Nonparametric estimation of a renewal reward process from discrete data

We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0, T]. We consider regimes where the sampling rate Δ = Δ _T → 0 as T → ∞. We propose an adaptive wavelet threshold density estimator and study its performance for the L _p loss, p ≥ 1, over Besov spaces. We achieve minimax rates of convergence for sampling rates Δ _T that vanish with T at arbitrary polynomial rate. In the same spirit as Buchmann and Grübel (2003) and Duval (2012), the estimation procedure is based on the inversion of the compounding operator. The inverse has no closed form expression and is approached with a fixed point technique.     

Bernoulli thinning of a Markov renewal process

A Bernoulli thinning of a Markov renewal process is investigated. The properties of the thinned process are considered and are related to the properties of the original process. The parameters, moments and equilibrium of the thinned process are determined in terms of the parameters defining the underlying Markov renewal process. Results are illustrated by examples. © 1998 John Wiley & Sons, Ltd.     

Nonparametric estimation with doubly censored and truncated data

Data from longitudinal studies in which an initiating event and a subsequent event occur in sequence are called doubly censored data if the time of both events is interval-censored. In some cases, the second event also suffers left-truncation. This article is concerned with using doubly censored and truncated data to estimate the distribution function F of the duration time, i.e. the elapsed time between the originating event and the subsequent event. An iterative procedure is proposed to obtain the estimate of F. A simulation study is conducted to investigate the performance of the proposed estimator. A modified data set is used to illustrate the proposed approach.     

Nonparametric estimation of a bivariate survivorship function with doubly censored data

The problem of nonparametric estimation of a bivariate survivorship function with doubly censored data is considered. A self-consistent estimator is developed by first “reducing” the problem to that of estimation in a singly and right censored situation. This estimator is shown to satisfy a likelihood equation, and its uniqueness is investigated. The results obtained naturally parallel those obtained by Campbell (1981) in the singly censored case.     

Nonparametric estimation of the integrated intensity of an unobservable transition in a Markov illness-death process

The transition between the two transient states in a nonhomogeneous Markov illness-death process is assumed to be unobservable. Information can only be collected directly about transitions to absorbing states and, by an appropriate experimental design, through premature discontinuation of observation of a nonabsorbed process. A nonparametric estimator of the cumulative intensity of the unobservable transition is proposed which is strongly consistent and asymptotically unbiased. An approximate variance is stated.     

Nonparametric estimation for censored lifetimes suffering from unknown selection bias

In a population of individuals, where the random variable (r.v.) σ denotes the birth time and X the lifetime, we consider the case, where an individual can be observed only if its life-line (σ, X) = {(σ + y, y), 0 ≤ y ≤ X} intersects a given Borel set S in ℝ × ℝ₊. Denoting by σ _S and X _S the birth time and lifetime for the observed individuals, we point out that the distribution function (d.f.) F _S of the r.v. X _S suffers from a selection bias in the sense that F _S = ∝ w d F/μ _S, where w and μ _S depend only on the distribution of σ and on F, the d.f. of X. Assuming in addition that the r.v. X _S is randomly right-censored as soon as the individual is selected, we construct a productlimit estimator for the d.f. F _S and a nonparametric estimator ŵ for the weight function w. We prove a consistency result for ŵ and a weak convergence result for . We establish in addition an exponential bound for .      

Replacement process decomposition for discounted Markov renewal programming

An iterative decomposition method is presented for computing the values in an infinite-horizon discounted Markov renewal program (DMRP). The states are partitioned intoM groups, with each iteration involving disaggregation of one group at a time, with the otherM–1 groups being collapsed intoM–1 singletons using the replacement process method. Each disaggregation also looks like a DMRP and can be performed by policy-iteration, value-iteration or linear programming. Anticipated benefits from the method include reduced computer time and memory requirements, scale-invariance and greater robustness to the starting point.     

Fisher information in window censored renewal process data and its applications

Suppose we have a renewal process observed over a fixed length of time starting from a random time point and only the times of renewals that occur within the observation window are recorded. Assuming a parametric model for the renewal time distribution with parameter θ, we obtain the likelihood of the observed data and describe the exact and asymptotic behavior of the Fisher information (FI) on θ contained in this window censored renewal process. We illustrate our results with exponential, gamma, and Weibull models for the renewal distribution. We use the FI matrix to determine optimal window length for designing experiments with recurring events when the total time of observation is fixed. Our results are useful in estimating the standard errors of the maximum likelihood estimators and in determining the sample size and duration of clinical trials that involve recurring events associated with diseases such as lupus.     

