Showing papers in "Methodology and Computing in Applied Probability in 2014"
TL;DR: In this paper, the authors consider a variant of the event ruin for a Levy risk process, where the surplus process is allowed to spend time under a pre-specified default level before ruin is recognized.
Abstract: We consider a similar variant of the event ruin for a Levy insurance risk process as in Czarna and Palmowski (J Appl Probab 48(4):984–1002, 2011) and Loeffen et al. (to appear, 2011) when the surplus process is allowed to spend time under a pre-specified default level before ruin is recognized. In these two articles, the ruin probability is examined when deterministic implementation delays are allowed. In this paper, we propose to capitalize on the idea of randomization and thus assume these delays are of a mixed Erlang nature. Together with the analytical interest of this problem, we will show through the development of new methodological tools that these stochastic delays lead to more explicit and computable results for various ruin-related quantities than their deterministic counterparts. Using the modern language of scale functions, we study the Laplace transform of this so-called Parisian time to ruin in an insurance risk model driven by a spectrally negative Levy process of bounded variation. In the process, a generalization of the two-sided exit problem for this class of processes is further obtained.
88 citations
TL;DR: The skew–normal copula is introduced, which is then extended to an infinite mixture model and an MCMC algorithm is developed to draw samples from the correct posterior distribution.
Abstract: The paper presents a general Bayesian nonparametric approach for estimating a high dimensional copula. We first introduce the skew–normal copula, which we then extend to an infinite mixture model. The skew–normal copula fixes some limitations in the Gaussian copula. An MCMC algorithm is developed to draw samples from the correct posterior distribution and the model is investigated using both simulated and real applications.
37 citations
TL;DR: In this article, the authors extend classical edge correction methods to the spatio-temporal setting and compare the performance of the related estimators for stationary/non-stationary and/or isotropic/anisotropic point patterns.
Abstract: Non-parametric estimates of the K-function and the pair correlation function play a fundamental role for exploratory and explanatory analysis of spatial and spatio-temporal point patterns. These estimates usually require information from outside of the study region, resulting to the so-called edge effects which have to be corrected. They also depend on first-order characteristics, which have to be estimated in practice. In this paper, we extend classical edge correction methods to the spatio-temporal setting and compare the performance of the related estimators for stationary/non-stationary and/or isotropic/anisotropic point patterns. Further, we explore the influence of the estimated intensity function on these estimators.
37 citations
TL;DR: Wang et al. as discussed by the authors proposed a method to use the National Natural Science Foundation of China [11171278] and Fundamental Research Funds for the Central Universities of China.
Abstract: National Natural Science Foundation of China [11171278]; Fundamental Research Funds for the Central Universities of China [2010121005]
36 citations
TL;DR: In this article, the exact simulation of jump-diffusion bridges has been studied in the context of statistical applications, and an algorithm that performs exact simulation for a class of SDEs has been proposed.
Abstract: Exact simulation of SDEs is a very important and challenging problem. In this paper we discuss exact simulation problems for jump-diffusion processes. Motivated by statistical applications, our main contribution is to propose an algorithm that performs exact simulation of a class of jump-diffusion bridges. We also present and discuss the existing methods for forward simulation and propose an extension of one of them to account for unbounded jump rate. Finally, the exact algorithms are compared to competing non-exact ones in some simulated examples.
31 citations
TL;DR: A modification of the delayed rejection algorithm is proposed, in which the direction of the different proposals is fixed once for all, and the Metropolis–Hastings accept-reject mechanism is used to select a proper scaling along the search direction.
Abstract: In this paper, we study the asymptotic efficiency of the delayed rejection strategy In particular, the efficiency of the delayed rejection Metropolis–Hastings algorithm is compared to that of the regular Metropolis algorithm To allow for a fair comparison, the study is carried under optimal mixing conditions for each of these algorithms After introducing optimal scaling results for the delayed rejection (DR) algorithm, we outline the fact that the second proposal after the first rejection is discarded, with a probability tending to 1 as the dimension of the target density increases To overcome this drawback, a modification of the delayed rejection algorithm is proposed, in which the direction of the different proposals is fixed once for all, and the Metropolis–Hastings accept-reject mechanism is used to select a proper scaling along the search direction It is shown that this strategy significantly outperforms the original DR and Metropolis algorithms, especially when the dimension becomes large We include numerical studies to validate these conclusions
27 citations
TL;DR: In this paper, the authors investigated the effect of inflation, truncation or censoring from below (use of a deductible) and truncation from above (using of a policy limit) on the entropy of losses of insurance policies.
