Brownian Bridge and Other Path-dependent Gaussian Processes Vectorial Simulation

The iterative simulation of the Brownian bridge is well known. In this article, we present a vectorial simulation alternative based on Gaussian processes for machine learning regression that is suitable for interpreted programming languages implementations. We extend the vectorial simulation of path-dependent trajectories to other Gaussian processes, namely, sequences of Brownian bridges, geometric Brownian motion, fractional Brownian motion, and Ornstein–Ulenbeck mean reversion process.     

Ruin problem of a two-dimensional fractional Brownian motion risk process

This paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity.     

A Latent Process Model for Joint Modeling of Events and Marker

The paper formulates joint modeling of a counting process and a sequence of longitudinal measurements, governed by a common latent stochastic process. The latent process is modeled as a function of explanatory variables and a Brownian motion process. The conditional likelihood given values of the latent process at the measurement times, has been drawn using Brownian bridge properties; then integrating over all possible values of the latent process at the measurement times leads to the desired joint likelihood. An estimation procedure using joint likelihood and a numerical optimization is described. The method is applied to the study of cognitive decline and Alzheimer's disease.     

Fractional Brownian motion and long term clinical trial recruitment

Prediction of recruitment in clinical trials has been a challenging task. Many methods have been studied, including models based on Poisson process and its large sample approximation by Brownian motion (BM); however, when the independent incremental structure is violated for BM model, we could use fractional Brownian motion to model and approximate the underlying Poisson processes with random rates. In this paper, fractional Brownian motion (FBM) is considered for such conditions and compared to BM model with illustrated examples from different trials and simulations.     

Approximation to two independent Gaussian processes from a unique Lévy process and applications

Abstract

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively.     

A quadratically convergent algorithm for first passage time distributions in the Markov-modulated Brownian motion

A Markov-modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian Motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As with Brownian Motion, the time-dependent analysis of the MMBM becomes easy once the first passage times between levels are determined. However, in the MMBM those distributions cannot be obtained explicitly, and we need efficient algorithms to compute them. In this article, we provide a powerful approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows without Brownian components that weakly converge to the MMBM. Our main result is a Riccati equation for an associated matrix of transforms that satisfies conditions for the Newton scheme to have quadratic convergence and thus yields a very practical tool. The solution of that Riccati equation determines needed first passage times in the MMBM without much additional work. The success of our approach, which is based essentially on first-order fluid flows and a stochastic limit process, is argued to be due to the way we have isolated certain terms involving the quadratic variation effects of the Brownian. As an illustration of our algorithm, we present a numerical example of time-dependent results for a MMBM considered by Asmussen for which he determined (only) the eventual first return probabilities which we use here as an accuracy check.     

Real-time reliability evaluation of two-phase Wiener degradation process

Reliability modeling and evaluation for the two-phase Wiener degradation process are studied. For many devices, the degradation rates could possibly increase or decrease in a non smooth manner at some point in time due to the change of degradation mechanism. A two-phase Wiener degradation process with an unobserved change point is used to model the degradation process. And we assume that the change point varies randomly from device to device. Furthermore, we integrate historical data and up-to-date observation data to improve the degradation modeling and evaluation based on Bayesian method. The change point between the two phases was obtained based on the Akaike information criterion (AIC) and the criterion of the residual sum of squares. Finally, a real example of liquid coupling devices (LCDs) and a numeric example are discussed to demonstrate the effectiveness of the proposed method. The results show that the proposed method is effective and efficient.     

Estimation of the Hurst parameter in some fractional processes

We propose to estimate the Hurst parameter involved in fractional processes via a method based on the Karhunen–Loève expansion of a Gaussian process. We specifically investigate the cases of the fractional Brownian motion, the fractional Ornstein–Uhlenbeck family and the fractional Brownian bridge. We numerically compare our results with the ones obtained by the maximum-likelihood method, which show the validity of our proposal.     

A tampered Brownian motion process model for partial step-stress accelerated life testing

In partial step-stress accelerated life testing, models extrapolating data obtained under more severe conditions to infer the lifetime distribution under normal use conditions are needed. Bhattacharyya (Invited paper for 46th session of the ISI, 1987) proposed a tampered Brownian motion process model and later derived the probability distribution from a decay process perspective without linear assumption. In this paper, the model is described and the features of the failure time distribution are discussed. The maximum likelihood estimates of the parameters in the model and their asymptotic properties are presented. An application of models for step-stress accelerated life test to fields other than engineering is described and illustrated by applying the tampered Brownian motion process model to data taken from a clinical trial.     

Residual life estimation based on bivariate Wiener degradation process with time-scale transformations

The issue of residual life (RL) estimation plays an important role for products while they are in use, especially for expensive and reliability-critical products. For many products, they may have two or more performance characteristics (PCs). Here, an adaptive method of RL estimation based on bivariate Wiener degradation process with time-scale transformations is presented. It is assumed that a product has two PCs, and that each PC is governed by a Wiener process with a time-scale transformation. The dependency of PCs is characterized by the Frank copula function. Parameters are estimated by using the Bayesian Markov chain Monte Carlo method. Once new degradation information is available, the RL is re-estimated in an adaptive manner. A numerical example about fatigue cracks is given to demonstrate the usefulness and validity of the proposed method.     

