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Recurrent events are often encountered in clinical and epidemiological studies where a terminal event is also observed. With recurrent events data it is of great interest to estimate the marginal mean of the cumulative number of recurrent events experienced prior to the terminal event. The standard nonparametric estimator was suggested in Cook and Lawless and further developed in Ghosh and Lin. We here investigate the efficiency of this estimator that, surprisingly, has not been studied before. We rewrite the standard estimator as an inverse probability of censoring weighted estimator. From this representation we derive an efficient augmented estimator using efficient estimation theory for right-censored data. We show that the standard estimator is efficient in settings with no heterogeneity. In other settings with different sources of heterogeneity, we show theoretically and by simulations that the efficiency can be greatly improved when an efficient augmented estimator based on dynamic predictions is employed, at no extra cost to robustness. The estimators are applied and compared to study the mean number of catheter-related bloodstream infections in heterogeneous patients with chronic intestinal failure who can possibly die, and the efficiency gain is highlighted in the resulting point-wise confidence intervals.

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We propose new resampling‐based approaches to construct asymptotically valid time‐simultaneous confidence bands for cumulative hazard functions in multistate Cox models. In particular, we exemplify the methodology in detail for the simple Cox model with time‐dependent covariates, where the data may be subject to independent right‐censoring or left‐truncation. We use simulations to investigate their finite sample behavior. Finally, the methods are utilized to analyze two empirical examples with survival and competing risks data.

In this paper, we propose a general framework to design accumulation scenarios that can be used to anticipate the impact of a massive cyber attack on an insurance portfolio. The aim is also to emphasize the role of countermeasures in stopping the spread of the attack over the portfolio, and to quantify the benefits of implementing such strategies of response. Our approach consists of separating the global dynamic of the cyber event (that can be described through compartmental epidemiological models), the effect on the portfolio, and the response strategy. This general framework allows us to obtain Gaussian approximations for the corresponding processes, and sharp confidence bounds for the losses. A detailed simulation study, which mimics the effects of a Wannacry scenario, illustrates the practical implementation of the method.

Among Mixed Poisson processes, counting processes having geometrically distributed increments can be obtained when the mixing random intensity is exponentially distributed. Dealing with shock models and compound counting models whose shocks and claims occur according to such counting processes, we provide various comparison results and aging properties concerning total claim amounts and random lifetimes. Furthermore, the main characteristic distributions and properties of these processes are recalled and proved through a direct approach, as an alternative to those available in the literature. We also provide closed-form expressions for the first-crossing-time problem through monotone nonincreasing boundaries, and numerical estimates of first-crossing-time densities through other suitable boundaries. Finally, we present several applications in seismology, software reliability and other fields.

We construct a news spreading model with a time dependent contact rate which generalizes the classical Susceptible–Infected model of epidemiology. In particular, we are interested on the time-dynamics of the sharing and diffusion process of news on the Internet. We focus on the counting process describing the number of connections to a given website, characterizing the cumulative density function of its inter-arrival times. Moreover, starting from the general form of the finite dimensional distribution of the process, we determine a formula for the time-variable rate of the connections and establish its relationship with the probability density function of the interarrival times. We finally show the effectiveness of our theoretical framework analyzing a real-world dataset, the Memetracker dataset, whose parameters characterizing the diffusion process are determined.

We propose a stochastic semi-Markovian framework for disability modelling in a multi-period discrete-time setting. The logistic transforms of disability inception and recovery probabilities are mod ...

We provide general conditions to derive posterior concentration rates for Aalen counting processes. The conditions are designed to resemble those proposed in the literature for the problem of density estimation, for instance in Ghosal et al. (2000), so that existing results on density estimation can be adapted to the present setting. We apply the general theorem to some prior models including Dirichlet process mixtures of uniform densities to estimate monotone non-increasing intensities and log-splines., 26 pages

A flexible approach for modeling both dynamic event counting and dynamic link-based networks based on counting processes is proposed, and estimation in these models is studied. We consider nonparametric likelihood based estimation of parameter functions via kernel smoothing. The asymptotic behavior of these estimators is rigorously analyzed in an asymptotic framework where the number of nodes tends to infinity. The finite sample performance of the estimators is illustrated through an empirical analysis of bike share data.

