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This paper establishes a general framework for measuring statistical divergence. Namely, with regard to a pair of random variables that share a common range of values: quantifying the distance of the statistical distribution of one random variable from that of the other. The general framework is then applied to the topics of socioeconomic inequality and renewal processes. The general framework and its applications are shown to yield and to relate to the following: f-divergence, Hellinger divergence, Renyi divergence, and Kullback–Leibler divergence (also known as relative entropy); the Lorenz curve and socioeconomic inequality indices; the Gini index and its generalizations; the divergence of renewal processes from the Poisson process; and the divergence of anomalous relaxation from regular relaxation. Presenting a 'fresh' perspective on statistical divergence, this paper offers its readers a simple and transparent construction of statistical-divergence gauges, as well as novel paths that lead from statistical divergence to the aforementioned topics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of latency aging on exchangeable (invariant under permutation of indices) inter-arrival times arising from mixed renewal point processes (statistical mixtures of point processes with renewal inter-arrival times) and explore the implications for reliability and survival analysis. We prove that aging preserves the exchangeability of inter-arrival times. Our data analysis, which includes both surrogate data and a Bayesian approach to high-frequency currency exchange-rate data, shows how aging impacts key survival analysis metrics such as failure survival, renewal, and hazard rate functions. [ABSTRACT FROM AUTHOR]

Perhaps the first nonparametric, asymptotically optimal prediction intervals are provided for univariate random walks, with applications to renewal processes. Perhaps the first nonparametric prediction regions are introduced for vector-valued random walks. This paper further derives nonparametric data-splitting prediction regions, which are underpinned by very simple theory. Some of the prediction regions can be used when the data distribution does not have first moments, and some can be used for high-dimensional data, where the number of predictors is larger than the sample size. The prediction regions can make use of many estimators of multivariate location and dispersion. [ABSTRACT FROM AUTHOR]

Perhaps the first nonparametric, asymptotically optimal prediction intervals are provided for univariate random walks, with applications to renewal processes. Perhaps the first nonparametric prediction regions are introduced for vector-valued random walks. This paper further derives nonparametric data-splitting prediction regions, which are underpinned by very simple theory. Some of the prediction regions can be used when the data distribution does not have first moments, and some can be used for high-dimensional data, where the number of predictors is larger than the sample size. The prediction regions can make use of many estimators of multivariate location and dispersion.

We consider the simple random walk (or Pólya walk) on the one-dimensional lattice subject to stochastic resetting to the origin with probability r at each time step. The focus is on the joint statistics of the numbers N t × of spontaneous returns of the walker to the origin and N t ∙ of resetting events up to some observation time t. These numbers are extensive in time in a strong sense: all their joint cumulants grow linearly in t, with explicitly computable amplitudes, and their fluctuations are described by a smooth bivariate large deviation function. A non-trivial crossover phenomenon takes place in the regime of weak resetting and late times. Remarkably, the time intervals between spontaneous returns to the origin of the reset random walk form a renewal process described in terms of a single 'dressed' probability distribution. These time intervals are probabilistic copies of the first one, the 'dressed' first-passage time. The present work follows a broader study, covered in a companion paper, on general nested renewal processes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of latency aging on exchangeable (invariant under permutation of indices) inter-arrival times arising from mixed renewal point processes (statistical mixtures of point processes with renewal inter-arrival times) and explore the implications for reliability and survival analysis. We prove that aging preserves the exchangeability of inter-arrival times. Our data analysis, which includes both surrogate data and a Bayesian approach to high-frequency currency exchange-rate data, shows how aging impacts key survival analysis metrics such as failure survival, renewal, and hazard rate functions.

