Fiducial inference in the pivotal family of distributions

Abstract

In this paper a family, called the pivotal family, of distributions is considered. A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given. The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.

Nonparametric Interval Estimation in One-Way Random-Effects Models

“…Inspired by the structural inference (Fraser, 1961(Fraser, , 1966, Hannig et al (2006) provided a structural method and used it to derive the fiducial generalized confidence interval for 2 . They also pointed out that this interval is the same as Weerahandi's generalized confidence interval and the fiducial interval which can be obtained directly from the fiducial distribution (see, for example, Dawid and Stone, 1982;Hannig, 2006;Xu and Li, 2006) of 2 . For other discussions on normal random-effects models based on the generalized inference or fiducial inference, the reader is referred to Li and Li (2005), Hannig et al (2006), , Burch (2007), , and Lidong et al (2008).…”

Generalized p-values for testing regression coefficients in partially linear models

“…In model (3), the sufficient statistic of (β, σ) is B = (X X) −1 X Y and S 2 = Y (I − P X )Y , where P X = X(X X) −1 X can be represented as the functional model [31] or the generalized pivotal model [32] B = β + σ(X X) −1 X E 1 ,…”

An asymptotics look at the generalized inference

“…In fact, the connection between FGPQs and Fisher's fiducial inference (see e.g. [5][6][7]4,34,10]) is established in their article, i.e., given a fiducial distribution for a parameter, there is a systematic procedure for constructing a FGPQ whose conditional distribution conditional on the observations is the same as the fiducial distribution. In addition, they proved that generalized confidence intervals/fiducial intervals of a scalar parameter have asymptotic frequentist coverage properties under mild conditions, and this is the first general result concerning the asymptotic behaviors of generalized confidence intervals and fiducial intervals.…”