In this paper, we discuss the problem of testing the homogeneity of several populations when the available data are progressively Type-II censored. Defining for each sample a univariate counting process, we can modify all the methods that were developed during the last two decades (see e.g. [P.K. Andersen, Ø. Borgan, R. Gill, N. Keiding, Statistical Models Based on Counting Processes, Springer, New York, 1993]) for use to this problem. An important aspect of these tests is that they are based on either linear or non-linear functionals of a discrepancy process (DP) based on the comparison of the cumulative hazard rate (chr) estimated from each sample with the chr estimated from the whole sample (viz., the aggregation of all the samples), leading to either linear tests or non-linear tests. Both these kinds of tests suffer from some serious drawbacks. For example, it is difficult to extend non-linear tests to the K-sample situation when K⩾3. For this reason, we propose here a new class of non-linear tests, based on a chi-square type functional of the DP, that can be applied to the K-sample problem for any K⩾2.

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal and van der Vaart (2007a). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data., Comment: With supplementary material

Statistical inference in competing risks models is often based on the famous Aalen-Johansen estimator. Since the corresponding limit process lacks independent increments, it is typically applied together with Lin’s (1997) resampling technique involving standard normal multipliers. Recently, it has been seen that this approach can be interpreted as a wild bootstrap technique and that other multipliers, e.g. centered Poissons, may lead to better finite sample performances, see Beyersmann et al. (2013). Since the latter is closely related to Efron’s classical bootstrap, the question arises whether this or more general weighted bootstrap versions of Aalen-Johansen processes lead to valid results. Here we analyze their asymptotic behaviour and it turns out that such weighted bootstrap versions in general possess the wrong covariance structure in the limit. However, we demonstrate that the weighted bootstrap can nevertheless be applied for specific null hypotheses of interest and also discuss its limitations for statistical inference. To this end, we introduce different consistent weighted bootstrap tests for the null hypothesis of stochastically ordered cumulative incidence functions and compare their finite sample performance in a simulation study.

IfX1, …,Xnare random variables we denote byX(1)⩽X(2)⩽…⩽X(n)their respective order statistics. In the case where the random variables are independent and identically distributed, one may demonstrate very strong notions of dependence between any two order statisticsX(i)andX(j). If in particular the random variables are independent with a common density or mass function, thenX(i)andX(j)areTP2dependent for anyiandj. In this paper we consider the situation in which the random variablesX1, …,Xnare independent but otherwise arbitrarily distributed. We show that for anyit∣X(i)>s] is an increasing function ofs. This is a stronger form of dependence betweenX(i)andX(j)than that of association, but we also show that among the hierarchy of notions of bivariate dependence this is the strongest possible under these circumstances. It is also shown that in this situation,P[X(j)>t∣X(i)>s] is a decreasing function ofi=1, …,nfor any fixeds

We consider a general high-dimensional additive hazards model in a non-asymptotic setting, including regression for censored-data. In this context, we consider a Lasso estimator with a fully data-driven ℓ1 penalization, which is tuned for the estimation problem at hand. We prove sharp oracle inequalities for this estimator. Our analysis involves a new “data-driven” Bernstein’s inequality, that is of independent interest, where the predictable variation is replaced by the optional variation.

