Extending the Accelerated Failure Conditionals Model to Location-Scale Families

TL;DR

This work extends the accelerated failure conditionals framework to real-valued outcomes by introducing a location function $\mu(x)$ and a scale function $\beta(x)$ in the conditional survival $P(Y>y|X>x)=\bar{F}_1\big(\frac{y-\mu(x)}{\beta(x)}\big)$, yielding a joint survival $P(X>x,Y>y)=\bar{F}_0(x)\bar{F}_1((y-\mu(x))/\beta(x))$. By pairing a Weibull marginal for $X$ with various location–scale families for $Y$ (Logistic, Gumbel, Laplace, Cauchy, Normal), the paper derives closed-form moments and explicit correlation bounds, and obtains the dependence strength via a dependence-scaling parameter $\tau$ together with the sign of dependence controlled by $\mu'(x)$. Simulation via Metropolis–Hastings demonstrates stable estimation for $\alpha,\lambda,\beta,\gamma,\tau$ across sample sizes, with estimated correlations converging to empirical Pearson correlations as $n$ grows. An applied dataset (log VIX vs log Nasdaq-100 returns) shows the Weibull-Laplace model yielding the best fit by AIC, consistent with the Laplace marginal’s fit to the data; the approach can accommodate $Y$ marginals on $\mathbb{R}$ and capture direction and strength of dependence through $\mu(x)$ and $\tau$. Overall, the framework provides a flexible, analytically tractable way to model dependence between a positive-support variable and a real-valued variable via conditional survival specifications in location–scale families.

Abstract

Arnold and Arvanitis (2020) introduced a novel class of bivariate conditionally specified distributions, in which dependence between two random variables is established by defining the distribution of one variable conditional on the other. This conditioning regime was formulated through survival functions and termed the accelerated failure conditionals model. Subsequently, Lakhani (2025) extended this conditioning framework to encompass distributional families whose marginal densities may exhibit unimodality and skewness, thereby moving beyond families with non-increasing densities. The present study builds on this line of work by proposing a conditional survival specification derived from a location-scale distributional family, where the dependence between $X$ and $Y$ arises not only through the acceleration function but also via a location function. An illustrative example of this new specification is developed using a Weibull marginal for $X$. The resulting models are fully characterized by closed-form expressions for their moments, and simulations are implemented using the Metropolis-Hastings algorithm. Finally, the model is applied to a dataset in which the empirical distribution of $Y$ lies on the real line, demonstrating the models' capacity to accommodate $Y$ marginals defined over $\mathbb{R}$.

Figures (3)

• Figure 1: Bivariate densities of the models with parameter values: $\alpha = 1$, $\beta = 2$, $\lambda = 3$, $\gamma = -4$, and $\tau = 0.5$. The left panel displays the three-dimensional bivariate density surfaces, while the right panel presents the corresponding contour plots (an arbitrary dataset of $n = 500$ used; data in black circles).
• Figure 2: Histogram of the logarithm of the VIX (left) with estimated marginal Weibull densities based on m.l.e's reported in Table \ref{['table: vix']}, and histogram of the logarithm of daily Nasdaq-100 (NQ) returns (right) with estimated marginal logistic, Laplace, and Cauchy densities, also based on the corresponding m.l.e's in Table \ref{['table: vix']}.
• Figure 3: Estimated bivariate densities of the models using m.l.e's from Table \ref{['table: vix']}. The left panel displays the estimated 3-D bivariate density surfaces, while the right panel presents the corresponding contour plots (observations in black circles).

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof