A non-parametric estimator for the doubly periodic Poisson intensity function

In a series of papers, J. Garrido and Y. Lu have proposed and investigated a doubly periodic Poisson model, and then applied it to analyze hurricane data. The authors have suggested several parametric models for the underlying intensity function. In the present paper we construct and analyze a non-parametric estimator for the doubly periodic intensity function. Assuming that only a single realization of the process is available in a bounded window, we show that the estimator is consistent and asymptotically normal when the window expands indefinitely. In addition, we calculate the asymptotic bias and variance of the estimator, and in this way gain helpful information for optimizing the performance of the estimator.     

Life expectancy of a bathtub shaped failure distribution

The mean residual life (MRL), or “life expectancy”, encapsulates the entire residual life of the product, and is thus of great interest in many fields, particularly those concerned with warranties, replacement policies, burn-in, and insurance. Estimating the profile of the MRL, let alone constructing confidence bands for it, is a difficult problem. We explore several approaches to estimating the variation in the profile of the mean residual life function, with special attention to a bathtub shaped failure distribution, called the modified Weibull distribution, which has generated considerable interest in reliability engineering. Parametric and non-parametric confidence bands are compared, and the theoretical coverage probabilities verified by simulation. An example application investigating the effectiveness of repairs is given, and possible further research discussed.