Learning the network structure of a large graph is computationally demanding, and dynamically monitoring the network over time for any changes in structure threatens to be more challenging still. This paper presents a two-stage method for anomaly detection in dynamic graphs: the first stage uses simple, conjugate Bayesian models for discrete time counting processes to track the pairwise links of all nodes in the graph to assess normality of behavior; the second stage applies standard network inference tools on a greatly reduced subset of potentially anomalous nodes. The utility of the method is demonstrated on simulated and real data sets., Comment: Published in at http://dx.doi.org/10.1214/10-AOAS329 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper considers the problem of multi-sample nonparametric comparison of counting processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For the problem considered, we construct two new classes of nonparametric test statistics based on the accumulated weighted differences between the rates of increase of the estimated mean functions of the counting processes over observation times, wherein the nonparametric maximum likelihood approach is used to estimate the mean function instead of the nonparametric maximum pseudo-likelihood. The asymptotic distributions of the proposed statistics are derived and their finite-sample properties are examined through Monte Carlo simulations. The simulation results show that the proposed methods work quite well and are more powerful than the existing test procedures. Two real data sets are analyzed and presented as illustrative examples., Published in at http://dx.doi.org/10.1214/08-AOS599 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A class of local linear kernel density estimators based on weighted least-squares kernel estimation is considered within the framework of Aalen's multiplicative intensity model. This model includes the filtered data model that, in turn, allows for truncation and/or censoring in addition to accommodating unusual patterns of exposure as well as occurrence. It is shown that the local linear estimators corresponding to all different weightings have the same pointwise asymptotic properties. However, the weighting previously used in the literature in the i.i.d. case is seen to be far from optimal when it comes to exposure robustness, and a simple alternative weighting is to be preferred. Indeed, this weighting has, effectively, to be well chosen in a 'pilot' estimator of the survival function as well as in the main estimator itself. We also investigate multiplicative and additive bias-correction methods within our framework. The multiplicative bias-correction method proves to be the best in a simulation study comparing the performance of the considered estimators. An example concerning old-age mortality demonstrates the importance of the improvements provided. A class of local linear kernel density estimators based on weighted least-squares kernel estimation is considered within the framework of Aalen's multiplicative intensity model. This model includes the filtered data model that, in turn, allows for truncation and/or censoring in addition to accommodating unusual patterns of exposure as well as occurrence. It is shown that the local linear estimators corresponding to all different weightings have the same pointwise asymptotic properties. However, the weighting previously used in the literature in the i.i.d. case is seen to be far from optimal when it comes to exposure robustness, and a simple alternative weighting is to be preferred. Indeed, this weighting has, effectively, to be well chosen in a 'pilot' estimator of the survival function as well as in the main estimator itself. We also investigate multiplicative and additive bias-correction methods within our framework. The multiplicative bias-correction method proves to be the best in a simulation study comparing the performance of the considered estimators. An example concerning old-age mortality demonstrates the importance of the improvements provided.

Marginal structural models were introduced in order to provide estimates of causal effects from interventions based on observational studies in epidemiological research. The key point is that this can be understood in terms of Girsanov's change of measure. This offers a mathematical interpretation of marginal structural models that has not been available before. We consider both a model of an observational study and a model of a hypothetical randomized trial. These models correspond to different martingale measures -- the observational measure and the randomized trial measure -- on some underlying space. We describe situations where the randomized trial measure is absolutely continuous with respect to the observational measure. The resulting continuous-time likelihood ratio process with respect to these two probability measures corresponds to the weights in discrete-time marginal structural models. In order to do inference for the hypothetical randomized trial, we can simulate samples using observational data weighted by this likelihood ratio., Published in at http://dx.doi.org/10.3150/10-BEJ303 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Estimation of a time-varying coefficient in a Cox-type parameterization of the stochastic intensity of a point process is considered. A sieve estimation procedure (Grenander, 1981) is used to estimate the coefficient. A rate of convergence in probability for the sieve estimator is given and a functional CLT for the integrated sieve estimator is proved.

International audience; We consider models based on multivariate counting processes, including multi‐state models. These models are specified semi‐parametrically by a set of functions and real parameters. We consider inference for these models based on coarsened observations, focusing on families of smooth estimators such as produced by penalized likelihood. An important issue is the choice of model structure, for instance, the choice between a Markov and some non‐Markov models. We define in a general context the expected Kullback–Leibler criterion and we show that the likelihood‐based cross-validation (LCV) is a nearly unbiased estimator of it. We give a general form of an approximate of the leave‐one‐out LCV. The approach is studied by simulations, and it is illustrated by estimating a Markov and two semi‐Markov illness–death models with application on dementia using data of a large cohort study.

