[Paper Review] On the Effect of Random Alternating Perturbations on Hazard Rates

Antonio Di Crescenzo, Barbara Martinucci|ArXiv.org|Jan 11, 2007

Probabilistic and Robust Engineering Design24 references3 citations

TL;DR

This paper proposes a stochastic model for hazard rates subject to random alternating perturbations using the telegraph process, modeling time-varying hazard rates as a perturbed version of a baseline function. The key contribution is the derivation of the distribution, mean, and variance of the resulting process, with application to survival and reliability data via confidence bands, showing the model fits real datasets when perturbation intensity is bounded.

ABSTRACT

We consider a model for systems perturbed by dichotomous noise, in which the hazard rate function of a random lifetime is subject to additive time-alternating perturbations described by the telegraph process. This leads us to define a real-valued continuous-time stochastic process of alternating type expressed in terms of the integrated telegraph process for which we obtain the probability distribution, mean and variance. An application to survival analysis and reliability data sets based on confidence bands for estimated hazard rate functions is also provided.

Motivation & Objective

• To model hazard rates subject to dichotomous noise with alternating behavior, reflecting real-world environmental variability.
• To analyze the impact of random alternating perturbations on hazard rate functions using the telegraph process as a noise model.
• To derive the probability distribution, mean, and variance of the resulting stochastic process for reliability and survival analysis.
• To develop a statistical procedure using confidence bands to assess model adequacy for real data sets.
• To demonstrate the model's applicability to constant and non-monotonic hazard rate functions in survival and reliability studies.

Proposed method

• The hazard rate function is modeled as a baseline function plus additive perturbations driven by the telegraph process, defined via a Poisson process with alternating signs.
• The integrated telegraph process is used to express the cumulative hazard, leading to a stochastic process X(t) with mixed-type distribution (discrete and absolutely continuous components).
• The moment generating function of the integrated telegraph process is derived to compute the first and second moments of X(t).
• Nonparametric kernel density estimation with the Epanechnikov kernel is used to estimate the baseline hazard rate and distribution function from data.
• Asymptotic confidence bands for estimated hazard rates are constructed to test model validity under perturbation.
• A hypothesis testing procedure is applied using the condition |r(t) − r̂(t)| ≤ [K/(nh f̂(t))]^{1/2} r̂(t) z_α − c to assess whether the perturbation level c is acceptable.

Experimental results

Research questions

• RQ1 How does the introduction of random alternating perturbations via the telegraph process affect the distribution and moments of the hazard rate function?
• RQ2 Can the proposed stochastic model with perturbed hazard rates adequately fit real survival and reliability data sets?
• RQ3 What is the statistical procedure to validate the model using confidence bands around estimated hazard rates?
• RQ4 How does the model perform when the baseline hazard rate is constant versus non-monotonic?
• RQ5 What is the maximum allowable perturbation intensity c for which the model remains statistically defensible for given data?

Key findings

• The distribution of the stochastic process X(t) has both discrete and absolutely continuous components, arising from the alternating nature of the telegraph process.
• The moment generating function of the integrated telegraph process is derived, enabling analytical computation of the mean and variance of X(t).
• For a constant baseline hazard rate r(t) = 0.0125, the model is defensible when the perturbation intensity c ≤ 0.0004, as confirmed by confidence bands.
• For a non-monotonic baseline hazard rate defined piecewise, the model is valid when c ≤ 0.00025, with the confidence band encompassing the perturbed hazard strip.
• The proposed confidence band procedure effectively rejects large values of c, providing a practical criterion for model validation in survival and reliability applications.
• The model demonstrates strong empirical fit to two real data sets: a melanoma survival study and a reliability service time study, using fixed bandwidths h=6 and h=75 respectively.