Extended generalized Marshall–Olkin model for dependent censoring

Traditional analyses of household service losses in personal injury and wrongful death litigation calculate the losses over the expected lifetime of the injured or deceased individual. In fact, the losses to the surviving family members are more accurately described by using joint survival probabilities of the injured or deceased person and their survivors, or a “joint life expectancy.” The use of joint probabilities will always serve to reduce expected household service losses and these reductions can be especially significant when the deceased is significantly younger than the surviving spouse or if the survivor has a relatively low remaining life expectancy. (This abstract was borrowed from another version of this item.)

In this paper, we consider the problem of dependent censoring models with a positive probability that the times of failure are equal. In this context, we propose to consider the Marshall–Olkin type model and studied some properties of the associated survival copula in its application to censored data. We also introduce estimators for the marginal distributions and the joint survival probabilities under different schemes and show their asymptotic normality under appropriate conditions. Finally, we evaluate the finite-sample performance of our approach relying on a small simulation study with synthetic data real data applications.

A reinsurance treaty involves two parties, an insurer and a reinsurer. The two parties have conflicting interests. Most existing optimal reinsurance treaties only consider the interest of one party. In this article, we consider the interests of both insurers and reinsurers and study the joint survival and profitable probabilities of insurers and reinsurers. We design the optimal reinsurance contracts that maximize the joint survival probability and the joint profitable probability. We first establish sufficient and necessary conditions for the existence of the optimal reinsurance retentions for the quota‐share reinsurance and the stop‐loss reinsurance under expected value reinsurance premium principle. We then derive sufficient conditions for the existence of the optimal reinsurance treaties in a wide class of reinsurance policies and under a general reinsurance premium principle. These conditions enable one to design optimal reinsurance contracts in different forms and under different premium principles. As applications, we design an optimal reinsurance contract in the form of a quota‐share reinsurance under the variance principle and an optimal reinsurance treaty in the form of a limited stop‐loss reinsurance under the expected value principle.

First passage models, where corporate assets undergo correlated random walks and a company defaults if its assets fall below a threshold provide an attractive framework for modeling the default process. Typical one year default correlations are small, i.e., of order a few percent, but nonetheless including correlations is very important for managing portfolio credit risk and pricing some credit derivatives (e.g. first to default baskets). In first passage models the exact dependence of the joint survival probability of more than two firms on their asset correlations is not known. We derive an expression for the dependence of the joint survival probability of n firms on their asset correlations using first order perturbation theory in the correlations. It includes all terms that are linear in the correlations but neglects effects of quadratic and higher order. For constant time independent correlations we compare the first passage model expression for the joint survival probability with what a multivariate normal Copula function gives. As a practical application of our results we calculate the dependence of the five year joint survival probability for five basic industrials on their asset correlations.

We consider the problem of controlling the drift and diffusion rate of the endowment processes of two firms such that the joint survival probability is maximized. We assume that the endowment processes are continuous diffusions, driven by independent Brownian motions, and that the aggregate endowment is a Brownian motion with constant drift and diffusion rate. Our results reveal that the maximal joint survival probability depends only on the aggregate risk-adjusted return and on the maximal risk-adjusted return that can be implemented in each firm. Here the risk-adjusted return is understood as the drift rate divided by the squared diffusion rate.

This article, which shows that the Marshall–Olkin model can be a viable alternative to the standard Gaussian copula, is organized as follows. Section 2 introduces the Marshall–Olkin model. Section 3 derives the copula function of default times. Section 4 studies the aggregate default distribution. Section 5 discusses the model calibration. Section 6 compares Marshall–Olkin with the Gaussian and t-copula, and Section 7 uses the Marshall–Olkin copula to reproduce the correlation skew in the collateralised debt obligation market.

The well-known Marshall–Olkin model is known for its extension of exponential distribution preserving lack of memory property. Based on shock models, a new generalization of the bivariate Marshall–Olkin exponential distribution is given. The proposed model allows wider range tail dependence which is appealing in modeling risky events. Moreover, a stochastic comparison according to this shock model and also some properties, such as association measures, tail dependence and Kendall distribution, are presented. The new shock model is analytically quite tractable, and it can be used quite effectively, to analyze discrete–continuous data. This has been shown on real data. Finally, we propose the multivariate extension of the Marshall–Olkin model that has some intersection with the well-known multivariate Archimax copulas.

