Uttara Naik-Nimbalkar

Unconditional Tests of Goodness of Fit for the Intensity of Time-Truncated Nonhomogeneous Poisson Processes

Procedures for testing trends in the intensity functions of nonhomogeneous Poisson processes are based mostly on conditioning on the number of failures observed in (0, t] with fixed t. We study an unconditional test based on the time-truncated data that enables meaningful asymptotics as t → ∞. We show that the asymptotic test is conservative and that its power quickly comes close to the power of the uniformly most powerful unbiased test for the power-law alternatives. Moreover, for the goodness of fit of a specified intensity, the exact test has more power than the test based on the conditional approach. We illustrate the procedure using a real dataset.

NONPARAMETRIC DENSITY ESTIMATION

This chapter describes the background material related to the nonparametric density estimation. Techniques such as histograms (together with its extension, known as ASH, see Sect. 2.3), Parzen windows and k-nearest neighbors are at the core of the applications of nonparametric density estimation. For that reason, we decided to include a chapter describing these for the sake of completeness and to allow less experienced readers develop their intuitions in terms of the nonparametric estimation. Most of the material is presented taking into account only the univariate case; extending the results to cover more than one variable, however, is often a straightforward task. The chapter is organized as follows: Sect. 2.2 presents a short overview of the fundamental concepts related to histograms. Section2.3 is devoted to a description of a smart extension of certain well-known histograms aimed at avoiding some of their drawbacks. Section2.4 presents basic concepts related to the nonparametric density estimation. Section2.5 is devoted to the Parzen windows, while Sect. 2.6 to the k-nearest neighbors approach.

On Competing Risks with Masked Failures

Competing risks data arise when the study units are exposed to several risks at the same time but it is assumed that the eventual failure of a unit is due to only one of these risks, which is called the “cause of failure”. Statistical inference procedures when the time to failure and the cause of failure are observed for each unit are well documented. In some applications, it is possible that the cause of failure is either missing or masked for some units. In this article, we review some statistical inference procedures used when the cause of failure is missing or masked for some units.