This note is a complement to the paper (Stoch. Process. Appl. 144:72–84, 2022) by Eberlein, Kabanov and Schmidt on the asymptotics of the ruin probability in a Sparre Andersen non-life insurance model with investments into a risky asset whose price follows a geometric Lévy process. Using techniques from the theory of semi-Markov processes, we extend the result of (Eberlein, Kabanov and Schmidt in Stoch. Process. Appl. 144:72–84, 2022) to the case of annuities and models with two-sided jumps. [ABSTRACT FROM AUTHOR]

For a stochastic process reset at random times, we discuss to what extent the probabilities of some orderings of observables associated with the intervals of time between resetting events are universal, i.e. independent of the choice of the observables, and in particular, to what extent universality depends on the choice of the distribution of these intervals. For Poissonian resetting, universality relies only on a combinatorial argument and on the statistical properties of Poisson points. For a generic distribution of time intervals between resets, universality no longer holds in general. [ABSTRACT FROM AUTHOR]

A large deviation principle (LDP) for a class of additive functionals of semi-Markov processes and their associated Markov renewal processes is studied via an almost sure functional central limit theorem. The rate function corresponding to the deviations from the paths of the corresponding empirical processes with logarithmic averaging is determined as a relative entropy with respect to the Wiener measure on D [ 0 , ∞). A martingale decomposition for additive functionals of Markov renewal processes is employed. [ABSTRACT FROM AUTHOR]

We establish a sharp large deviation principle for renewal–reward processes, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. In fact, we demonstrate a weak large deviation principle without assuming any exponential moment condition on the law of waiting times and rewards by resorting to a sharp version of Cramér's theory. We also exhibit sufficient conditions for exponential tightness of renewal–reward processes, which leads to a full large deviation principle. [ABSTRACT FROM AUTHOR]

We develop new models for imperfect repair and the corresponding generalized renewal processes for stochastic description of repairable items that fail when their degradation reaches the specified deterministic or random threshold. The discussion is based on the recently suggested notion of a random virtual age and is applied to monotone processes of degradation with independent increments. Imperfect repair reduces degradation of an item on failure to some intermediate level. However, for the nonhomogeneous processes, the corresponding age reduction, which sets back the 'clock' of the process, is also performed. Some relevant stochastic comparisons are obtained. It is shown that the cycles of the corresponding generalized imperfect renewal process are stochastically decreasing/increasing depending on the monotonicity properties of the failure rate that describes the random failure threshold of an item. [ABSTRACT FROM AUTHOR]

In this article, the observations of literary scholar K.Kahramanov on his assessment of the life and work of modern literary figures who were victims of repression in Uzbek literary studies and criticism of the independence period as a methodological update. [ABSTRACT FROM AUTHOR]

Screening for chronic diseases, such as cancer, is an important public health priority, but traditionally only the frequency or rate of screening has received attention. In this work, we study the importance of adhering to recommended screening policies and develop new methodology to better optimize screening policies when adherence is imperfect. We consider a progressive disease model with four states (healthy, undetectable preclinical, detectable preclinical, clinical), and overlay this with a stochastic screening–behavior model using the theory of renewal processes that allows us to capture imperfect adherence to screening programs in a transparent way. We show that decreased adherence leads to reduced efficacy of screening programs, quantified here using elements of the lead time distribution (i.e., the time between screening diagnosis and when diagnosis would have occurred clinically in the absence of screening). Under the assumption of an inverse relationship between prescribed screening frequency and individual adherence, we show that the optimal screening frequency generally decreases with increasing levels of non-adherence. We apply this model to an example in breast cancer screening, demonstrating how accounting for imperfect adherence affects the recommended screening frequency. [ABSTRACT FROM AUTHOR]

This article is a study of vector-valued renewal-reward processes on R d . The jumps of the process are assumed to be independent and identically distributed nonnegative random vectors with mutually dependent components, each of which may be either discrete or continuous (or a mixture of discrete and continuous components). Each component of the process has a fixed threshold. Operational calculus techniques and symmetries with respect to permutations are used to find a general result for the probability of an arbitrary weak ordering of threshold crossings. The analytic and numerical tractability of the result are demonstrated by an application to the reliability of stochastic networks and some other special cases. Results are shown to agree with empirical probabilities generated through simulation of the process. [ABSTRACT FROM AUTHOR]

