First hitting time of integral diffusions and applications

We study the first hitting time of integral functionals of time-homogeneous diffusions, and characterize their Laplace transforms through a stochastic time change. We obtain explicit expressions of the Laplace transforms for the geometric Brownian motion (GBM) and the mean-reverting GBM process. We also introduce a novel probability identity based on an independent exponential randomization and obtain explicit Laplace transforms of the price of arithmetic Asian options and other derivative prices that non-linearly depend on the integral diffusions. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.     

Comparing first hitting time and proportional hazards regression models

Cox's widely used semi-parametric proportional hazards (PH) regression model places restrictions on the possible shapes of the hazard function. Models based on the first hitting time (FHT) of a stochastic process are among the alternatives and have the attractive feature of being based on a model of the underlying process. We review and compare the PH model and an FHT model based on a Wiener process which leads to an inverse Gaussian (IG) regression model. This particular model can also represent a “cured fraction” or long-term survivors. A case study of survival after coronary artery bypass grafting is used to examine the interpretation of the IG model, especially in relation to covariates that affect both of its parameters.     

First hitting probabilities for semi-Markov chains and estimation

We first consider a stochastic system described by an absorbing semi-Markov chain (SMC) with finite state space, and we introduce the absorption probability to a class of recurrent states. Afterwards, we study the first hitting probability to a subset of states for an irreducible SMC. In the latter case, a non-parametric estimator for the first hitting probability is proposed and the asymptotic properties of strong consistency and asymptotic normality are proven. Finally, a numerical application on a five-state system is presented to illustrate the performance of this estimator.     

On Transition and First Hitting Time Densities and Moments of the Ornstein–Uhlenbeck Process

This paper derives transition and first hitting time densities and moments for the Ornstein–Uhlenbeck Process (OUP) between exponential thresholds. The densities are obtained by simplifying the process via Doob’s representation into Brownian motion between affine thresholds. The densities in this paper also offer easy-to-use and fast small-time approximations for the densities of OUP between constant thresholds given that exponential thresholds are virtually constant for a small time. This is of interest for estimation with high-frequency data given that extant approaches for constant thresholds impose a large demand on computing power. The moments of the transition distribution up to order n are derived within a closed-form recursive formula that offers valuable information for management. Expressions for the moments of the first hitting time distribution are also obtained in closed form by simplifying integrals via series expansions.     

Modeling of semi-competing risks by means of first passage times of a stochastic process

In semi-competing risks one considers a terminal event, such as death of a person, and a non-terminal event, such as disease recurrence. We present a model where the time to the terminal event is the first passage time to a fixed level c in a stochastic process, while the time to the non-terminal event is represented by the first passage time of the same process to a stochastic threshold S, assumed to be independent of the stochastic process. In order to be explicit, we let the stochastic process be a gamma process, but other processes with independent increments may alternatively be used. For semi-competing risks this appears to be a new modeling approach, being an alternative to traditional approaches based on illness-death models and copula models. In this paper we consider a fully parametric approach. The likelihood function is derived and statistical inference in the model is illustrated on both simulated and real data.     

Failure Time of Non Homogeneous Gamma Process

We consider the non homogeneous gamma process as a degradation model. The hitting time of a deterministic or random level is studied here. We provide its distribution (both cdf and pdf) explicitly in the first case and in the second case when the threshold is exponentially or gamma distributed. The general case for a random threshold can be approximated by considering mixtures of Erlang distributions. Aging properties are also discussed in this article.     

Parameter identification for a stochastic logistic growth model with extinction

We consider a stochastic logistic growth model given by a stochastic differential equation, for which extinction can occur. We first propose appropriate adaptation of some standard inference methods when the process is observed at discrete time. Second, we show that the individual birth and death can be identified separately to some extent.     

Estimating fault hitting rates by recapture sampling

For the recapture debugging design introduced by Nayak (1988) we consider the problem of estimating the hitting rates of the faults remaining in a system. In the context of a conditional likelihood, moment estimators are derived and are shown to be asymptotically normal and fully efficient. Fixed sample properties of the moment estimators are compared, through simulation, with those of the conditional maximum likelihood estimators. Also considered is a procedure for testing the assumption that faults have identical hitting rates; this provides a test of fit of the Jelinski-Moranda (1972) model. It is assumed that the residual hitting rates follow a log linear rate model and that the testing process is truncated when the gaps between the detection of new errors exceed a fixed amount of time.     

Extremal limit theorems for observations separated by random power law waiting times

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.     

First-passage densities of controlled Gaussian processes

Whittle has proved a theorem that gives the optimal control of Gaussian processes in terms of the mathematical expectation of a function of the time and the place where the uncontrolled processes hit the boundary of the stopping region for the first time. In this paper we obtain formulae for the joint probability density function of the first hitting time and place and, in the time-invariant case, for the moment generating function of the first exit time of the optimally controlled processes. Two particular one-dimensional cases are considered.     

Dynamic path analysis—a new approach to analyzing time-dependent covariates

In this article we introduce a general approach to dynamic path analysis. This is an extension of classical path analysis to the situation where variables may be time-dependent and where the outcome of main interest is a stochastic process. In particular we will focus on the survival and event history analysis setting where the main outcome is a counting process. Our approach will be especially fruitful for analyzing event history data with internal time-dependent covariates, where an ordinary regression analysis may fail. The approach enables us to describe how the effect of a fixed covariate partly is working directly and partly indirectly through internal time-dependent covariates. For the sequence of times of event, we define a sequence of path analysis models. At each time of an event, ordinary linear regression is used to estimate the relation between the covariates, while the additive hazard model is used for the regression of the counting process on the covariates. The methodology is illustrated using data from a randomized trial on survival for patients with liver cirrhosis.     