On a new stochastic global optimization algorithm based on censored observations

In this paper a new algorithm is proposed for global optimization problems. The main idea is that of modifying a standard clustering approach by sequentially sampling the objective function while adaptively deciding an appropriate sample size. Theoretical as well as computational results are presented.     

Nonparametric estimation and adaptive control in a class of finite Markov decision chains

We consider a class of Markov decision processes withfinite state and action spaces which, essentially, is determined by the following condition: The state space isirreducible under the action of any stationary policy. However, except by this restriction, the transition law iscompletely unknown to the controller. In this context, we find a set of policies under which thefrequency estimators of the transition law are strongly consistent and then, this result is applied to constructadaptive asymptotically discount-optimal policies.Dedicated to Professor Truman O. Lewis, on the occasion of his sixtieth birthdayThis research was supported in part by the Third World Academy of Sciences (TWAS) under Grant TWAS RG MP 898-152, and in part by the Consejo Nacional de Ciencia y Tecnología (CONACYT) under Grant A128CCOEO550 (MT-2).     

Nonparametric estimation of the stationary density and the transition density of a Markov chain

In this paper, we study first the problem of nonparametric estimation of the stationary density f$f$ of a discrete-time Markov chain (_Xi)$\left(X_i\right)$. We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density g$g$ of (_Xi,_Xi+1)$(X_i,X_{i+1})$ and so to provide an adaptive estimator of the transition density π=g/f$\pi =g/f$. We give bounds in L²$L^2$ norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.     

Nonparametric prediction intervals for progressive Type-II censored order statistics based on k-records

In this paper, we consider the prediction problem in two-sample case and study the non-parametric predicting future progressively Type-II censored order statistics based on observed $k$ -records from the same distribution. Also, prediction intervals for progressively Type-II censored spacings are obtained based on $k$ -record spacings. It is shown that the coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Moreover, optimal prediction intervals are derived for each case. Finally, for illustrating the proposed procedure, we consider a real data set and numerical computations are given. The results of Ahmadi and Balakrishnan (Statistics 44:417–430, 2010) can be achieved as special cases of our results.     

On the posterior distribution of a dirichlet process given randomly right censored observations

The purpose of this note is to show that the conditional distribution of a Dirichlet process P given n independent observations X₁… X_n from P and belonging to measurable sets A₁,… A_n with A₁ ? A₁₊₁ for i=1,… n=1 is a mixture of Dirichlet processes as introduced by Antoniak. It is also shown that this result is applicable in Bayesin decision problems concerning a random survival distribution under Dirichlet process priors.     

Nonparametric likelihood based estimation for a multivariate Lipschitz density

We consider a problem of nonparametric density estimation under shape restrictions. We deal with the case where the density belongs to a class of Lipschitz functions. Devroye [L. Devroye, A Course in Density Estimation, in: Progress in Probability and Statistics, vol. 14, Birkhäuser Boston Inc., Boston, MA, 1987] considered these classes of estimates as tailor-made estimates, in contrast in some way to universally consistent estimates. In our framework we get the existence and uniqueness of the maximum likelihood estimate as well as strong consistency. This NPMLE can be easily characterized but it is not easy to compute. Some simpler approximations are also considered.     

Nonparametric estimation of the location and scale parameters based on density estimation

Summary By representing the location and scale parameters of an absolutely continuous distribution as functionals of the usually unknown probability density function, it is possible to provide estimates of these parameters in terms of estimates of the unknown functionals. Using the properties of well-known methods of density estimates, it is shown that the proposed estimates possess nice large sample properties and it is indicated that they are also robust against dependence in the sample. The estimates perform well against other estimates of location and scale parameters.     

On the temporal analysis of special GERT networks using a modified Markov renewal process

In this paper we show how to represent a STEOR network (GERT network with only nodes of the stochastic exclusive-or Type) by a modified Markov renewal process. The computation of the renewal kernel of this process yields the temporal analysis of the network.

---

Zusammenfassung Es wird gezeigt, wie ein STEOR-Netzplan (GERT-Netzplan, der nur Knoten vom Typ stochastisch exklusiv-oder enthält) durch einen modifizierten Markowschen Erneuerungsprozeß beschrieben werden kann. Die Bestimmung der Erneuerungsfunktionen dieses Prozesses liefert die Auswertung des STEOR-Netzplans (im Sinne der Zeitplanung).