Confidence Distributions and Related Themes

Abstract

This is the guest editors' general introduction to a Special Issue of the Journal of Statistical Planning and Inference, dedicated to confidence distributions and related themes. Confidence distributions (CDs) are distributions for parameters of interest, constructed via a statistical model after analysing the data. As such they serve the same purpose for the frequentist statisticians as the posterior distributions for the Bayesians. There have been several attempts in the literature to put up a clear theory for such confidence distributions, from Fisher's fiducial inference and onwards. There are certain obstacles and difficulties involved in these attempts, both conceptually and operationally, which have contributed to the CDs being slow in entering statistical mainstream. Recently there is a renewed surge of interest in CDs and various related themes, however, reflected in both series of new methodological research, advanced applications to substantive sciences, and dissemination and communication via workshops and conferences. The present special issue of the JSPI is a collection of papers emanating from the {\it Inference With Confidence} workshop in Oslo, May 2015. Several of the papers appearing here were first presented at that workshop. The present collection includes however also new research papers from other scholars in the field.

Figures (5)

• Figure 3.1: Confidence curve for the probability $p$ that there would be a 100 m race of 9.72 or better, in the course of 2008, as seen from January 1 that year. The point estimate is 0.034, and the 90% confidence interval is $[0,0.189]$. The dotted curve is a fine-tuned version of (\ref{['eq:CDapprox3']}), via Bartletting.
• Figure 4.1: The dashed lines are the confidence curves for the risk inflation parameter $\gamma$ from each of the six studies, from the model (\ref{['eq:poissonmodel']}) with the lidocaine data of Table \ref{['table:lido']}. The thick black curve is the optimal combined confidence curve, while the virtually identical dashed curve is the combined confidence curve based on the II-CC-FF methods of Section 5, without using the Poisson model properties per se.
• Figure 4.2: Left panel: The expected mean lifelength for women born in Norway (full curve), Sweden (dashed curve), Denmark (dotted curve), in calendar years 1960, 1970, 1980, 1990, 200, 2010, 2015. Right panel: The CD $C(\tau,{\mathcal{D}})$ for the spread parameter in the model $\beta_1,\beta_2,\beta_3\sim{\rm N}(\beta_0,\tau^2)$ for the three regression slope parameters; for men (dashed curve, starting at 0.603 at zero) and for women (full curve, starting at 0.021 at zero). 95% intervals for $\tau$ are $[0,0.060]$ for men and $[0.003,0.053]$ for women.
• Figure 6.1: Confidence curves ${\rm cc}(q)$ for deciles 0.1, 0.3, 0.5, 0.7, 0.9 of birthweight distributions, for boys ($n=548$) and girls ($n=480$) born in Oslo 2001--2008.
• Figure 7.1: Left panel: with average $x_0$ bodyweight (in kg) and average brainweight $y_0$ (in g), for 28 species of land animals, the plot gives $(x,y)=(\log x_0,\log y_0)$. Right panel: two confidence curves for the correlation coefficient $\rho$, based on maximum likelihood (estimate 0.779) and one using the robust BHHJ method (estimate 0.819).