Non-parametric Confidence Intervals for the Renewal Function and the Point Availability
提出了一种基于线性化和经验分布函数弱收敛性的非参数方法，用于构建更新函数和点可用度的置信区间，该方法计算简单且区间更窄，并通过未公开数据进行了验证。
A large sample non-parametric method for constructing confidence intervals for the renewal function and the point availability is investigated. The method is based on a linearization and on the fact that the empirical distribution function converges weakly to a Gaussian process as the sample size increases. The technique is illustrated by the analysis of some hitherto unpublished data. Two of the most important functions arising in renewal theory are the renewal function, the expected number of renewals in a given interval, and the point availability, the probability that a system modelled by an alternating renewal process (ARP) is in a particular state at a specified time. See, for example, Karlin & Taylor (1975, Ch. 5), Ross (1970, Ch. 3) and Cox (1962) for a discussion of applications of these functions. If the functional forms of the distribution functions of the random variables generating the processes are known, and observations of the random variables are available, point estimates of these functions are readily constructed. Further, approximate (large sample) confidence intervals may, in principle, be calculated by an application of the delta method, assuming that the parameter estimates are asymptotically normally distributed. If, however, as is sometimes the case, the functional forms of the underlying distribution functions are unknown, a non-parametric approach is required. Frees (1986a, b, 1988) discussed some non-parametric estimators of the renewal function and constructed a non-parametric confidence interval for this quantity. See Schneider et al. (1990) for a study of these estimators. In this paper, we propose an alternative non-parametric confidence interval for the renewal function which is easier to compute than that of Frees (1986a) and which is appreciably narrower. In addition, we derive an analogous non-parametric confidence interval for the point availability. To the best, of our knowledge, this is the first non-parametric interval estimator of the point availability to have ,been proposed. Our methodology is based on the analysis of Harel et al. (1994), who prove that the empirical renewal function converges weakly to a Gaussian process as the sample size increases. A numerical study shows that our proposed confidence intervals are easy to compute, requiring only a few seconds of CPU time on a Sun Sparc Station, and are fairly narrow for moderate sample sizes.