Abstract: The best policy for an insurance company is that which lasts for a long period of time and is less uncertain with reference to its claims. In information theory, entropy is a measure of the uncertainty associated with a random variable. It is a descriptive quantity as it belongs to the class of measures of variability, such as the variance and the standard deviation. The purpose of this paper is to investigate the effect of inflation, truncation or censoring from below (use of a deductible) and truncation or censoring from above (use of a policy limit) on the entropy of losses of insurance policies. Losses are differentiated between per-payment and per-loss (franchise deductible). In this context we study the properties of the resulting entropies such as the residual loss entropy and the past loss entropy which are the result of use of a deductible and a policy limit, respectively. Interesting relationships between these entropies are presented. The combined effect of a deductible and a policy limit is also studied. We also investigate residual and past entropies for survival models. Finally, an application is presented involving the well-known Danish data set on fire losses.
25 citations
TL;DR: The main results obtained in this paper generalize some related ones in recent literature about the signature of a k-out-of-n coherent system consisting of n modules.
Abstract: In this paper, we study how to compute the signature of a k-out-of-n coherent system consisting of n modules. Formulas for computing the signature and the minimal signature of this kind of systems based on those of their modules are derived. Examples are presented to demonstrate the applications of our formulas. The main results obtained in this paper generalize some related ones in recent literature.
21 citations
TL;DR: This article reviews some nonparametric serial independence tests based on measures of divergence between densities, and the well-known Kullback–Leibler, Hellinger, Tsallis, and Rosenblatt divergences are analyzed.
Abstract: This article reviews some nonparametric serial independence tests based on measures of divergence between densities. Among others, the well-known Kullback–Leibler, Hellinger, Tsallis, and Rosenblatt divergences are analyzed. Moreover, their copula-based version is taken into account. Via a wide simulation study, the performances of the considered serial independence tests are compared under different settings. Both single-lag and multiple-lag testing procedures are investigated to find out the best “omnibus” solution.
16 citations
TL;DR: In this paper, the authors study generalizations of the tax identity of Albrecher and Hipp (2007) from the classical risk model to more general risk processes driven by spectrally negative MAPs.
Abstract: Taxed risk processes, i.e. processes which change their drift when reaching new maxima, represent a certain type of generalizations of Levy and of Markov additive processes (MAP), since the times at which their Markovian mechanism changes are allowed to depend on the current position. In this paper we study generalizations of the tax identity of Albrecher and Hipp (2007) from the classical risk model to more general risk processes driven by spectrally-negative MAPs. We use the Sparre Andersen risk processes with phase-type interarrivals to illustrate the ideas in their simplest form.
15 citations
TL;DR: In this paper, the authors introduced a flexible inhomogeneous space-time shot-noise Cox process model and derived a two-step estimation procedure for it in the first step the inhomogeneity is estimated by means of a Poisson score estimating equation and in the second step the minimum contrast estimation based on second order properties to obtain estimates of the clustering parameters.
Abstract: In the paper we introduce a flexible inhomogeneous space-time shot-noise Cox process model and derive a two-step estimation procedure for it In the first step the inhomogeneity is estimated by means of a Poisson score estimating equation and in the second step we use minimum contrast estimation based on second order properties to obtain estimates of the clustering parameters The suggested model is not separable but it has a special interaction structure which enables to use the spatial and temporal projections of the process for parameter estimation Efficiency of the introduced method is investigated by means of a simulation study and it is compared to a previously used method
TL;DR: In this article, the authors investigated the stochastic behavior and reliability properties of the residual lifetime of live components in coherent systems under the assumption that the system fails at time t.
Abstract: Some coherent systems are such that failure of the system does not mean that all components fail. This paper investigates the stochastic behavior and reliability properties of the residual lifetime of live components in coherent systems under the assumption that the system fails at time t. We also investigate the stochastic properties of inactivity time of failed components in coherent systems where failure of some components does not cause the failure of the complete system.
TL;DR: In this paper, for heavy-tailed models and through the use of probability weighted moments based on the largest observations, the authors deal essentially with the semi-parametric estimation of the Value-at-Risk at a level p, where the size of the loss occurred with a small probability p, as well as the dual problem of estimating the probability of exceedance of a high level x.