We study a Sparre Andersen model in which the business activity of the company is described by a compound renewal process with drift assuming that the capital reserves are invested in a risky asset. The price of the latter is assumed to evolve according to a geometric Lévy process. We prove that the asymptotic behavior of the ruin probability depends to a large extent only on the properties of the price process. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and justify a new, basic approach to modelling the imperfect repair and the corresponding imperfect repair processes for items with observable degradation. We consider monotone processes of degradation with independent increments. Imperfect repair reduces degradation of an item on failure to some intermediate level. To define the state of an item after this imperfect repair, the random virtual age is introduced. Some stochastic properties describing the corresponding remaining lifetime are considered. The generalized renewal process based on the suggested notion of imperfect repair is described, and some of its properties are studied. An alternative approach that considers the imperfect repair process defined in the degradation scale is outlined. [ABSTRACT FROM AUTHOR]

The second order statistical properties of time point processes (PP) are described by the time coincidence function (CF) and the frequency Bartlett spectrum (BS). For PPs recorded by pulses appearing at random time instants, as in photodetection experiments, the CF can be measured by various physical devices showing in particular the famous bunching effect of photons. On the other hand, for PPs recorded by the intervals between successive points (lifetimes), especially for renewal processes, there is no usual procedure for the estimation of the CF, and the aim of this paper is to describe an approach of this problem. The starting point is a mathematical relation between the CF and the set of probabilities density functions of the lifetimes of any order of the PP. As a consequence, the CF can be obtained by processing the results of the multiple normalized histograms of these lifetimes. In the cases, relatively rare, where the mathematical expression of the CF is known in closed form, the correct behavior of the procedure is verified by an experimental analysis of simulated data. The method is extended in order to verify the relationship between the CF and the BS. [ABSTRACT FROM AUTHOR]

We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning. [ABSTRACT FROM AUTHOR]

Fifty years ago P. Erdős and A. Rényi published their famous paper on the new law of large numbers. In this survey, we describe numerous results and achievements which are related with this paper or motivated by it during these years. [ABSTRACT FROM AUTHOR]

This article is a study of vector-valued renewal-reward processes on Rd. The jumps of the process are assumed to be independent and identically distributed nonnegative random vectors with mutually dependent components, each of which may be either discrete or continuous (or a mixture of discrete and continuous components). Each component of the process has a fixed threshold. Operational calculus techniques and symmetries with respect to permutations are used to find a general result for the probability of an arbitrary weak ordering of threshold crossings. The analytic and numerical tractability of the result are demonstrated by an application to the reliability of stochastic networks and some other special cases. Results are shown to agree with empirical probabilities generated through simulation of the process.

Abstract This paper investigates a nonstandard renewal counting process with dependent inter-arrival times-web renewal process. Several limit properties, including the tail of the exponential moment which is a crucial condition in many situations, are obtained. Then the results are applied in insurance to derive precise large deviations and moderate deviation formulas for the aggregate amount of claims.

We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the consideration of particular models and highlight the essentials. Besides giving a unified presentation of the theory, this work investigates facets so far unexplored in previous studies. Specifically, we show how the study of the class of random allocation models and zero-range processes can serve as a backdrop for the study of the two other classes of processes central to the present work—tied-down and free renewal processes. We then present new insights on the extreme value statistics of these three classes of processes which allow a deeper understanding of the mechanism of condensation and the quantitative analysis of the fluctuations of the condensate. [ABSTRACT FROM AUTHOR]

We discuss two methods for measuring the effectiveness of earthquake prediction algorithms: The information score based on the likelihood ratio and error diagrams. For both of these methods, closed form expressions are obtained for the renewal process based on the gamma and lognormal distributions. The error diagram is more informative than the likelihood ratio and uniquely specifies the information score. We derive an expression connecting the information score and error diagrams. We then obtain the estimate of the region bounds in the error diagram for any value of the information score. We discuss how these preliminary results can be extended for more realistic models of earthquake occurrence.

Simple, controllable models play an important role in learning how to manipulate and control quantum resources. We focus here on quantum non-Markovianity and model the evolution of open quantum systems by quantum renewal processes. This class of quantum dynamics provides us with a phenomenological approach to characterise dynamics with a variety of non-Markovian behaviours, here described in terms of the trace distance between two reduced states. By adopting a trajectory picture for the open quantum system evolution, we analyse how non-Markovianity is influenced by the constituents defining the quantum renewal process, namely the time-continuous part of the dynamics, the type of jumps and the waiting time distributions. We focus not only on the mere value of the non-Markovianity measure, but also on how different features of the trace distance evolution are altered, including times and number of revivals.

