Analysis of rounded exponential data

Abstract

The problem of inference based on a rounded random sample from the exponential distribution is treated. The main results are given by an explicit expression for the maximum-likelihood estimator, a confidence interval with a guaranteed level of confidence, and a conjugate class of distributions for Bayesian analysis. These results are exemplified on two concrete examples. The large and increasing body of results on the topic of grouped data has been mostly focused on the effect on the estimators. The methods and results for the derivation of confidence intervals here are hence of some general theoretical value as a model approach for other parametric models. The Bayesian credibility interval recommended in cases with a lack of other prior information follows by letting the prior equal the inverted exponential with a scale equal to one divided by the resolution. It is shown that this corresponds to the standard non-informative prior for the scale in the case of non-rounded data. For cases with the absence of explicit prior information it is argued that the inverted exponential prior with a scale given by the resolution is a reasonable choice for more general digitized scale families also.

Keywords:

• quantization error
• objective prior
• digitized data
• rounded data
• grouped data
• displaying error
• finite resolution error
• coarsened data
• censored data
• interval-censored
• latent variable
• nonlinear measurement model
• ISO GUM

Acknowledgements

The authors of the paper 4 are acknowledged for making their data available. The contributions from three reviewers have also been most helpful.