Improved techniques for parametric and nonparametric evaluations of the first‐passage time for degradation processes

For degradation data in reliability analysis, estimation of the first‐passage time (FPT) distribution to a threshold provides valuable information on reliability characteristics. Recently, Balakrishnan and Qin (2019; Applied Stochastic Models in Business and Industry, 35:571–590) studied a nonparametric method to approximate the FPT distribution of such degradation processes if the underlying process type is unknown. In this article, we propose some improved techniques based on saddlepoint approximation, which enhance those existing methods. Numerical examples and Monte Carlo simulation studies are used to illustrate the advantages of the proposed techniques. Limitations of the improved techniques are discussed and some possible solutions to such are proposed. Some concluding remarks and practical recommendations are provided based on the results.     

A note on the first passage time of diffusions with holding and jumping boundary

In this note we are concerned with the first passage time (FPT) of diffusions with holding and jumping boundary (DHJ) in one dimensional case. We first show that the Laplace transform of FPT of DHJ can be represented explicitly by the behavior of the killed process for one holding and jumping point. The results are then extended to the Laplace transform of FPT of DHJ with two end points. Finally, we demonstrate how the results are applied to a Wiener-type neuronal model in the presence of exponential refractoriness.     

含估计参数的加权经验过程

在讨论(广义)非参数似然比拟合优度检验时,加权经验过程理论是一个非常重要的基础.但对含估计参数的加权经验过程理论,目前文献中很少讨论.对含估计参数的加权经验过程的上界型和积分型两种统计量的近似分布进行了讨论,给出了较一般的结果.所得结论叮以为进一步讨论复合零假设下(广义)非参似然比拟合优度检验提供理论基础.     

A recursive approach to mortality-linked derivative pricing

In this paper, we develop a recursive method to derive an exact numerical and nearly analytical representation of the Laplace transform of the transition density function with respect to the time variable for time-homogeneous diffusion processes. We further apply this recursion algorithm to the pricing of mortality-linked derivatives. Given an arbitrary stochastic future lifetime T, the probability distribution function of the present value of a cash flow depending on T can be approximated by a mixture of exponentials, based on Jacobi polynomial expansions. In case of mortality-linked derivative pricing, the required Laplace inversion can be avoided by introducing this mixture of exponentials as an approximation of the distribution of the survival time T in the recursion scheme. This approximation significantly improves the efficiency of the algorithm.     

Empirical processes with estimated parameters under auxiliary information

Empirical processes with estimated parameters are a well established subject in nonparametric statistics. In the classical theory they are based on the empirical distribution function which is the nonparametric maximum likelihood estimator for a completely unknown distribution function. In the presence of some “nonparametric” auxiliary information about the distribution, like a known mean or a known median, for example, the nonparametric maximum likelihood estimator is a modified empirical distribution function which puts random masses on the observations in order to take the available information into account [see Owen, Biometrika 75 (1988) 237–249, Ann. Statist. 18 (1990) 90–120, Empirical Likelihood, Chapman & Hall/CRC, London/Boca Raton, FL; Qin and Lawless, Ann. Statist. 22 (1994) 300–325]. Zhang [Metrika 46 (1997) 221–244] has proved a functional central limit theorem for the empirical process pertaining to this modified empirical distribution function. We will consider the corresponding empirical process with estimated parameters here and derive its asymptotic distribution. The limiting process is a centered Gaussian process with a complicated covariance function depending on the unknown parameter. The result becomes useful in practice through the bootstrap, which is shown to be consistent in case of a known mean. The performance of the resulting bootstrap goodness-of-fit test based on the Kolmogorov–Smirnov statistic is studied through simulations.     

光滑分布函数分位数估计的注记(英)

文中通过光滑经验分布函数构造了分位数估计，建立该估计的Bahadu－强弱表示定理，并由Bahadur表示定理证明了该分估计估的重对数律和渐近正态性等深刻结果．     

LAPLACE TRANSFORM OF THE SURVIVAL PROBABILITY UNDER SPARRE ANDERSEN MODEL

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the survival probability is well given when the claim amount distribution is Erlang distribution or mixed Erlang distribution. The expressions for moments of the time to ruin with the model above are given.     

Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform

This paper considers two flexible classes of omnibus goodness-of-fit tests for the inverse Gaussian distribution. The test statistics are weighted integrals over the squared modulus of some measure of deviation of the empirical distribution of given data from the family of inverse Gaussian laws, expressed by means of the empirical Laplace transform. Both classes of statistics are connected to the first nonzero component of Neyman's smooth test for the inverse Gaussian distribution. The tests, when implemented via the parametric bootstrap, maintain a nominal level of significance very closely. A large-scale simulation study shows that the new tests compare favorably with classical goodness-of-fit tests for the inverse Gaussian distribution, based on the empirical distribution function.     

An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tests of fit for the Rayleigh distribution based on the empirical Laplace transform

In this paper a class of goodness-of-fit tests for the Rayleigh distribution is proposed. The tests are based on a weighted integral involving the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity the test statistics approach limit values. In a particular case the resulting limit statistic is related to the first nonzero component of Neyman’s smooth test for this distribution. The new tests are compared with other omnibus tests for the Rayleigh distribution.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.     

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump‐diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises. Copyright © 2008 John Wiley & Sons, Ltd.     