We are mainly interested in estimating the stress–strength parameter, say , when the parent distribution follows the well-known Marshall–Olkin model and the accessible data has the form of the progressively Type-II censored samples. In this case, the stress–strength parameter is free of the base distribution employed in the Marshall–Olkin model. Thus, we use the exponential distribution for simplicity. The maximum likelihood methods as well as some Bayesian approaches are used for the estimation purpose. The corresponding estimators of the latter approach are obtained by using Lindley's approximation and Gibbs sampling methods since the Bayesian estimator of cannot be obtained as an explicit form. Moreover, some confidence intervals of various types are derived for and then compared via a Monte Carlo simulation. Finally, the survival times of head and neck cancer patients are analyzed by two therapies for illustrating purposes.

Explicit expressions for the probability of joint survival up to time x of the cedent and the reinsurer, under an excess of loss reinsurance contract with a limiting and a retention level are obtained, under the reasonably general assumptions of any non-decreasing premium income function, Poisson claim arrivals and continuous claim amounts, modelled by any joint distribution. By stating appropriate optimality problems, we show that these results can be used to set the limiting and the retention levels in an optimal way with respect to the probability of joint survival. Alternatively, for fixed retention and limiting levels, the results yield an optimal split of the total premium income between the two parties in the excess of loss contract. This methodology is illustrated numerically on several examples of independent and dependent claim severities. The latter are modelled by a copula function. The effect of varying its dependence parameter and the marginals, on the solutions of the optimality problems and the joint survival probability, has also been explored.

The pseudo-observations approach has been gaining popularity as a method to estimate covariate effects on censored survival data. It is used regularly to estimate covariate effects on quantities such as survival probabilities, restricted mean life, cumulative incidence, and others. In this work, we propose to generalize the pseudo-observations approach to situations where a bivariate failure-time variable is observed, subject to right censoring. The idea is to first estimate the joint survival function of both failure times and then use it to define the relevant pseudo-observations. Once the pseudo-observations are calculated, they are used as the response in a generalized linear model. We consider 2 common nonparametric estimators of the joint survival function: the estimator of Lin and Ying (1993) and the Dabrowska estimator (Dabrowska, 1988). For both estimators, we show that our bivariate pseudo-observations approach produces regression estimates that are consistent and asymptotically normal. Our proposed method enables estimation of covariate effects on quantities such as the joint survival probability at a fixed bivariate time point or simultaneously at several time points and, consequentially, can estimate covariate-adjusted conditional survival probabilities. We demonstrate the method using simulations and an analysis of 2 real-world datasets.

We propose an extension of the generalized bivariate Marshall–Olkin model assuming dependence between the random variables involved. Probabilistic, aging properties, and survival copula representation of the extended model are obtained and illustrated by examples. Bayesian analysis is performed and possible applications are discussed. A dual version of extended Marshall–Olkin model is introduced and related stochastic order comparisons are presented.

23 Multivariate exponential distributions and their applications in reliability

Joint survival probability via truncated invariant copula

In limited overs cricket, the goal of a batsman is to score a maximum number of runs within a limited number of balls. Therefore, the number of runs scored and the number of balls faced are the two key statistics used to evaluate the performance of a batsman. In cricket, as the batsmen play as pairs, having longer partnerships is also key to building strong innings. Moreover, having a steady opening partnership is extremely important as a team aims to build such a stronger innings. In this study, we have shown a way to evaluate the performance of opening partnerships in Twenty20 (T20) cricket and the performance of individual batsmen in One Day International Cricket (ODI) by modeling the joint distribution of runs scored and balls faced using copula functions. The joint survival probabilities derived from this approach are then used to evaluate the batting performance of opening partnerships and individual batsmen for different stages of the innings. Results of the study have shown that cricket managers and team officials can use the proposed method in selecting appropriate partnership pairs and individual batsmen in an efficient manner for specific situations in the match.

Thispaper provides a simple model for valuing a credit derivativewhose payoff depends on the identity (or identities) of the first(or first two) to occur of a given list of credit events, suchas defaults. The joint survival probability of occurrence timesof credit events is formulated in terms of stochastic intensityprocesses under the assumption of conditional independence. Basedon the joint survival probability, we can easily obtain the pricingformulas of such credit derivatives under the risk-neutral valuationframework. When the default intensity processes follow the extendedVasicek model, closed-form solutions of the pricing formulasare given.