We study the problem of nonparametric, completely data-driven estimation of the intensity of counting processes satisfying the Aalen multiplicative intensity model. To do so, we use model selection techniques and, specifically, penalized projection estimators for a random inner product. For histogram estimators, under some assumptions on the process, we obtain adaptive results for the minimax risk. In general, for more intricate (predictable) models, we only obtain oracle inequalities. The study is complemented by some simulations in the right-censoring model.

We consider a filtering problem when the state process is a reflected Brownian motion X t and the observation process is its local time Λ s , for s ≤ t . For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λ t . The convergence of the approximating filter to the original one combined with an explicit construction of the approximating filter allows us to derive the explicit form of the original filter. The last result can be obtained also by means of the Azema martingale.

International audience; We consider first the mixed discrete-continuous scheme of observation in multistate models; this is a classical pattern in epidemiology because very often clinical status is assessed at discrete visit times while times of death or other events are observed exactly. A heuristic likelihood can be written for such models, at least for Markov models; however, a formal proof is not easy and has not been given yet. We present a general class of possibly non-Markov multistate models which can be represented naturally as multivariate counting processes. We give a rigorous derivation of the likelihood based on applying Jacod's formula for the full likelihood and taking conditional expectation for the observed likelihood. A local description of the likelihood allows us to extend the result to a more general coarsening observation scheme proposed by Commenges & Gégout-Petit. The approach is illustrated by considering models for dementia, institutionalization and death.

We show, under regularity conditions, that a nonnegative nondecreasing real-valued stochastic process satisfies a large deviation principle (LDP) with nonlinear scaling if and only if its inverse process does. We also determine how the associated scaling and rate functions must be related. A key condition for the LDP equivalence is for the composition of two of the scaling functions to be regularly varying with nonnegative index. We apply the LDP equivalence to develop equivalent characterizations of the asymptotic decay rate in nonexponential asymptotics for queue-length tail probabilities. These alternative characterizations can be useful to estimate the asymptotic decay constant from systems measurements.

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.

We apply an extended contraction principle and superexponential convergence in probability to show that a functional large deviation principle for a sequence of stochastic processes implies a corresponding functional large deviation principle for an associated sequence of first-passage-time or inverse processes. Large deviation principles are established for both inverse processes and centered inverse processes, based on corresponding results for the original process. We apply these results to obtain functional large deviation principles for renewal processes and superpositions of independent renewal processes.

We describe models for infectious disease attack rates outside Aalen's multiplicative class and incorporating heterogeneity and interactions between subjects. Large-sample theory for the Nelson-Aalen estimator is developed, and its relevance examined in a simulation study. Planning for randomized, controlled clinical trials of prophylactic HIV vaccines partly motivated this work.

It has been over 60 years since Kolmogorov introduced a distribution-free omnibus test for the simple null hypothesis that a distribution function coincides with a given distribution function. Doob subsequently observed that Kolmogorov's approach could be simplified by transforming the empirical process to an empirical process based on uniform random variables. Recent use of more sophisticated transformations has led to the construction of asymptotically distribution-free omnibus tests when unknown parameters are present. The purpose of the present paper is to use the transformation approach to construct an asymptotically distribution-free omnibus test for independence of a survival time from a covariate. The test statistic is obtained from a certain test statistic process (indexed by time and covariate), which is shown to converge in distribution to a Brownian sheet. A simulation study is carried out to investigate the finite sample properties of the proposed test and an application to data from the British Medical Research Council's 4th myelomatosis trial is given.

The frailty model is a generalization of Cox's proportional hazards model which includes a random effect. Nielsen, Gill, Andersen and Sorensen (1992) proposed an EM algorithm to estimate the cumulative baseline hazard and the variance of the random effect. Here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance.