Abstract: In this paper, for heavy-tailed models and through the use of probability weighted moments based on the largest observations, we deal essentially with the semi-parametric estimation of the Value-at-Risk at a level p, the size of the loss occurred with a small probability p, as well as the dual problem of estimation of the probability of exceedance of a high level x. These estimation procedures depend crucially on the estimation of the extreme value index, the primary parameter in Statistics of Extremes, also done on the basis of the same weighted moments. Under regular variation conditions on the right-tail of the underlying distribution function F, we prove the consistency and asymptotic normality of the estimators under consideration in this paper, through the usual link of their asymptotic behaviour to the one of the extreme value index estimator they are based on. The performance of these estimators, for finite samples, is illustrated through Monte-Carlo simulations. An adaptive choice of thresholds is put forward. Applications to a real data set in the field of insurance as well as to simulated data are also provided.
TL;DR: The joint limiting distribution of standardisedmaximum of a continuous, stationary Gaussian process and the standardised maximum of this process sampled at discrete time points are derived and it is proved that these two random sequences are asymptotically complete dependent if the grid of the discreteTime points is sufficiently dense, and asymPTotically independent if theGrid is sufficiently sparse.
Abstract: With motivation from Husler (Extremes 7:179–190, 2004) and Piterbarg (Extremes 7:161–177, 2004) in this paper we derive the joint limiting distribution of standardised maximum of a continuous, stationary Gaussian process and the standardised maximum of this process sampled at discrete time points. We prove that these two random sequences are asymptotically complete dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the grid is sufficiently sparse. We show that our results are relevant for computational problems related to discrete time approximation of the continuous time maximum.
TL;DR: In this paper, the distances between flats of a Poisson k-flat process in the d-dimensional Euclidean space with k < d/2 are discussed, and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous point process on the positive real half-axis.
Abstract: The distances between flats of a Poisson k-flat process in the d-dimensional Euclidean space with k < d/2 are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the m-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real half-axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener–Ito chaos decomposition and the Malliavin–Stein method.
TL;DR: In this paper, local stereological estimators of Minkowski tensors defined on convex bodies in Ω d ≥ 3 were presented, and the performance of some of these estimators for centres of gravity and volume tensors of rank two was investigated by simulation.
Abstract: In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in ℝ
d
. Special cases cover a number of well-known local stereological estimators of volume and surface area in ℝ3, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation.
TL;DR: In this article, a non-standard risk model defined on a fixed time interval [0,t] was revisited, where the arrival times of claims are distributed as the order statistics of n i.i.d. random variables and the ultimate non-ruin probabilities were derived as a limit.
Abstract: Recently, Lefevre and Picard (Insur Math Econ 49:512–519, 2011) revisited a non-standard risk model defined on a fixed time interval [0,t]. The key assumption is that, if n claims occur during [0,t], their arrival times are distributed as the order statistics of n i.i.d. random variables with distribution function F
t
(s), 0 ≤ s ≤ t. The present paper is concerned with two particular cases of that model, namely when F
t
(s) is of linear form (as for a (mixed) Poisson process), or of exponential form (as for a linear birth process with immigration or a linear death-counting process). Our main purpose is to obtain, in these cases, an expression for the non-ruin probabilities over [0,t]. This is done by exploiting properties of an underlying family of Appell polynomials. The ultimate non-ruin probabilities are then derived as a limit.
TL;DR: In this paper, the first passage distributions of semi-Markov processes are calculated using the inverse discrete Fourier transform for lattice distributions, which is shown to be fast and accurate.
Abstract: First passage distributions of semi-Markov processes are of interest in fields such as reliability, survival analysis, and many others. Finding or computing first passage distributions is, in general, quite challenging. We take the approach of using characteristic functions (or Fourier transforms) and inverting them to numerically calculate the first passage distribution. Numerical inversion of characteristic functions can be unstable for a general probability measure. However, we show they can be quickly and accurately calculated using the inverse discrete Fourier transform for lattice distributions. Using the fast Fourier transform algorithm these computations can be extremely fast. In addition to the speed of this approach, we are able to prove a few useful bounds for the numerical inversion error of the characteristic functions. These error bounds rely on the existence of a first or second moment of the distribution, or on an eventual monotonicity condition. We demonstrate these techniques with two examples.
TL;DR: In this article, the truncated second moment of the innovations (l(x)=\textsf{E} [u_1^2I\{|u_ 1|\le x\}] is a slowly varying function at ∞, which may tend to infinity as x → ∞.
