Maths Project Repository Database

Suppose data $x_1, \ldots, x_n$ is thought to be generated from a distribution with density $f(x|\theta)$ but a few observations may have density $g(x|\theta,\delta)$ (often $g(x|\theta,\delta)=f(x|\theta + \delta)$ or $f(x|\theta \times \delta)$. If g is more likely to produce observations at the extremes of the sample, this may be used to model the situation when we have outliers in the data. To examine the possibility of outliers we can compare a model that says all observations have density $f$ to one that says some specific observations have distribution $g$. This comparison might be done classically by a likelihood ratio, leading to many tests for outliers described, for example, in Barnett and Lewis [1], or in a Bayesian approach by a Bayes factor. If the prior distribution is proper this can be fairly straightforward (e.g. Pettit and Smith [3], Pettit [4] but an improper prior leads to complications in defining the Bayes factor. There have been a number of suggestions for dealing with these, (e.g. Spiegelhalter and Smith [8], O’Hagan [2]) with applications to identification of outliers from various distributions in Pettit [5, 6] and Sothinathan and Pettit [7].
In this project the first aim would be to review and explain these ideas. A second aim would be to extend these ideas to a new distribution such as the Pareto or Uniform.

1. V. Barnett, T. Lewis, Outliers in Statistical Data. Third edition. Wiley, Chichester, 1994.
2. A. O’Hagan, Fractional Bayes factors for models comparison (with discussion), J. Roy. Statist. Soc. B 57 (1995), 99–138.
3. L.I. Pettit, A.F.M. Smith, Outliers and Influential Observations in Linear Models (with discussion), in Bayesian Statistics 2, Edited by J.M. Bernardo, M.H. DeGroot, D.V. Lindley, A.F.M. Smith. North-Holland, Amsterdam, 1985, 473–494.

4. L.I. Pettit, Bayes methods for outliers in exponential samples, J. Roy. Statist. Soc. B 50 (1988): 371–380.

5. L.I. Pettit, Bayes factors for outlier models using the device of imaginary observations, J. Amer. Statist. Assoc. 87 (1992): 541–545.

6. L.I. Pettit, Bayesian approaches to the detection of outliers in Poisson samples, Communications in Statistics, Theory and Methods 23 (1994): 1785–1795

7. N. Soothinathan, L.I. Pettit, Bayes methods for outliers in binomial samples, Communications in Statistics, Theory and Methods 34 (2005): 351–366.

8. D.J. Spiegelhalter, A.F.M. Smith, Bayes factors for linear and log-linear models with vague prior information, J. Roy. Statist. Soc. B 44 (1982): 377–387.