Uniform Laws of Large Numbers for Triangular Arrays of Function-indexed Processes under Random Entropy Conditions

Mean Glivenko Cantelli Theorems are established for triangular arrays of rowwise independent processes. Methods developed by Pollard (1990) are combined with a truncation method essentially due to Alexander (1987). By this, applicability to partial sum processes in particular is achieved, for which Pollard’s truncation method fails. Nevertheless, the metric entropy condition appearing here is kept as weak as Pollard’s by means of application of Hoffmann-Jørgensen’s inequality, which has not been used so far in this context. The main theorem of the paper contains Pollard’s theorem as well as former results by Giné and Zinn (1984) and proves applicable to so-called random measure processes, certain function-indexed processes including empirical processes, partial-sum processes, the sequential empirical process and certain types of smoothed empirical processes. Statistical applications include nonparametric regression and the estimation of the intensity measure of a spatial Poisson process (Poisson point process).     

Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect

In this paper, we obtain approximate‐analytical solutions of a cancer chemotherapy effect model involving fractional derivatives with exponential kernel and with general Mittag‐Leffler function. Laplace homotopy perturbation method and the modified homotopy analysis transform method were applied. The first method is based on a combination of the Laplace transform and homotopy methods, while the second method is an analytical technique based on homotopy polynomial. The cancer chemotherapy effect equations are solved numerically and analytically using the aforesaid methods. Illustrative examples are included to demonstrate the validity and applicability of the presented technique with new fractional‐order derivatives with exponential decay law and with general Mittag‐Leffler law.     

Laplace approximation for stochastic line integrals

We prove a precision of large deviation principle for current-valued processes such as shown in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) for mean empirical measures. The class of processes we consider is determined by the martingale part of stochastic line integrals of 1-forms on a compact Riemannian manifold. For the pair of the current-valued process and mean empirical measures, we give an asymptotic evaluation of a nonlinear Laplace transform under a nondegeneracy assumption on the Hessian of the exponent at equilibrium states. As a direct consequence, our result implies the Laplace approximation for stochastic line integrals or periodic diffusions. In particular, we recover a result in Bolthausen et al. (Ann Probab 23(1):236–267, 1995) in our framework.     

Lévy risk model with two-sided jumps and a barrier dividend strategy

In this paper, we consider a general Lévy risk model with two-sided jumps and a constant dividend barrier. We connect the ruin problem of the ex-dividend risk process with the first passage problem of the Lévy process reflected at its running maximum. We prove that if the positive jumps of the risk model form a compound Poisson process and the remaining part is a spectrally negative Lévy process with unbounded variation, the Laplace transform (as a function of the initial surplus) of the upward entrance time of the reflected (at the running infimum) Lévy process exhibits the smooth pasting property at the reflecting barrier. When the surplus process is described by a double exponential jump diffusion in the absence of dividend payment, we derive some explicit expressions for the Laplace transform of the ruin time, the distribution of the deficit at ruin, and the total expected discounted dividends. Numerical experiments concerning the optimal barrier strategy are performed and new empirical findings are presented.     

Exit times for a class of piecewise exponential Markov processes with two-sided jumps

We consider first passage times for piecewise exponential Markov processes that may be viewed as Ornstein–Uhlenbeck processes driven by compound Poisson processes. We allow for two-sided jumps and as a main result we derive the joint Laplace transform of the first passage time of a lower level and the resulting undershoot when passage happens as a consequence of a downward (negative) jump. The Laplace transform is determined using complex contour integrals and we illustrate how the choice of contours depends in a crucial manner on the particular form of the negative jump part, which is allowed to belong to a dense class of probabilities. We give extensions of the main result to two-sided exit problems where the negative jumps are as before but now it is also required that the positive jumps have a distribution of the same type. Further, extensions are given for the case where the driving Lévy process is the sum of a compound Poisson process and an independent Brownian motion. Examples are used to illustrate the theoretical results and include the numerical evaluation of some concrete exit probabilities. Also, some of the examples show that for specific values of the model parameters it is possible to obtain closed form expressions for the Laplace transform, as is the case when residue calculus may be used for evaluating the relevant contour integrals.     

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Brownian motion and normal distribution have been widely used in Cox-Ingersoll-Ross interest rate framework to model the instantaneous interest rate dynamics. However, empirical studies have also shown that the return distribution of interest rate has a higher peak and two fatter tails than those of the normal distribution. Meanwhile, when the rare catastrophic shocks occur or the regime shifts in the economy and finance, the money market may have jumps. In this paper, we will consider a class of reflected Cox-Ingersoll-Ross interest rate models with noise. Furthermore, we shall continue to supply the Laplace transform of the stationary distribution about this reflected diffusion process with jumps.     

A new method for proving weak convergence results applied to nonparametric estimators in survival analysis

Using the limit theorem for stochastic integral obtained by Jakubowski et al. (Probab. Theory Related Fields 81 (1989) 111–137), we introduce in this paper a new method for proving weak convergence results of empirical processes by a martingale method which allows discontinuities for the underlying distribution. This is applied to Nelson–Aalen and Kaplan–Meier processes. We also prove that the same conclusion can be drawn for Hjort's nonparametric Bayes estimators of the cumulative distribution function and cumulative hazard rate.