigated with stress and strength marginally distributed as non-identical Dagum r.v.s and their dependence described by a Frank copula. In [9] the problem of estimation of the reliability parameter is considered when the Farlie-Gumbel-Morgenstern copula is used to link stress and strength variables, whose marginal distributions both belong to the Burr system.…”

Estimation ofP(X&gt;Y) whenXandYare dependent random variables using different bivariate sampling schemes

“…Currently, many authors have tried to estimate θ in the case where X and Y are dependent random variables. For example Barbiero (2012) assumed that (X, Y) are jointly normally distributed; Rubio and Steel (2013) assumed that X and Y are marginally distributed as a skewed scale mixture of normal and constructed the corresponding joint distribution using a Gaussian copula; Domma and Giordano (2013) constructed the joint distribution of (X, Y) using a Farlie-Gumbel-Morgenstern copula with marginal distributions belonging to the Burr system; Domma and Giordano (2012) considered Dagum distributed marginals and constructed their joint distribution using a Frank copula; among others (Gupta et al, 2013;Nadarajah, 2005). In these papers, the importance of taking the assumption of dependence between X and Y into consideration is illustrated using simulated and real data sets.…”