Reliability Evaluation of a Load-sharing Parallel System with Failure Dependence

In a parallel structure load-sharing system, the failure rate of the operating components will usually increase, due to the additional loading induced by the other components' failure. Hence failure dependency exists among components. To quantify the failure dependency, a dependence function is introduced. Under the assumptions that the repair time distributions of components are arbitrary and life times are exponential distributions whose failure rates vary with the number of operating components, a new load-sharing parallel system with failure dependency is proposed. To model the stochastic behavior of the system, the Semi-Markov process induced by it is given. The Semi-Markov kernel associated with the process is also presented. The availability and the time to the first system failure are obtained by employing Markov renewal theory. A numerical example is presented to illustrate the results obtained in the paper. The impact of the failure dependence on the system is also considered.     

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.     

Approximating the distribution of a renewal process using generalized poisson distributions

The exact distribution of a renewal counting process is not easy to compute and is rarely of closed form. In this article, we approximate the distribution of a renewal process using families of generalized Poisson distributions. We first compute approximations to the first several moments of the renewal process. In some cases, a closed form approximation is obtained. It is found that each family considered has its own strengths and weaknesses. Some new families of generalized Poisson distributions are recommended. Theorems are obtained determining when these variance to mean ratios are less than (or exceed) one without having to find the mean and variance. Some numerical comparisons are also made.     

Gaussian models for degradation processes-part I: Methods for the analysis of biomarker data

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.     

Efficient volatility estimation in a two-factor model

We statistically analyze a multivariate Heath-Jarrow-Morton diffusion model with stochastic volatility. The volatility process of the first factor is left totally unspecified while the volatility of the second factor is the product of an unknown process and an exponential function of time to maturity. This exponential term includes some real parameter measuring the rate of increase of the second factor as time goes to maturity. From historical data, we efficiently estimate the time to maturity parameter in the sense of constructing an estimator that achieves an optimal information bound in a semiparametric setting. We also nonparametrically identify the paths of the volatility processes and achieve minimax bounds. We address the problem of degeneracy that occurs when the dimension of the process is greater than two, and give in particular optimal limit theorems under suitable regularity assumptions on the drift process. We consistently analyze the numerical behavior of our estimators on simulated and real datasets of prices of forward contracts on electricity markets.     

On Comparison of Stopping Times in Sequential Procedures for Exponential Families of Stochastic Processes

For curved ( k + 1), k -exponential families of stochastic processes a natural and often studied sequential procedure is to stop observation when a linear combination of the coordinates of the canonical process crosses a prescribed level. For such procedures the model is, approximately or exactly, a non-curved exponential family. Subfamilies of these stopping rules defined by having the same Fisher (expected) information are considered. Within a subfamily the Bartlett correction for a point hypothesis is also constant. Methods for comparing the durations of the sampling periods for the stopping rules in such a subfamily are discussed. It turns out that some stopping times tend to be smaller than others. For exponential families of diffusions and of counting processes the probability that one such stopping time is smaller than another can be given explicity. More generally, an Edgeworth expansion of this probability is given     

CURRENT STATUS AND RIGHT-CENSORED DATA STRUCTURES WHEN OBSERVING A MARKER AT THE CENSORING TIME

We study nonparametric estimation with two types of data structures. In the first data structure n i.i.d. copies of (C, N(C)) are observed, where N is a finite state counting process jumping at time-variables of interest and C a random monitoring time. In the second data structure n i.i.d. copies of (C ∧ T, I (T ≤ C), N(C ∧ T)) are observed, where N is a counting process with a final jump at time T (e.g., death). This data structure includes observing right-censored data on T and a marker variable at the censoring time.In these data structures, easy to compute estimators, namely (weighted)-pool-adjacent-violator estimators for the marginal distributions of the unobservable time variables, and the Kaplan-Meier estimator for the time T till the final observable event, are available. These estimators ignore seemingly important information in the data. In this paper we prove that, at many continuous data generating distributions the ad hoc estimators yield asymptotically efficient estimators of [Formula: see text]-estimable parameters.     

BACKWARD ESTIMATION OF STOCHASTIC PROCESSES WITH FAILURE EVENTS AS TIME ORIGINS

Stochastic processes often exhibit sudden systematic changes in pattern a short time before certain failure events. Examples include increase in medical costs before death and decrease in CD4 counts before AIDS diagnosis. To study such terminal behavior of stochastic processes, a natural and direct way is to align the processes using failure events as time origins. This paper studies backward stochastic processes counting time backward from failure events, and proposes one-sample nonparametric estimation of the mean of backward processes when follow-up is subject to left truncation and right censoring. We will discuss benefits of including prevalent cohort data to enlarge the identifiable region and large sample properties of the proposed estimator with related extensions. A SEER-Medicare linked data set is used to illustrate the proposed methodologies.     

Martingales without tears

This pedagogical paper presents a casual introduction to martingales, or fair gambling processes. Our objective is to describe the concept of a martingale and its application to common statistical tests used in the analysis of survival data, but without the mathematical rigor required for formal proofs.We use heuristic arguements to demonstrate that the logrank statistic evaluated over followup time is a fair gambling process, and introduce some mathematical notation and terminology along the way. We then employ the counting process approach to show that the logrank statistic computed over followup time can be expressed as the difference of two martingale transforms, and thus is a martingale. These ideas are first introduced in the context of a discrete time process, and are then generalized to a continuous time process. With slight modifications, the same idea extends from the logrank to other weighted Mantel-Haenszel statistics computed over time.