Abstract: An asymptotic theory was given by Phillips and Magdalinos (J Econom 136(1):115–130, 2007) for autoregressive time series Yt = ρYt−1 + ut, t = 1,...,n, with ρ = ρn = 1 + c/kn, under (2 + δ)-order moment condition for the innovations ut, where δ > 0 when c 0, {ut} is a sequence of independent and identically distributed random variables, and (kn)n ∈ ℕ is a deterministic sequence increasing to infinity at a rate slower than n. In the present paper, we established similar results when the truncated second moment of the innovations \(l(x)=\textsf{E} [u_1^2I\{|u_1|\le x\}]\) is a slowly varying function at ∞, which may tend to infinity as x → ∞. More interestingly, we proposed a new pivotal for the coefficient ρ in case c < 0, and formally proved that it has an asymptotically standard normal distribution and is nuisance parameter free. Our numerical simulation results show that the distribution of this pivotal approximates the standard normal distribution well under normal innovations.
TL;DR: In this paper, the authors studied the waiting time distribution for random variables related to the first occurrence of an almost perfect run in a sequence of Bernoulli trials, using an appropriate Markov chain embedding approach.
Abstract: A natural and intuitively appealing generalization of the runs principle arises if instead of looking at fixed-length strings with all their positions occupied by successes, we allow the appearance of a small number of failures. Therefore, the focus is on clusters of consecutive trials which contain large proportion of successes. Such a formation is traditionally called “scan” or alternatively, due to the high concentration of successes within it, almost perfect (success) run. In the present paper, we study in detail the waiting time distribution for random variables related to the first occurrence of an almost perfect run in a sequence of Bernoulli trials. Using an appropriate Markov chain embedding approach we present an efficient recursive scheme that permits the construction of the associated transition probability matrix in an algorithmically efficient way. It is worth mentioning that, the suggested methodology, is applicable not only in the case of almost perfect runs, but can tackle the general discrete scan case as well. Two interesting applications in statistical process control are also discussed.
TL;DR: Aggregate Monte Carlo (AMC) as mentioned in this paper is a fast alternative to the standard CMC algorithm, which is suitable for efficient simulation of continuous-time Markov chains that are nearly-completely decomposable.
Abstract: A methodology is proposed that is suitable for efficient simulation of continuous-time Markov chains that are nearly-completely decomposable. For such Markov chains the effort to adequately explore the state space via Crude Monte Carlo (CMC) simulation can be extremely large. The purpose of this paper is to provide a fast alternative to the standard CMC algorithm, which we call Aggregate Monte Carlo (AMC). The idea of the AMC algorithm is to reduce the jumping back and forth of the Markov chain in small subregions of the state space. We accomplish this by aggregating such problem regions into single states. We discuss two methods to identify collections of states where the Markov chain may become ‘trapped’: the stochastic watershed segmentation from image analysis, and a graph-theoretic decomposition method. As a motivating application, we consider the problem of estimating the charge carrier mobility of disordered organic semiconductors, which contain low-energy regions in which the charge carrier can quickly become stuck. It is shown that the AMC estimator for the charge carrier mobility reduces computational costs by several orders of magnitude compared to the CMC estimator.
TL;DR: In this article, the authors considered the M/M/c retrial queues with multiclass of customers and showed that the stationary joint distribution for the number of customers in service facility and orbit converges to those of the ordinary M/m/c with discriminatory random order service (DROS) policy as retrial rate tends to infinity.
Abstract: We consider the M/M/c retrial queues with multiclass of customers. We show that the stationary joint distribution for the number of customers in service facility and orbit converges to those of the ordinary M/M/c with discriminatory random order service (DROS) policy as retrial rate tends to infinity. Approximation formulae for the distributions of the number of customers in service facility, the mean number of customers in orbit and the sojourn time distribution of a customer are presented. The approximations are compared with exact and simulation results.
TL;DR: Guo as discussed by the authors derived the Laplace transform of the first passage time in a 2-state Markov-switching model and gave one of the pioneering works improving the analytical tractability of Markov switching models.
Abstract: Guo (Methodol Comput Appl Probab 3(2):135–143, 2001a) derived the Laplace transform of the first-passage time in a 2-state Markov-switching model and gave one of the pioneering works improving the analytical tractability of Markov-switching models. However, the Laplace transforms in her paper are wrong. This short note provides the correct expression and an alternative proof using the matrix Wiener–Hopf technique.
TL;DR: In this article, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered, and an interesting application to risk theory is discussed.
Abstract: In this paper, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered. First, we obtain an error bound using the method of exponents and it is compared with existing ones. It is known that Kerstan’s method is more powerful in compound approximation problems. We employ Kerstan’s method to obtain better estimates, using higher-order approximations. These bounds are of higher-order accuracy and improve upon some of the known results in the literature. Finally, an interesting application to risk theory is discussed.
