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Renewal theory

About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.


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Journal ArticleDOI
TL;DR: In this article, the authors established strong consistency (i.e., almost sure convergence) of infinitesimal perturbation analysis (IPA) estimators of derivatives of steady-state means for a broad class of systems.
Abstract: We establish strong consistency (i.e., almost sure convergence) of infinitesimal perturbation analysis (IPA) estimators of derivatives of steady-state means for a broad class of systems. Our results substantially extend previously available results on steady-state derivative estimation via IPA.Our basic assumption is that the process under study is regenerative, but our analysis uses regenerative structure in an indirect way: IPA estimators are typically biased over regenerative cycles, so straightforward differentiation of the regenerative ratio formula does not necessarily yield a valid estimator of the derivative of a steady-state mean. Instead, we use regeneration to pass from unbiasedness over fixed, finite time horizons to convergence as the time horizon grows. This provides a systematic way of extending results on unbiasedness to strong consistency.Given that the underlying process regenerates, we provide conditions under which a certain augmented process is also regenerative. The augmented process includes additional information needed to evaluate derivatives; derivatives of time averages of the original process are time averages of the augmented process. Thus, through this augmentation we are able to apply standard renewal theory results to the convergence of derivatives.

35 citations

Proceedings Article
07 Dec 2009
TL;DR: It is shown that for any modulated renewal process model, the log-likelihood is concave in the linear filter parameters only under certain restrictive conditions on the renewal density, suggesting that real-time history effects are easier to estimate than non-Poisson renewal properties.
Abstract: Recent work on the statistical modeling of neural responses has focused on modulated renewal processes in which the spike rate is a function of the stimulus and recent spiking history. Typically, these models incorporate spike-history dependencies via either: (A) a conditionally-Poisson process with rate dependent on a linear projection of the spike train history (e.g., generalized linear model); or (B) a modulated non-Poisson renewal process (e.g., inhomogeneous gamma process). Here we show that the two approaches can be combined, resulting in a conditional renewal (CR) model for neural spike trains. This model captures both real-time and rescaled-time history effects, and can be fit by maximum likelihood using a simple application of the time-rescaling theorem [1]. We show that for any modulated renewal process model, the log-likelihood is concave in the linear filter parameters only under certain restrictive conditions on the renewal density (ruling out many popular choices, e.g. gamma with shape k ≠ 1), suggesting that real-time history effects are easier to estimate than non-Poisson renewal properties. Moreover, we show that goodness-of-fit tests based on the time-rescaling theorem [1] quantify relative-time effects, but do not reliably assess accuracy in spike prediction or stimulus-response modeling. We illustrate the CR model with applications to both real and simulated neural data.

35 citations

Journal ArticleDOI
TL;DR: Four approximation techniques for obtaining confidence intervals for parameters associated with the steady-state distribution when the simulation does not contain an embedded renewal process.
Abstract: The previous papers in this series developed a methodology for obtaining from certain simulations confidence intervals for parameters associated with the steady-state distribution. This methodology required the simulations to contain an embedded renewal process at whose epochs the simulation started from scratch. The present paper contains four approximation techniques for obtaining confidence intervals when the simulation does not contain the required renewal process.

35 citations

Book ChapterDOI
01 Jan 1999
TL;DR: In this article, the authors consider the class of so-called trend-renewal processes (TRP), which have both the NHPP and the renewal process as special cases.
Abstract: A repairable system can briefly be characterized as a system which is repaired rather than replaced after a failure. The most commonly used models for the failure process of a repairable system are nonhomogeneous Poisson processes (NHPP), corresponding to minimal repairs, and renewal processes (RP), corresponding to perfect repairs. The paper reviews models for more general repair actions, often called “better-than-minimal repair” models. In particular we study the class of so called trend-renewal processes (TRP), which has both the NHPP and the RP as special cases. Parametric inference in TRP models is considered, including cases with several systems involving unobserved heterogeneity. Trend testing is discussed when the null hypothesis is that the failure process is an RP. It is shown how Monte Carlo trend tests for this case can be made from the commonly used trend tests for the null hypothesis of a homogeneous Poisson process (e.g. the Laplace test and the Military Handbook test). Simulations show that the Monte Carlo tests have favorable properties when the sample sizes are not too small.

35 citations

Journal ArticleDOI
TL;DR: In this paper, the authors introduce the minimal maximally predictive models of processes generated by hidden semi-Markov models whose causal states are either discrete, mixed, or continuous random variables and transitions are described by partial differential equations.
Abstract: We introduce the minimal maximally predictive models ( $$\epsilon \text{-machines }$$ ) of processes generated by certain hidden semi-Markov models Their causal states are either discrete, mixed, or continuous random variables and causal-state transitions are described by partial differential equations As an application, we present a complete analysis of the $$\epsilon \text{-machines }$$ of continuous-time renewal processes This leads to closed-form expressions for their entropy rate, statistical complexity, excess entropy, and differential information anatomy rates

35 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202327
202260
202173
202083
201973
201886