Exponential Marginal Distributions Research Articles

Clayton copula for survival data with dependent censoring: An application to a tuberculosis treatment adherence data.

Ignoring the presence of dependent censoring in data analysis can lead to biased estimates, for example, not considering the effect of abandonment of the tuberculosis treatment may influence inferences about the cure probability. In order to assess the relationship between cure and abandonment outcomes, we propose a copula Bayesian approach. Therefore, the main objective of this work is to introduce a Bayesian survival regression model, capable of taking into account the dependent censoring in the adjustment. So, this proposed approach is based on Clayton's copula, to provide the relation between survival and dependent censoring times. In addition, the Weibull and the piecewise exponential marginal distributions are considered in order to fit the times. A simulation study is carried out to perform comparisons between different scenarios of dependence, different specifications of prior distributions, and comparisons with the maximum likelihood inference. Finally, we apply the proposed approach to a tuberculosis treatment adherence dataset of an HIV cohort from Alvorada-RS, Brazil. Results show that cure and abandonment outcomes are negatively correlated, that is, as long as the chance of abandoning the treatment increases, the chance of tuberculosis cure decreases.

Modeling Point Referenced Spatial Count Data: A Poisson Process Approach

Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and mathematical tractability. However, this assumption seems to be restrictive when dealing with counting data. To deal with this situation, we propose a random field with a Poisson marginal distribution considering a sequence of independent copies of a random field with an exponential marginal distribution as “inter-arrival times” in the counting renewal processes framework. Our proposal can be viewed as a spatial generalization of the Poisson counting process. Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal generates a (non)-stationary random field that is mean square continuous and with Poisson marginal distributions. For the proposed Poisson spatial random field, analytic expressions for the covariance function and the bivariate distribution are provided. In an extensive simulation study, we investigate the weighted pairwise likelihood as a method for estimating the Poisson random field parameters. Finally, the effectiveness of our methodology is illustrated by an analysis of reindeer pellet-group survey data, where a zero-inflated version of the proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson Gaussian copula models. Supplementary materials for this article, including technical proofs and R code for reproducing the work, are available as an online supplement.

Defaultable Bond Pricing under the Jump Diffusion Model with Copula Dependence Structure

We study the pricing of a defaultable bond under various dependence structure captured by copulas. For that purpose, we use a bivariate jump-diffusion process to represent a bond issuer’s default intensity and the market short rate of interest. We assume that each jump of both variables occur simultaneously, and that their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process and three copulas, which are a Farlie-Gumbel- Morgenstern copula, a Gaussian copula, and a Student t-copula are used, respectively. We use the joint Laplace transform of the integrated risk processes to obtain the expression of the defaultable bond price with copula-dependent jump sizes. Assuming exponential marginal distributions, we compute the zero coupon defaultable bond prices and their yields using the three copulas to illustrate the bond. We found that the bond price values are the lowest under the Student-t copula, suggesting that a dependence structure under the Student-t copula could be a suitable candidate to depict a riskier environment. Additionally, the hypothetical term structure of interest rates under the risky environment are also upward sloping, albeit with yields greater than 100%, reflecting a higher compensation required by investors to lend funds for a longer period when the financial market is volatile.

Joint survival probability via truncated invariant copula

Given an intensity-based credit risk model, this paper studies dependence structure between default intensities. To model this structure, we use a multivariate shot noise intensity process, where jumps occur simultaneously and their sizes are correlated. Through very lengthy algebra, we obtain explicitly the joint survival probability of the integrated intensities by using the truncated invariant Farlie–Gumbel–Morgenstern copula with exponential marginal distributions. We also apply our theoretical result to pricing basket default swap spreads. This result can provide a useful guide for credit risk management.

Modeling the Dependency Structure of Integrated Intensity Processes.

This paper studies an important issue of dependence structure. To model this structure, the intensities within the Cox processes are driven by dependent shot noise processes, where jumps occur simultaneously and their sizes are correlated. The joint survival probability of the integrated intensities is explicitly obtained from the copula with exponential marginal distributions. Subsequently, this result can provide a very useful guide for credit risk management.

A MULTIVARIATE JUMP DIFFUSION PROCESS FOR COUNTERPARTY RISK IN CDS RATES

We consider counterparty risk in CDS rates. To do so, we use a multivariate jump diffusion process for obligors’ default intensity, where jumps (i.e. magnitude of contribution of primary events to default intensities) occur simultaneously and their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process. We apply copuladependent default intensities of multivariate Cox process to derive the joint Laplace transform that provides us with joint survival/default probability and other relevant joint probabilities. For that purpose, the piecewise deterministic Markov process (PDMP) theory developed in [7] and the martingale methodology in [6] are used. We compute survival/default probability using three copulas, which are Farlie-Gumbel-Morgenstern (FGM), Gaussian and Student-t copulas, with exponential marginal distributions. We then apply the results to calculate CDS rates assuming deterministic rate of interest and recovery rate. We also conduct sensitivity analysis for the CDS rates by changing the relevant parameters and provide their figures.

Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims

We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001, 2, 98–110), which takes the form of the Volterra integral equation (VIE), we used the method of successive approximation to derive the Neumann series of the recursive moments. We then compute the first two moments of aggregate discounted claims, i.e., its mean and variance, based on the Neumann series expression, where the dependence structure is captured by a Farlie–Gumbel–Morgenstern (FGM) copula, a Gaussian copula and a Gumbel copula with exponential marginal distributions. Insurance premium calculations with their figures are also illustrated.

Characterization of upper comonotonicity via tail convex order

In this paper, we show a characterization of upper comonotonicity via tail convex order. For any given marginal distributions, a maximal random vector with respect to tail convex order is proved to be upper comonotonic under suitable conditions. As an application, we consider the computation of the Haezendonck risk measure of the sum of upper comonotonic random variables with exponential marginal distributions.

On the asymptotic behaviour of Lévy processes, Part I: Subexponential and exponential processes

We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.

Discontinuities in indirect estimation: An application to EAR models

Among the simulation-based methods, indirect estimation techniques like Indirect Inference (I NDI NF) and Efficient Method of Moments (E MM) provide a simple solution to many computational problems associated with intractable Likelihood functions. Optimisation of the objective function can be critical in presence of not continuous response variables like, for instance, binary choice or discrete choice models, limited dependent variables, switching regime models. In particular, gradient-based optimisation algorithms can face difficulties when the not continuous response involves discontinuities in the objective function. A simple computational tool is suggested to “empirically” solve the problem. The case study is E MM applied to the autoregressive model with exponential marginal distribution (E AR). The proposed solution is also compared with the performance of the Conditional Least Squares estimation, suitable for this autoregressive model, by a set of Monte Carlo experiments.

MSurvPow: a FORTRAN program to calculate the sample size and power for cluster-randomized clinical trials with survival outcomes

Manatunga and Chen [A.K. Manatunga, S. Chen, Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes, Biometrics 56 (2000) 616-621] proposed a method to estimate sample size and power for cluster-randomized studies where the primary outcome variable was survival time. The sample size formula was constructed by considering a bivariate marginal distribution (Clayton-Oakes model) with univariate exponential marginal distributions. In this paper, a user-friendly FORTRAN 90 program was provided to implement this method and a simple example was used to illustrate the features of the program.

Simulation of non-Gaussian processes using fractile correlation

The difficulties of simulating non-Gaussian stochastic processes to follow arbitrary product–moment covariance models and arbitrary non-Gaussian marginal distributions are well known. This paper proposes to circumvent these difficulties by prescribing a fractile correlation function, rather than the usual product–moment covariance function. This fractile correlation can be related to the product–moment correlation of a Gaussian process analytically. A Gaussian process with the requisite product–moment correlation can be simulated using the Karhunen–Loeve (K–L) expansion and transformed to satisfy any arbitrary marginal distribution using the usual CDF mapping. The fractile correlation of the non-Gaussian process will be identical to that of the underlying Gaussian process because it is invariant to monotone transforms. This permits the K–L expansion to be extended in a very general way to any second-order non-Gaussian processes. The simplicity of the proposed approach is illustrated numerically using a stationary squared exponential and a non-stationary Brown–Bridge fractile correlation function in conjunction with a shifted lognormal and a shifted exponential marginal distribution.

A Generalized Pareto intensity‐duration model of storm rainfall exploiting 2‐Copulas

Stochastic models of rainfall, usually based on Poisson arrivals of rectangular pulses, generally assume exponential marginal distributions for both storm duration and average rainfall intensity, and the statistical independence between these variables. However, the advent of stochastic multifractals made it clear that rainfall statistical properties are better characterized by heavy tailed Pareto‐like distributions, and also the independence between duration and intensity turned out to be a nonrealistic assumption. In this paper an improved intensity‐duration model is considered, which describes the dependence between these variables by means of a suitable 2‐Copula, and introduces Generalized Pareto marginals for both the storm duration and the average storm intensity. Several theoretical results are derived: in particular, we show how the use of 2‐Copulas allows reproducing not only the marginal variability of both storm average intensity and storm duration, but also their joint variability by describing their statistical dependence; in addition, we point out how the use of heavy tailed Generalized Pareto laws gives the possibility of modeling both the presence of extreme values and the scaling features of the rainfall process, and has interesting connections with the statistical structure of the process of rainfall maxima, which is naturally endowed with a Generalized Extreme Value law. Finally, a case study considering rainfall data is shown, which illustrates how the theoretical results derived in the paper are supported by the practical analysis.

