Inference for Exponential Order Statistic Models Based on an Integrated Likelihood Function

Abstract

Methods of statistical inference are developed for the exponential order statistic (EOS) model, where only a subset of order statistics from a collection of N iid exponential random detection times is observable. When the rate parameter for detections is unknown, the maximum likelihood estimator (MLE) of the unknown integer parameter N can be infinite with substantial probability. Inference for N is developed using a pseudolikelihood function obtained by integrating out the rate parameter. The estimator that maximizes this function, called the integrated likelihood estimator (ILE), is shown to be finite and to have better sampling properties than the MLE. Parameter-based asymptotics are developed for the case where N is large. Application of the methodology is illustrated using two datasets.

Key Words:

• Integrated likelihood
• Likelihood ratio
• Parameter-based asymptotics
• Profile likelihood
• Software reliability