The probability of successfully spending twice the same bitcoins is considered. A double-spending attack consists in issuing two transactions transferring the same bitcoins. The first transaction, from the fraudster to a merchant, is included in a block of the public chain. The second transaction, from the fraudster to himself, is recorded in a block that integrates a private chain, exact copy of the public chain up to substituting the fraudster-to-merchant transaction by the fraudster-to-fraudster transaction. The double-spending hack is completed once the private chain reaches the length of the public chain, in which case it replaces it. The growth of both chains are modelled by two independent counting processes. The probability distribution of the time at which the malicious chain catches up with the honest chain, or, equivalently, the time at which the two counting processes meet each other, is studied. The merchant is supposed to await the discovery of a given number of blocks after the one containing the transaction before delivering the goods. This grants a head start to the honest chain in the race against the dishonest chain. [ABSTRACT FROM AUTHOR]

Abstract Deterioration modelling has been a bottlenecking step towards risk-informed asset management of municipal water distribution networks. To close the gap, we proposed a two-time-scale (TTS) point process model on a pipe level for modelling and prediction of water main breaks. This paper presents the characterization, statistical parameter estimation, probabilistic features, and application of the model. Combining Poisson and renewal models into one, the proposed TTS process is characterized by a conditional intensity function of two time variables—one in a pipe clock for overall pipe aging and the other in a repair clock for local renewal. As a result, different aging patterns including the complicated bathtub-type behaviour can be modelled. A novel statistical method that combines data augmentation and Markov Chain Monte Carlo was developed for model estimation to deal with partially missing event histories. A case study using real-life data collected from a regional municipality in Canada was presented to illustrate the application of the proposed model. The modelling process ranging from model estimation, verification, validation, and updating to application in asset management was thoroughly demonstrated. This study also demonstrated that one must use the full distributions of the parameters to obtain an unbiased prediction of mean number of water main breaks. The proposed model was also compared with the Poisson process model in terms of break intensity, survival probability, mean cumulative number of breaks, and mean annual number of breaks. The implications of the different results to asset management were carefully discussed as well. Last, the ability of the proposed model to capture the maintenance effectiveness of pipe repair was proven. This work represents a solid advancement towards holistic assessment of the aging risk of a municipal water distribution network. Highlights • A two-time-scale point process model was proposed for modelling of water main breaks. • The model captures various aging patterns including bathtub behaviour. • A MCMC/DC algorithm considering partial missing break histories was proposed for parameter estimation. • A real life case study was presented to demonstrate the whole modelling process. [ABSTRACT FROM AUTHOR]

Abstract Complement activation is associated with regional inflammation during acute gastrointestinal injury (AGI). This study is designed to explore how intracellular C3 activation in Paneth cells (PCs) affects regeneration of intestinal epithelium during AGI. AGI was induced in wildtype C57BL/6 mice, with sham operation employed as control. Exogenous C3 (1 mg, I.P.) was applied at 6 h post-surgery. Intestinal crypts harvested from ileum were cultured with presence or absence of C3 (20 μg/ml), with small interfering RNA against BST1 and complement activation inhibitor selectively applied in vitro. The intestinal integrity, percentage of PCs and intestinal stem cells (ISCs) were evaluated. Importantly, cADPR, C3 fragments, and S6-related proteins were detected in PCs to inspect the mammalian target of rapamycin complex 1 (mTORC1) signaling. AGI caused breakdown of intestinal mucosa integrity and regional inflammation. Exogenous C3 by itself failed to promote the growth of intestinal epithelium, but distinctly enhanced the activity of PCs via intracellular activation, which subsequently supported the expansion of ISCs inside of intestinal crypts. Inhibition of C3 activation was associated with decreased expressions of S6, S6K1 and cADPR, with blocking BST1 found to depress cADPR only. Collectively, these data confirmed intracellular activation of C3 in PCs enhanced expansion of ISCs in response to acute injury. The mTORC1 signaling pathway in PCs contributed to this crosstalk during exogenous C3 treatment. Highlights • Functional PCs by intracellular C3 activation are attributed to ISCs expansion and intestinal regeneration during AGI. • Intracellular C3 activation regulates mTORC1 signaling in PCs to promote expansion of ISCs in response to such injury. • Exogenous C3 therapy for AGI would not cause uncontrolled inflammation, but rather enhance intestinal epithelial repair. [ABSTRACT FROM AUTHOR]

