Some results on renewal equations

In this paper solutions of renewal-type integral equations are studied. It is proved that a recursively spades;efined approximation to the solution has some nice convergence properties. Some simple bounds and other results on the renewal function and the renewal spades;,ensity are obtained.     

FIXED VERSUS RANDOM SAMPLING OF CERTAIN CONTINUOUS PARAMETER PROCESSES

Let {Z(t)} be a stochastic point process. When {Z(t)} is Poisson and it is desired to estimate the intensity A, it is shown that the optimal (in terms of Fisher information) discrete sampling scheme is to sample {Z(t)} at predetermined fixed time points. On the other hand, when {Z(t)} is a pure birth process and a maximum likelihood estimator of the birth rate is desired, it is sometimes better to sample at random time points, according to a renewal process. An application of these ideas is given in the estimation of bacterial density in a liquid, by the method of dilutions.     

Estimating the Cost of a Warranty

The cost of certain types of warranties is closely related to functions that arise in renewal theory. The problem of estimating the warranty cost for a random sample of size n can be reduced to estimating these functions. In an earlier paper, I gave several methods of estimating the expected number of renewals, called the renewal function. This answered an important accounting question of how to arrive at a good approximation of the expected warranty cost. In this article, estimation of the renewal function is reviewed and several extensions are given. In particular, a resampling estimator of the renewal function is introduced. Further, I argue that managers may wish to examine other summary measures of the warranty cost, in particular the variability. To estimate this variability, I introduce estimators, both parametric and nonparametric, of the variance associated with the number of renewals. Several numerical examples are provided.     

Some results on the mean residual waiting time of records

Often, in reliability theory, risk analysis, renewal processes and actuarial studies, mean residual life function or life expectancy plays an important role in studying the conditional tail measure of lifetime data. In this paper, we introduce the notion of the mean residual waiting time of records and present some monotonic and aging properties. Sharp bounds for the mean residual waiting time of records are also investigated.     

Some bounds related to Harris family of distributions

As a lifetime distribution, Harris family of distributions are applied to the lifetime of a series system with random number of components. In this paper, properties of various ageing classes of mixtures of Harris family of distributions, where the tilt parameter of a Harris distribution is taken as a random variable, are studied. We obtain an upper bound for maximum error in evaluating its reliability function. Two bounds are also presented for survival function and expectation of the mixed Harris family. We also provide some interesting bounds for its residual survival function. Our results generalize several previous findings in this connection. Some illustrative examples are also provided.     

Truncated Sequential Change-point Detection based on Renewal Counting Processes

The typical approach in change-point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the "normal" behaviour. Based on the, perhaps, more realistic situation that the process can only be partially observed, we consider the counting process related to the original process observed at equidistant time points, after which action is taken or not depending on the number of observations between those time points. In order for the procedure to stop also when everything is in order, we introduce a fixed time horizon n at which we stop declaring "no change" if the observed data did not suggest any action until then. We propose some stopping rules and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes and extreme value asymptotics for Gaussian processes.     

Bounds for L-Statistics from Weakly Dependent Samples of Random Length

ABSTRACT

Sharp bounds on expected values of L-statistics based on a sample of possibly dependent, identically distributed random variables are given in the case when the sample size is a random variable with values in the set {0, 1, 2,…}. The dependence among observations is modeled by copulas and mixing. The bounds are attainable and provide characterizations of some non trivial distributions.     

Bayesian prediction with a random sample size for the burr lifetime distribution

This paper is concerned with the problem of deriving Bayesian prediction bounds for the Burr distribution when the sample size is a random variable. Prediction bounds for both the future observations (the case of two-sample prediction) and the remaining observations in the same sample (the case of one-sample prediction) will be derived. The analysis will depend mainly on assuming that the size of the sample is a random variable having the Poisson distribution. Finally, numerical examples are given to illustrate the results.     

Sharp bounds for L-statistics from dependent samples of random length

Several distribution-free bounds on expected values of L-statistics based on the sample of possibly dependent and nonidentically distributed random variables are given in the case when the sample size is a random variable, possibly dependent on the observations, with values in the set {1,2,…}. Some bounds extend the results of Papadatos (2001a) to the case of random sample size. The others provide new evaluations even if the sample size is nonrandom. Some applications of the presented bounds are also indicated.     

Change-Point Tests for the Error Distribution in Non-parametric Regression

Abstract.  Several testing procedures are proposed that can detect change-points in the error distribution of non-parametric regression models. Different settings are considered where the change-point either occurs at some time point or at some value of the covariate. Fixed as well as random covariates are considered. Weak convergence of the suggested difference of sequential empirical processes based on non-parametrically estimated residuals to a Gaussian process is proved under the null hypothesis of no change-point. In the case of testing for a change in the error distribution that occurs with increasing time in a model with random covariates the test statistic is asymptotically distribution free and the asymptotic quantiles can be used for the test. This special test statistic can also detect a change in the regression function. In all other cases the asymptotic distribution depends on unknown features of the data-generating process and a bootstrap procedure is proposed in these cases. The small sample performances of the proposed tests are investigated by means of a simulation study and the tests are applied to a data example.     

Bounds and approximations for pairwise comparisons of independent multinormal means

Properties and relationships of some commonly used probability bounds, along with other recently developed bounds and approximations, are evaluated for their performance with pairwise comparisons. The comparisons are of independent sample means obtained from normal random variables with a common variance. Computational methods are presented and numerical results are used to further evaluate the performance of the bounds.     