Weighted Approximations to Continuous Time Martingales with Applications

A weighted approximation to a sequence of continuous time martingales by a time transformed Wiener process is established. The basic tool of proof is the Skorohod imbedding for martingale difference sequences. As an application of the main result a useful weighted approximation to the randomly weighted uniform empirical process is derived. A number of other applications are also discussed.     

Some observations on the KMT dyadic scheme

We point out some useful properties of the KMT dyadic scheme, which was used to construct a Brownian bridge approximation to the uniform empirical process.     

A month carlo approximation of the distributions of the maximum of various brownjan bridges

A Brownian bridge of order a is the weak limit of a residual partial sum obtained from regression fitting. When q=0, the process is teh usual Brownian bridge, and the distribution of the maximum is known analytically. For q>1, the reflection principle does not easily apply. The supremum distributions are approximated by a Monte Carlo technique. Tables of these distributions, as well as a finite sample size correction are given.     

Approximations for weighted Kolmogorov–Smirnov distributions via boundary crossing probabilities

A statistical application to Gene Set Enrichment Analysis implies calculating the distribution of the maximum of a certain Gaussian process, which is a modification of the standard Brownian bridge. Using the transformation into a boundary crossing problem for the Brownian motion and a piecewise linear boundary, it is proved that the desired distribution can be approximated by an n-dimensional Gaussian integral. Fast approximations are defined and validated by Monte Carlo simulation. The performance of the method for the genomics application is discussed.     

A note on the residual median process

We study the residual median process, defined as the median of those observations which are greater than a number t. Using appropriate limit theorems, it is shown that the stochastic process converges in law to a Gaussian process defined in terms of a Brownian bridge.     

Repeated Confidence Intervals Under Fractional Brownian Motion in Long-Term Clinical Trials

Repeated confidence interval (RCI) is an important tool for design and monitoring of group sequential trials according to which we do not need to stop the trial with planned statistical stopping rules. In this article, we derive RCIs when data from each stage of the trial are not independent thus it is no longer a Brownian motion (BM) process. Under this assumption, a larger class of stochastic processes fractional Brownian motion (FBM) is considered. Comparisons of RCI width and sample size requirement are made to those under Brownian motion for different analysis times, Type I error rates and number of interim analysis. Power family spending functions including Pocock, O'Brien-Fleming design types are considered for these simulations. Interim data from BHAT and oncology trials is used to illustrate how to derive RCIs under FBM for efficacy and futility monitoring.     

The uniform local asymptotics for a Lévy process and its overshoot and undershoot

ABSTRACT

In this article, we obtain the uniform local asymptotics for a Lévy process with a heavy-tailed Lévy measure and for the overshoot and undershoot of the Lévy process. As applications, we get the uniform asymptotics of the finite-time ruin probability and the local ruin probability for the Lévy risk model with a heavy-tailed Lévy measure. By the above results, we find that in the compound Poisson model perturbed by a Brownian motion, the effect of the Brownian component on the asymptotics of the finite-time ruin probability and the local ruin probability washes out.     

Mis-specification Analyses of Nonlinear Wiener Process-based Degradation Models

For some highly reliable products, degradation data have been studied quite extensively to evaluate their reliability characteristics. However, the accuracy of evaluation results depends strongly on the suitability of the proposed degradation model for capturing the degradation over time. If the degradation model is mis-specified, it may result in inaccurate results. In this work, we focus on the issue of model mis-specification between nonlinear Wiener process-based degradation models in which both the product-to-product variability and the temporal uncertainty of the degradation can be considered simultaneously with the nonlinearity in degradation paths. Specifically, a generalized Wiener process-based degradation model is wrongly fitted by its two limiting cases. The effects of model mis-specification in such situations on the MTTF (mean-time-to-failure) of the product are measured with the relative bias and the relative variability. Results from a numerical example concerning fatigue cracks show that the effect of mis-specification is serious under some parameter settings, i.e., the relative bias departs from 0, and the relative variability significantly departs from 1, if the generalized Wiener degradation process is wrongly assumed to be its limiting cases.     

Slowing time: Markov-modulated Brownian motions with a sticky boundary

We define a new family of stochastic processes called Markov modulated Brownian motions with a sticky boundary at zero. Intuitively, each process is a regulated Markov-modulated Brownian motion whose boundary behavior is modified to slow down at level zero.

To determine the stationary distribution of a sticky MMBM, we follow a Markov-regenerative approach similar to the one developed with great success in the context of quasi-birth-and-death processes and fluid queues. Our analysis also relies on recent work showing that Markov-modulated Brownian motions arise as limits of a parametrized family of fluid queues.