Variability of spiking activity is ubiquitous throughout the brain but little is known about its contextual dependance. Trial-to-trial spike count variability, estimated by the Fano Factor (FF), and within-trial spike time irregularity, quantified by the coefficient of variation (CV), reflect variability on long and short time scales, respectively. We co-analyzed FF and the local coefficient of variation (CV2) in monkey motor cortex comparing two behavioral contexts, movement preparation (wait) and execution (movement). We find that the FF significantly decreases from wait to movement, while the CV2 increases. The more regular firing (expressed by a low CV2) during wait is related to an increased power of local field potential (LFP) beta oscillations and phase locking of spikes to these oscillations. In renewal processes, a widely used model for spiking activity under stationary input conditions, both measures are related as FF ≈ CV2. This expectation was met during movement, but not during wait where FF ≫ CV22. Our interpretation is that during movement preparation, ongoing brain processes result in changing network states and thus in high trial-to-trial variability (expressed by a high FF). During movement execution, the network is recruited for performing the stereotyped motor task, resulting in reliable single neuron output. Our interpretation is in the light of recent computational models that generate non-stationary network conditions.

For a general renewal process N (allowing delay, defect and multiple simultaneous arrivals) the independence of the first renewal epochs of the marked processes got from N by Bernoulli 0/1 thinning is characterized. This independence is well-known to hold true in the case of homogeneous Poisson processes; by way of corollary one obtains the interesting observation that, when coupled with some minimal extra conditions, it in fact already identifies them. The proof is analytic in character. [ABSTRACT FROM AUTHOR]

Variability of spiking activity is ubiquitous throughout the brain but little is known about its contextual dependance. Trial-to-trial spike count variability, estimated by the Fano Factor (FF), and within-trial spike time irregularity, quantified by the coefficient of variation (CV), reflect variability on long and short time scales, respectively. We co-analyzed FF and the local coefficient of variation (CV2) in monkey motor cortex comparing two behavioral contexts, movement preparation (wait) and execution (movement). We find that the FF significantly decreases from wait to movement, while the CV2 increases. The more regular firing (expressed by a low CV2) during wait is related to an increased power of local field potential (LFP) beta oscillations and phase locking of spikes to these oscillations. In renewal processes, a widely used model for spiking activity under stationary input conditions, both measures are related as FF ≈ CV2. This expectation was met during movement, but not during wait where FF ≫ CV22. Our interpretation is that during movement preparation, ongoing brain processes result in changing network states and thus in high trial-to-trial variability (expressed by a high FF). During movement execution, the network is recruited for performing the stereotyped motor task, resulting in reliable single neuron output. Our interpretation is in the light of recent computational models that generate non-stationary network conditions. [ABSTRACT FROM AUTHOR]

Summary: Assessment of the regularity of a sequence of events over time is important for clinical decision‐making as well as informing public health policy. Our motivating example involves determining the effect of an intervention on the regularity of HIV self‐testing behavior among high‐risk individuals when exact self‐testing times are not recorded. Assuming that these unobserved testing times follow a renewal process, the goals of this work are to develop suitable methods for estimating its distributional parameters when only the presence or absence of at least one event per subject in each of several observation windows is recorded. We propose two approaches to estimation and inference: a likelihood‐based discrete survival model using only time to first event; and a potentially more efficient quasi‐likelihood approach based on the forward recurrence time distribution using all available data. Regularity is quantified and estimated by the coefficient of variation (CV) of the interevent time distribution. Focusing on the gamma renewal process, where the shape parameter of the corresponding interevent time distribution has a monotone relationship with its CV, we conduct simulation studies to evaluate the performance of the proposed methods. We then apply them to our motivating example, concluding that the use of text message reminders significantly improves the regularity of self‐testing, but not its frequency. A discussion on interesting directions for further research is provided. [ABSTRACT FROM AUTHOR]

