Inference for exponential order statistic models based on an integrated likelihood function

Methods of statistical inference are developed for the exponential order st atistic (EOS) model, where only a subset of order statistics from a collect ion of N lid exponential random detection times is observable. When the rat e parameter for detections is unknown, the maximum likelihood estimator (ML E) of the unknown integer parameter N can be infinite with substantial prob ability. inference for N is developed using a pseudolikelihood function obt ained by integrating out the rate parameter. The estimator that maximizes t his function, called the integrated likelihood estimator (ILE), is shown to be finite and to have better sampling properties than the MLE. Parameter-b ased asymptotics are developed for the case where N is large. Application o f the methodology is illustrated using two datasets.