Prediction in trend-renewal processes for repairable systems

Some problems of point and interval prediction in a trend-renewal process (TRP) are considered. TRP’s, whose realizations depend on a renewal distribution as well as on a trend function, comprise the non-homogeneous Poisson and renewal processes and serve as useful reliability models for repairable systems. For these processes, some possible ideas and methods for constructing the predicted next failure time and the prediction interval for the next failure time are presented. A method of constructing the predictors is also presented in the case when the renewal distribution of a TRP is unknown (and consequently, the likelihood function of this process is unknown). Using the prediction methods proposed, simulations are conducted to compare the predicted times and prediction intervals for a TRP with completely unknown renewal distribution with the corresponding results for the TRP with a Weibull renewal distribution and power law type trend function. The prediction methods are also applied to some real data.     

Bayesian inferences given a type-2 censored sample from a burr distribution

A Bayesian approach is used to make inferences given a random sample of observations from a Burr distribution. Complete and type-2 censored samples are considered and inferences are made on the unknown parameters and the reliability function. In the case of a type-2 censored sample prediction bounds are derived for the unobserved sample values.     

On the Gerber–Shiu discounted penalty function in a risk model with delayed claims

In this paper, we consider an extension to the continuous time risk model for which the occurrence of the claim may be delayed and the time of delay for the claim is assumed to be random. Two types of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim. The time of occurrence of a by-claim is later than that of its associate main claim and the time of delay for the occurrence of a by-claim is random. An integro-differential equations system for the Gerber–Shiu discounted penalty function is established using the auxiliary risk models. Both the system of Laplace transforms of the Gerber–Shiu discounted penalty functions and the Gerber–Shiu discounted penalty functions with zero initial surplus are obtained. From Lagrange interpolating theorem, we prove that the Gerber–Shiu discounted penalty function satisfies a defective renewal equation. Exact representation for the solution of this equation is derived through an associated compound geometric distribution. Finally, examples are given with claim sizes that have exponential and a mixture of exponential distributions.     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.     

Asymptotics for the Finite Time Ruin Probability in the Renewal Model with Consistent Variation

Abstract

This paper investigates the finite time ruin probability in the renewal risk model. Under some mild assumptions on the tail probabilities of the claim size and of the inter-occurrence time, a simple asymptotic relation is established as the initial surplus increases. In particular, this asymptotic relation is requested to hold uniformly for the horizon varying in a relevant infinite interval. The uniformity allows us to consider that the horizon flexibly varies as a function of the initial surplus, or to change the horizon into any nonnegative random variable as long as it is independent of the risk system.     

Detecting change in a hazard regression model with right-censoring

The hazard function plays an important role in survival analysis and reliability, since it quantifies the instantaneous failure rate of an individual at a given time point t, given that this individual has not failed before t. In some applications, abrupt changes in the hazard function are observed, and it is of interest to detect the location of such a change. In this paper, we consider testing of existence of a change in the parameters of an exponential regression model, based on a sample of right-censored survival times and the corresponding covariates. Likelihood ratio type tests are proposed and non-asymptotic bounds for the type II error probability are obtained. When the tests lead to acceptance of a change, estimators for the location of the change are proposed. Non-asymptotic upper bounds of the underestimation and overestimation probabilities are obtained. A short simulation study illustrates these results.     

Uniform asymptotics for ruin probability of a two-dimensional dependent renewal risk model

ABSTRACT

In this paper we consider the tail behavior of a two-dimensional dependent renewal risk model with two dependent classes of insurance business, in which the claim sizes are governed by a common renewal counting process, and their inter-arrival times are dependent, identically distributed. For the case that the claim size distribution belongs to the intersection of long-tailed distribution class and dominant variation class, we obtain an asymptotic formula, which holds uniformly for all time in an infinite interval. Moreover, we point out that the formula still holds uniformly for all time in an infinite interval for widely dependent random variables (r.v.s) under some conditions.     

Bounding sample size projections for the area under a ROC curve

Studies of diagnostic tests are often designed with the goal of estimating the area under the receiver operating characteristic curve (AUC) because the AUC is a natural summary of a test's overall diagnostic ability. However, sample size projections dealing with AUCs are very sensitive to assumptions about the variance of the empirical AUC estimator, which depends on two correlation parameters. While these correlation parameters can be estimated from the available data, in practice it is hard to find reliable estimates before the study is conducted. Here we derive achievable bounds on the projected sample size that are free of these two correlation parameters. The lower bound is the smallest sample size that would yield the desired level of precision for some model, while the upper bound is the smallest sample size that would yield the desired level of precision for all models. These bounds are important reference points when designing a single or multi-arm study; they are the absolute minimum and maximum sample size that would ever be required. When the study design includes multiple readers or interpreters of the test, we derive bounds pertaining to the average reader AUC and the ‘pooled’ or overall AUC for the population of readers. These upper bounds for multireader studies are not too conservative when several readers are involved.     

Two-dimensional Renewal Function Approximation

Two-dimensional renewal functions, which are naturally extensions of one-dimensional renewal functions, have wide applicability in areas where two random variables are needed to characterize the underlying process. These functions satisfy the renewal equation, which is not amenable for analytical solutions. This paper proposes a simple approximation for the computation of the two- dimensional renewal function based only on the first two moments and the correlation coefficient of the variables. The approximation yields exact values of renewal function for bivariate exponential distribution function. Illustrations are presented to compare our approximation with that of Iskandar (1991) who provided a computational procedure which requires the use of the bivariate distribution function of the two variables. A two-dimensional warranty model is used to illustrate the approximation.