The frailty model is a generalization of Cox's proportional hazards model which includes a random effect. Nielsen, Gill, Andersen and Sorensen proposed an EM algorithm to estimate the cumulative baseline hazard and the variance of the random effect. Here existence and consistency of the estimators are proved. An example using truncated and censored data is considered.

To check the validity of an assumed parametric model for survival data, one may compare $\hat{A}(t)$, the nonparametric Nelson-Aalen plot of the cumulative hazard rate, with $A(t, \hat{\theta})$, the estimated parametric cumulative hazard rate, $\hat{\theta}$ being for example the maximum likelihood estimator. Convergence in distribution of $\sqrt n (\hat{A}(t) - A(t, \hat{\theta}))$ and more general processes is studied in the present paper, employing the general framework of counting processes, which allows for quite general models for life history data and for quite general censoring schemes. The results are applied to the construction of $\chi^2$-type statistics for goodness of fit. Cramer-von Mises and Kolmogorov-Smirnov type tests are presented in the case where the unknown parameter is one-dimensional. Power considerations are also included, and some optimality results are reached. Finally tests are constructed for the hypothesis that the unspecified hazard rate part in Cox's regression model follows a parametric form.

Time-sequential point estimation is studied in the model of fully parametric censored data and Cox's regression model. Both are investigated in the context of counting processes through estimating equations defined by martingales. The concept of information and a related inequality developed in estimating function theory by Godambe are adapted to these models. These suggest some optimality criteria for choosing stopping times as well as estimators. These lead naturally to some sequential procedures, which are shown to be asymptotically efficient with respect to the above criteria.

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.

Random left truncation is modelled by the conditional distribution of the random variable $X$ of interest, given that it is larger than the truncating random variable $Y$; usually $X$ and $Y$ are assumed independent. The present paper is based on a simple reparametrization of the left truncation model as a three-state Markov process. The derivation of a nonparametric estimator is a distribution function under random truncation is then a special case of results on the statistical theory of counting processes by Aalen and Johansen. This framework also clarifies the status of the estimator as a nonparametric maximum likelihood estimator, and consistency, asymptotic normality and efficiency may be derived directly as special cases of Aalen and Johansen's general theorems and later work. Although we do not carry through these here, we note that the present framework also allows several generalizations: censoring may be incorporated; the independence hypothesis underlying the truncation models may be tested; ties (occurring when the distributions of $F$ and $G$ have discrete components) may be handled.

D. Commenges and H. Jacqmin-Gadda (1997, J. Roy. Statist. Soc. B59, 157–171) considered generalized score tests of homogeneity that accommodate parametric and semiparametric regression models arising from multivariate random effects with common variance and known correlations. In this paper we give some sufficient hypotheses under which the asymptotic behavior of the test statistic is standard normal for the proportional hazards model with right censored data. Several examples of applicability are studied.

It is assumed that we observe one realization of an r dimensional counting process with intensities that are products of predictable and observable weight processes, a common function of time, and predictable functions that depend on an unknown parameter θ. Given that the realization brings increasing information on θ as the observed time grows asymptotic results are proved for the distributions of parameter estimates, certain test statistics for parametric hypothesis, and goodness-of-fit tests.

In this paper, for right censored competing risks data, a class of tests developed for comparing the cumulative incidence of a particular type of failure among different groups. The tests are based on comparing weighted averages of the hazards of the subdistribution for the failure type of interest. Asymptotic results are derived by expressing the statistics in terms of counting processes and using martingale central limit theory. It is proposed that weight functions very similar to those for the $G^p$ tests from ordinary survival analysis be used. Simulation results indicate that the asymptotic distributions provide adequate approximations in moderate sized samples.

