Models with Accelerated Failure Conditionals

TL;DR

This work extends the accelerated failure conditionals framework to allow flexible, unimodal and skewed marginal densities while preserving a closed-form, copula-augmented joint structure: $P(X>x,Y>y)=\bar{F}_0(x)\bar{F}_1(\beta(x)y)$. By selecting beta-acceleration forms that saturate derived bounds, it yields tractable moments and explicit correlation bounds for exponential, Lomax, Weibull, log-logistic, half-Cauchy, and Gamma families, with dependence controlled by a single scalar $\tau\in[0,1]$. Copula-based simulations and Metropolis-Hastings likelihood estimation are used to assess finite-sample behavior, revealing generally stable estimation with larger samples but some instability for the Lomax and mixed-marginal settings. Empirical applications to unimodal, skewed data (TikTok/song metrics and VIX/gold prices) show improved fit for flexible marginals over monotone-density alternatives, while underscoring the importance of selecting marginals consistent with data; the approach also points to potential gains from heterogeneous-marginal (mixture) models. The framework offers a versatile, analytically tractable toolbox for modeling dependent lifetimes under accelerated conditioning with broad applicability in reliability and finance contexts.

Abstract

Arnold and Arvanitis (2020) introduced a novel bivariate conditionally specified distribution, a distribution in which dependence between two random variables is established by defining the distribution of one variable conditional on the other. This novel conditioning regime was achieved through the use of survival functions, and the approach was termed the accelerated failure conditionals model. In their work, the conditioning framework was constructed using the exponential distribution. Although further generalization was proposed, challenges emerged in deriving the necessary and sufficient conditions for valid joint survival functions. The present study achieves such generalization, extending the conditioning framework to encompass distributional families whose marginal densities may exhibit unimodality and skewness, moving beyond distributional families whose marginal densities are non-increasing. The resulting models are fully specified through closed-form expressions for their moments, with simulations implemented using either a copula-based procedure or the Metropolis-Hastings algorithm. Empirical applications to two datasets, each featuring variables which are unimodal and skewed, demonstrate that the models with flexible, non-monotonic marginal densities yield a superior fit relative to those models with marginal densities restricted to monotonically decaying forms.

Figures (3)

• Figure 1: Bivariate densities of models given $\alpha = 1, \gamma = 2, \lambda =3, \nu = 4$ and $\tau = 0.5$.
• Figure 2: Contour plots of estimated bivariate densities of models on TikTok dataset (observations in black dots).
• Figure 3: Contour plots of estimated bivariate densities of models on VIX/GC dataset (observations in black dots).

Theorems & Definitions (14)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Remark 2
• Theorem 3
• proof
• Theorem 4
• proof