A reversibility relationship for two Markovian time series models by stationary exponential tailed distribution

The continuous autoregressive and minification stationary non-negative time series models discussed by Chernick et al. (1988) are generalized to model marginal distributions which have atoms of mass at zero. The reversibility theorem relating these processes with exponential marginal distributions is extended to the case where the marginal distribution has exponential tail.

A reversibility relationship for two Markovian time series models by stationary exponential tailed distribution

The continuous autoregressive and minification stationary non-negative time series models discussed by Chernick et al. (1988) are generalized to model marginal distributions which have atoms of mass at zero. The reversibility theorem relating these processes with exponential marginal distributions is extended to the case where the marginal distribution has exponential tail.

Small sample comparison of six bivariate survival curve estimators1

We study the performance of six proposed bivariate survival curve estimators on simulated right censored data. The performance of the estimators is compared for data generated by three bivariate models with exponential marginal distributions. The estimators are compared in their ability to estimate correlations and survival functions probabilities. Simulated data results are presented so that the proposed estimators in this relatively new area of analysis can be explicitly compared to the known distribution of the data and the parameters of the underlying model. The results show clear differences in the performance of the estimators.

Two examples of nonlinear processes with a mixed exponential marginal distribution

The simple nonlinear models of autoregressive and moving average structure with a mixed exponential marginal distribution are described.

A stochastic model for the distribution of HIV latency time based on T4 counts

SUMMARY A mathematical model is proposed for the description of the T4 level, as a function of the time since initial infection, in the blood of an HIv-infected individual. Two random variables are defined for each infected individual, the time T from the moment of infection to the moment when the infection is first discovered, and the level R of T4 cells in the blood at the time of the discovery. Here T is the length of the latency period, and is not typically observable. On the basis of a stochastic model employing a stationary Gaussian process, the joint density of these two random variables is derived, and hence the conditional distribution of T, given R. It is shown that if T has an exponential marginal distribution, then the conditional distribution of T is a censored normal distribution. The theory is applied to data collected from intravenous drug users in New York City.

Bivariate extreme value theory: Models and estimation

SUMMARY Bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima. No natural parametric family exists for the dependence between the marginal distributions, but there are considerable restrictions on the dependence structure. We consider modelling the dependence function with parametric models, for which two new models are presented. Tests for independence, and discriminating between models, are also given. The estimation procedure, and the flexibility of the new models, are illustrated with an application to sea level data. Extreme value theory has recently been an area of much theoretical and practical work. Univariate theory is a well documented area, whereas bivariate/multivariate extreme value theory has, until recently, received surprisingly little attention. In the multivariate case, no natural parametric family exists for the dependence structure, so this must be modelled in some way. In the analysis of environmental extreme value data, there is a need for models of dependence between extremes from different sources: for example at various sea ports, or at various -points of a river. In this paper we consider bivariate extreme value distributions. We assume, without loss of generality, that we have exponential marginal distributions with unit means. The class of bivariate exponential distributions in which we are interested, satisfy a strong stability relation. Exponential variables (X, Y) satisfy the stability relation if and only if W = min (aX, bY) is also exponentially distributed for all a, b > 0 (Pickands, 1981). Therefore, the models we will consider have particular application in reliability and survival analysis. One approach to modelling the dependence structure is via parametric models. This requires a flexible family of models which satisfy certain constraints. Models are of two kinds: either differentiable, or nondifferentiable. All nondifferentiable models give distri- butions which are singular, with nonzero probability concentrated on a certain subspace. The differentiable models have densities, but the existing models are symmetric which leads to the variables being exchangeable. Here, we present two new asymmetric differenti- able models, which have increased flexibility. Properties of the differentiable models are examined. Estimation of the parametric models has previously been by ad hoc methods, because there is a nonregular estimation problem when the margins are independent. For the

Modelling and Residual Analysis of Nonlinear Autoregressive Time Series in Exponential Variables

SUMMARY An approach to modelling and residual analysis of nonlinear autoregressive time series in exponential variables is presented; the approach is illustrated by an analysis of a long series of wind velocity data which has first been detrended and then transformed into a stationary series with an exponential marginal distribution. The stationary series is modelled with a newly developed type of second order autoregressive process with random coefficients, called the NEAR(2) model; it has a second order autoregressive correlation structure but is nonlinear because its coefficients are random. The exponential distributional assumptions involved in this model highlight a very broad four parameter structure which combines five exponential random variables into a sixth exponential random variable; other applications of this structure are briefly considered. Dependency in the NEAR(2) process not accounted for by standard autocorrelations is explored by developing a residual analysis for time series having autoregressive correlation structure; this involves defining linear uncorrelated residuals which are dependent, and then assessing this higher order dependence by standard time series computations. The application of this residual analysis to the wind velocity data illustrates both the utility and difficulty of nonlinear time series modelling.