TL;DR: In this article, a regime-switching reflected stochastic process with two-sided barriers is proposed for modeling asset price dynamics in a regulated market, where the macroeconomic environment is taken into account.
Abstract: This paper considers asset dynamics in a regulated (controlled) market, where the macroeconomic environment is taken into account. A regime-switching reflected stochastic process with two-sided barriers is proposed for modeling asset price dynamics. We study a default problem with the default time being defined as the first passage time of the price dynamics. By solving a pair of interacting ordinary differential equations (ODEs), we obtain an explicit formula for the Laplace transform (LT) of the default time. Some numerical results are given for illustration.
TL;DR: In this paper, a graphical and formal approach is introduced to distinguish different types of inhomogeneity on Neyman-Scott point processes, such as inhomogeneous cluster centers, second order intensity reweighted stationarity, location dependent scaling and growing clusters.
Abstract: In this paper we introduce a graphical and formal approach to distinguishing different typed of inhomogeneity on Neyman–Scott point processes. The assumed types of inhomogeneity are (1) inhomogeneous cluster centers, (2) second order intensity reweighted stationarity, (3) location dependent scaling and a new type (4) growing clusters. The performance of the method is studied via a simulation study. This work has been motivated and illustrated by ecological studies of the spatial distribution of fish in an inland reservoir.
TL;DR: In this paper, the authors studied the problem of optimal approximation of a fractional Brownian motion by martingales and proved that there exists a unique martingale closest to fractional brownian motion in a specific sense.
Abstract: We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exists a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.
TL;DR: This paper treats a class of models with translationally invariant migration and use Fourier transforms for computing these quantities and shows how the QE approach is related to other methods based on conditional kinship coefficients between subpopulations under mutation-migration-drift equilibrium.
Abstract: For populations with geographic substructure and selectively neutral genetic data, the short term dynamics is a balance between migration and genetic drift. Before fixation of any allele, the system enters into a quasi equilibrium (QE) state. Hossjer and Ryman (2012) developed a general QE methodology for computing approximations of spatial autocorrelations of allele frequencies between subpopulations, subpopulation differentiation (fixation indexes) and variance effective population sizes. In this paper we treat a class of models with translationally invariant migration and use Fourier transforms for computing these quantities. We show how the QE approach is related to other methods based on conditional kinship coefficients between subpopulations under mutation-migration-drift equilibrium. We also verify that QE autocorrelations of allele frequencies are closely related to the expected value of Moran’s autocorrelation function and treat limits of continuous spatial location (isolation by distance) and an infinite lattice of subpopulations. The theory is illustrated with several examples including island models, circular and torus stepping stone models, von Mises models, hierarchical island models and Gaussian models. It is well known that the fixation index contains information about the effective number of migrants. The spatial autocorrelations are complementary and typically reveal the type of migration (local or global).
TL;DR: In this article, a lower bound for the critical value of the supremum of a Chi-square process is given, which can be approximated using a MCQMC simulation.
Abstract: We describe a lower bound for the critical value of the supremum of a Chi-Square process. This bound can be approximated using a MCQMC simulation. We compare numerically this bound with the upper bound given by Davies, only suitable for a regular Chi-Square process. In a second part, we focus a non regular Chi-Square process : the Ornstein-Uhlenbeck Chi-Square process. Recently, Rabier et al. (2009) have shown that this process has an application in genetics : it is the limiting process of the likelihood ratio test process related to the test of a gene on an interval representing a chromosome. Using results from \citet{del}, we propose a theoretical formula for the supremum of such a process and we compare it in particular with our simulated lower bound.
TL;DR: In this article, the authors study the fringe of random recursive trees, by analyzing the joint distribution of the counts of uncorrelated motifs, and show that these numbers have a limiting joint multivariate normal distribution.
Abstract: We study the fringe of random recursive trees, by analyzing the joint distribution of the counts of uncorrelated motifs Our approach allows for finite and countably infinite collections To be able to deal with the collection when it is infinitely countable, we use measure-theoretic themes Each member of a collection of motifs occurs a certain number of times on the fringe We show that these numbers, under appropriate normalization, have a limiting joint multivariate normal distribution We give a complete characterization of the asymptotic covariance matrix The methods of proof include contraction in a metric space of distribution functions to a fixed-point solution (limit distribution) We discuss two examples: the finite collection of all possible motifs of size four, and the infinite collection of rooted stars We conclude with remarks to compare fringe-analysis with matching motifs everywhere in the tree