General Counting Process Research Articles

On increasing-failure-rate random variables

We provide sufficient conditions for the following types of random variable to have the increasing-failure-rate (IFR) property: sums of a random number of random variables; the time at which a Markov chain crosses a random threshold; the time until a random number of events have occurred in an inhomogeneous Poisson process; and the number of events of a renewal process, and of a general counting process, that have occurred by a randomly distributed time.

UNIFORM ESTIMATES FOR THE TAIL PROBABILITY OF MAXIMA OVER FINITE HORIZONS WITH SUBEXPONENTIAL TAILS

Let F be the common distribution function of the increments of a random walk {Sn, n ≥ 0} with S0 = 0 and a negative drift and let {N(t), t ≥ 0} be a general counting process, independent of {Sn, n ≥ 0}. This article investigates the tail probability, denoted by ψ(x; t), of the maximum of SN(v) over a finite horizon 0 ≤ v ≤ t. When F is strongly subexponential, some asymptotics for ψ(x; t) are derived as x → ∞. The merit is that all of the obtained asymptotics are uniform for t in a finite or infinite time interval.

A study of different ageing classes via total time on test transform and Lorenz curves

A device subject to shocks occurring randomly according to a general counting process and failing when the cumulative damage exceeds a fixed threshold is considered. Sufficient conditions for the lifetime of this device belonging to the HNBUE class are given and as a consequence a procedure to obtain an HNBUE sample is performed. A computational program is implemented to obtain an HNBUE sample. Moreover, another computational program has been implemented to check whether these values belong to a smaller class than HNBUE by means of known graphic procedures, namely Lorenz and TTT curves. © 1998 John Wiley & Sons, Ltd.

A study of different ageing classes via total time on test transform and Lorenz curves

A device subject to shocks occurring randomly according to a general counting process and failing when the cumulative damage exceeds a fixed threshold is considered. Sufficient conditions for the lifetime of this device belonging to the HNBUE class are given and as a consequence a procedure to obtain an HNBUE sample is performed. A computational program is implemented to obtain an HNBUE sample. Moreover, another computational program has been implemented to check whether these values belong to a smaller class than HNBUE by means of known graphic procedures, namely Lorenz and TTT curves. © 1998 John Wiley & Sons, Ltd.

Smoothed Cox regression

Nonparametric regression was shown by Beran and McKeague and Utikal to provide a flexible method for analysis of censored failure times and more general counting processes models in the presence of covariates. We discuss application of kernel smoothing towards estimation in a generalized Cox regression model with baseline intensity dependent on a covariate. Under regularity conditions we show that estimates of the regression parameters are asymptotically normal at rate root-n, and we also discuss estimation of the baseline cumulative hazard function and related parameters.

Hnbue property in a shock model with cumulative damage threshold

A device subjected to shocks occurring randomly according to a general counting process and that fails when the cu~nulative damage exceeds a fixed threshold is considered. Sufficient conditions for the lifetime of an HNBTJE class device are given and it is shown that the HNBUE property for the shock model depends only on the arrival process. A means of obtaining distribution functions of the HNBUE class in parametric form is given and the characteristic function of the accumulated damage process is obtained.

Mean residual life and increasing convex comparison of shock models

Two devices are subjected to shocks arriving according to a general counting process. Let M 1 and M 2 be the random number of shocks that cause the failure of the first and the second device, respectively. We find conditions on the counting process such that the mean residual life ordering, the increasing convex ordering and the expectation ordering between M 1 and M 2 are preserved in the random lifetimes of the two devices.

Hattendorff's theorem and Thiele's differential equation generalized

Abstract Hattendorff's classical result on zero means an uncorrelatedness of the losses created in disjoint time intervals by a life insurance policy is an immediate consequence of the very definition of the concept of loss. Thus, the result is formulated and proved here in a setting with quite general payments, discount function, and time intervals, all stochastic. A general representation is given for the variances of the losses. They are easy to compute when sufficient structure is added to the model. The traditional continuous time Markov chain model is given special consideration. A stochastic Thiele's differential equation is obtained in a fairly general counting process framework.

Inference for a Nonlinear Counting Process Regression Model

Martingale and counting process techniques are applied to the problem of inference for general conditional hazard functions. This problem was first studied by Beran, who introduced a class of estimators for the conditional cumulative hazard and survival functions in the special case of time-independent covariates. Here the covariate can be time dependent; the classical i.i.d. assumptions are relaxed by replacing them with certain asymptotic stability assumptions, and models involving recurrent failures are included. This is done within the framework of a general nonparametric counting process regression model. Important examples of the model include right-censored survival data, semi-Markov processes, an illness-death process with duration dependence, and age-dependent birth and death processes.