Dear colleagues, We are glad to invite you to the seminar that will take place on the 8th of September, at 16.00: - in presence: room 101 of the campus Perrone, via Perrone, 18, Novara Università del Piemonte Orientale, Dipartimento di Studi per l’Economia e l’Impresa. - on-line at the link: https://meet.google.com/xaz-bwct-dqo

*An Introduction to Saddlepoint Approximations*

*Prof. Elvezio Ronchetti* Research Center for Statistics and Geneva School of Economics and Management University of Geneva, Switzerland Elvezio.Ronchetti@unige.ch www.unige.ch/gsem/en/research/faculty/honorary-professors/elvezio-ronchetti/

*Abstract:* Classical inference in statistics is typically carried out by means of standard (first-order) asymptotic theory. However, the asymptotic distribution of estimators and test statistics can provide a poor approximation of tail areas especially when the sample size is moderate to small. This can lead to inaccurate p-values and confidence intervals.

Several techniques, both parametric and nonparametric, have been devised to improve first-order asymptotic approximations, including e.g. Edgeworth expansions, Bartlett's corrections, and bootstrap methods. Here we focus on saddlepoint techniques, introduced into statistics by H. Daniels, and more generally on small sample asymptotic techniques, an expression coined by F. Hampel to express the spirit of these methods. Indeed they provide very accurate approximations of tail probabilities down to small sample sizes and /or out in the tails. Moreover, these approximations exhibit a relative error of order 1/n, an improvement with respect to other available approximations obtained by means of Edgeworth expansions and similar techniques.

We will review the basic ideas, show the link with other nonparametric methods such as empirical likelihood, and outline some connections to information theory and optimal transportation.

Best wishes, Enea Bongiorno