Many population processes in demography, epidemiology and other fields can be represented by a time-continuous Markov chain model with a finite state space. If we have complete information on the life history of a cohort, the intensities of the Markov model may be estimated by the occurrence/ exposure rates or by nonparametric techniques. In many situations, however, we have only incomplete information. In this paper we consider the special, but important, case where the occurrences and the total exposure are known, but not the distribution of the latter over the various separate statuses. Methods for handling such data, so-called demographic incidence rates, and methods for estimating the intensities from this kind of data, are known in the literature. However, their statistical properties are only vaguely known. The present paper gives a thorough presentation of the theory of these methods, and provide rigorous proofs of their statistical properties using stochastic process theory. 1. Introduction. In demography and fields with a similar methodology, a population process on the individual level can frequently be represented by a time-continuous Markov chain with a finite state space. The states of the Markov chain represent the demographic statuses and the jumps between the states correspond to the demographic events. The time parameter of the Markov process may be the age of an individual, the time since a specific event, or some such quantity. We will call it seniority. Given this general framework, the population phenomena may be described by the transition intensities of the model. Consequently, it is of great interest to estimate these functions. In principle, this is an easy task if the individuals studied are watched continuously over some period of time. One may then use the classical methods based on occurrence/exposure rates (for a review, see Hoem, 1976), or for small populations or samples, nonparametric techniques developed by Aalen (1978) and others. Quite often, however, one does not obtain the sufficiently detailed data required to use any of these methods. This fact has inhibited the use of Markov chain models in demography and related fields. The development and study of statistical methods for situations with incomplete data is therefore of considerable interest. There seems to be no general solution to the problem of estimating the intensities of partially observed Markov chains. In the present paper we consider a special, but important, type of incomplete observation. Namely, that transitions within a subset of states K are observed in detail, while counts of transitions out

On observe une realisation d'un processus de comptage r-dimensionnel avec des intensites qui sont des produits d'un processus de poids predictible, d'une fonction commune du temps et de parametres β 1 , i=1, 2, …, r qui distinguent les composantes. A condition que la realisation observee apporte une information croissante sur β quand le temps observe augmente, on montre la convergence forte d'un estimateur de maximum de vraisemblance partiel. Pour de pareilles realisations, il est montre ainsi que l'estimation, apres application d'une normalisation aleatoire, est asymptotiquement normale

Weak convergence results are proved for the product-limit estimator on the whole line. Applications are given to confidence band construction, estimation of mean lifetime, and to the theory of $q$-functions. The results are obtained using stochastic calculus and in probability linear bounds for empirical processes.

Under extremely general patterns of patient-arrival, allocation to treatment and loss to follow-up in (randomized) clinical trial settings, the sequentially computed logrank statistic (Mantel, 1966) is shown (under the null hypothesis of identically distributed lifetimes) to have exactly uncorrelated increments, and is shown via Rebolledo's (1980) martingale invariance principle to satisfy a functional central limit theorem, justifying sequential logrank tests of Jones and Whitehead (1979). Generalizations are made to other two-sample rank tests for censored survival data, and practical applicability to real randomized clinical trials is discussed.

The kernel function method developed during the last twenty-five years to estimate a probability density function essentially is a way of smoothing the empirical distribution function. This paper shows how one can generalize this method to estimate counting process intensities using kernel functions to smooth the nonparametric Nelson estimator for the cumulative intensity. The properties of the estimator for the intensity itself are investigated, and uniform consistency and asymptotic normality are proved. We also give an illustrative numerical example.

The Fleming-Harrington class for right-censored data was first introduced by Harrington and Fleming (1982). This class is widely used in survival analysis studies and it is a subset of the so-called weighted logrank test statistics. Recently, Oller and Gómez (2012) proposed an extension of this class for interval-censored data. This paper introduces the R package FHtest, which implements the Fleming-Harrington class for right-censored and interval-censored survival data. It provides an integrated approach for performing two-sample, k-sample and trend tests based on either counting process theory, likelihood theory, or permutation distributions. In this paper, we summarize the main aspects of the theory framework and present several examples with R codes to illustrate the usage of the main functions of FHtest.