Cell switch-off (CSO) is an important approach to reducing energy consumption in cellular networks during off-peak periods. CSO addresses the research question of which cells to switch off when. Whereas online CSO, based on immediate user demands and channel states, is problematic to implement and difficult to model, off-line CSO is more practical and tractable. Furthermore, it is known that regular cell layouts generally provide the best coverage and spectral efficiency, which leads us to prefer regular static (off-line) CSO. We introduce sector-based regular CSO patterns for the first time. We organize the existing and newly introduced patterns using a systematic nomenclature; studying 26 patterns in total. We compare these patterns in terms of energy efficiency and the average number of users supported, via a combination of analysis and simulation. We also compare the performance of CSO with two benchmark algorithms. We show that the average number of users can be captured by one parameter. Moreover, we find that the distribution of the number of users is close to Gaussian, with a tractable variance. Our results demonstrate that several patterns that activate only one out of three sectors are particularly beneficial; such CSO patterns have not been studied before. [ABSTRACT FROM PUBLISHER]

Abstract: We consider non‐parametric estimation for interarrival times density of a renewal process. For continuous time observation, a projection estimator in the orthonormal Laguerre basis is built. Nonstandard decompositions lead to bounds on the mean integrated squared error (MISE), from which rates of convergence on Sobolev–Laguerre spaces are deduced, when the length of the observation interval gets large. The more realistic setting of discrete time observation is more difficult to handle. A first strategy consists in neglecting the discretization error. A more precise strategy aims at taking into account the convolution structure of the data. Under a simplifying ‘dead‐zone’ condition, the corresponding MISE is given for any sampling step. In the three cases, an automatic model selection procedure is described and gives the best MISE, up to a logarithmic term. The results are illustrated through a simulation study. [ABSTRACT FROM AUTHOR]

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and 1 / f noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes.

This paper studies energy efficiency (EE) and average throughput maximization for cognitive radio systems in the presence of unslotted primary users. It is assumed that primary user activity follows an ON–OFF alternating renewal process. Secondary users first sense the channel possibly with errors in the form of miss detections and false alarms, and then, start the data transmission only if no primary user activity is detected. The secondary user transmission is subject to constraints on collision duration ratio, which is defined as the ratio of average collision duration to transmission duration. In this setting, the optimal power control policy which maximizes the EE of the secondary users or maximizes the average throughput while satisfying a minimum required EE under average/peak transmit power and average interference power constraints is derived. Subsequently, low-complexity algorithms for jointly determining the optimal power level and frame duration are proposed. The impact of probabilities of detection and false alarm, transmit and interference power constraints on the EE, average throughput of the secondary users, optimal transmission power, and the collisions with primary user transmissions are evaluated. In addition, some important properties of the collision duration ratio are investigated. The tradeoff between the EE and average throughput under imperfect sensing decisions and different primary user traffic are further analyzed. [ABSTRACT FROM PUBLISHER]

A multiple filter test for the analysis and detection of rate change points in point processes on the line has been proposed recently. The underlying statistical test investigates the null hypothesis of constant rate. For that purpose, multiple filtered derivative processes are observed simultaneously. Under the null hypothesis, each process G asymptotically takes the form while L is a zero-mean Gaussian process with unit variance. This result is used to derive a rejection threshold for statistical hypothesis testing. The purpose of this paper is to describe the behavior of G under the alternative hypothesis of rate changes and potential simultaneous variance changes. We derive the approximation with deterministic functions $$\Delta $$ and $$\Lambda $$ . The function $$\Lambda $$ accounts for the systematic deviation of G in the neighborhood of a change point. When only the rate changes, $$\Lambda $$ is hat shaped. When also the variance changes, $$\Lambda $$ takes the form of a shark's fin. In addition, the parameter estimates required in practical application are not consistent in the neighborhood of a change point. Therefore, we derive the factor $$\Delta $$ termed here the distortion function. It accounts for the lack in consistency and describes the local parameter estimating process relative to the true scaling of the filtered derivative process. [ABSTRACT FROM AUTHOR]