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D. Krpelík and T. Basu
bounding the type I error in hypothesis testing procedures. Bayesian procedures
can only comply with these asymptotically and can be severely biased by the
information supplied through the prior distribution in cases when only a small
number of observations is available. Conversely, the results of Bayesian procedures,
the posterior distributions, can be propagated in a straightforward manner to obtain
assertions about the derived quantities, f (X). The problem of obtaining similar
distributional estimate in the frequentist framework was studied by Fisher in his
work on fiducial inference [23]. It later inspired the development of the well-known
theory of confidence intervals and hypothesis tests. Nevertheless, fiducial inference
has suffered from various, justified drawbacks and has mostly been forgotten by
mainstream statisticians. Several attempts had been made on the revival of ideas,
and it seems that in order to do so, the inferential results have to be modelled by IP
distributions instead of precise ones.
The recent advancements were enabled by Dempster [16, 17] by his development
of the evidence theory and its application to statistical problems. In [29], a statistical
framework for constructing random set structures, which can be used to obtain
valid confidence intervals on any level of significance and to conduct hypothesis
tests, is presented. The results of these inferences are generally random sets, beliefs
and plausibility functions which can be used to bound the inferences about the
investigated RV. The observations are modelled via a pivotal model, where we
assume that the observed value is a deterministic function of some ancillary RVs
with known distribution, thus extending [24, 47], who aimed at constructing precise
frequentist distributional estimates. Ryan et al. used this pivotal relation to propagate
a random set prediction of the ancillary RV, by which they obtained statistical
procedures with superior properties. Aside from that, they allow us to develop
derived methods for situations with additional knowledge, propagate the resulting
random sets to obtain assessments about derived quantities, and naturally analyse
imprecise observations without additional modelling assumptions.
As another example of a frequentist inferential method, we would like to
present the non-parametric predictive inference (NPI) [11]. The method assumes
exchangeability of the observations and bases its indifference principle on Hill’s
assumption—indifference among all possible orderings. Formally, after the n realvalued observations
x = {x 1 , . . . , x n }, the NPI constructs a predictive random set
for the next observation X n+1 by placing mass 1/(n + 1) on each of the intervals
(x i , x i+1 ), i = 0 . . . n, where x 0 , x n+1 are some bounds of the X n+1 support
(possibly infinite). The corresponding lower and upper probabilities may be derived
by Eqs. (2.12) and (2.13). It was shown that the result of the inference is an ∞monotone capacity and an F-probability [2]. The NPI has also been extended to
include situations with censored observations without the need of including any
restrictive censoring assumptions. An example of NPI lower and upper CDFs is
shown in Fig. 2.9 compared with the empirical distribution.
D. Krpelík and T. Basu
bounding the type I error in hypothesis testing procedures. Bayesian procedures
can only comply with these asymptotically and can be severely biased by the
information supplied through the prior distribution in cases when only a small
number of observations is available. Conversely, the results of Bayesian procedures,
the posterior distributions, can be propagated in a straightforward manner to obtain
assertions about the derived quantities, f (X). The problem of obtaining similar
distributional estimate in the frequentist framework was studied by Fisher in his
work on fiducial inference [23]. It later inspired the development of the well-known
theory of confidence intervals and hypothesis tests. Nevertheless, fiducial inference
has suffered from various, justified drawbacks and has mostly been forgotten by
mainstream statisticians. Several attempts had been made on the revival of ideas,
and it seems that in order to do so, the inferential results have to be modelled by IP
distributions instead of precise ones.
The recent advancements were enabled by Dempster [16, 17] by his development
of the evidence theory and its application to statistical problems. In [29], a statistical
framework for constructing random set structures, which can be used to obtain
valid confidence intervals on any level of significance and to conduct hypothesis
tests, is presented. The results of these inferences are generally random sets, beliefs
and plausibility functions which can be used to bound the inferences about the
investigated RV. The observations are modelled via a pivotal model, where we
assume that the observed value is a deterministic function of some ancillary RVs
with known distribution, thus extending [24, 47], who aimed at constructing precise
frequentist distributional estimates. Ryan et al. used this pivotal relation to propagate
a random set prediction of the ancillary RV, by which they obtained statistical
procedures with superior properties. Aside from that, they allow us to develop
derived methods for situations with additional knowledge, propagate the resulting
random sets to obtain assessments about derived quantities, and naturally analyse
imprecise observations without additional modelling assumptions.
As another example of a frequentist inferential method, we would like to
present the non-parametric predictive inference (NPI) [11]. The method assumes
exchangeability of the observations and bases its indifference principle on Hill’s
assumption—indifference among all possible orderings. Formally, after the n realvalued observations
x = {x 1 , . . . , x n }, the NPI constructs a predictive random set
for the next observation X n+1 by placing mass 1/(n + 1) on each of the intervals
(x i , x i+1 ), i = 0 . . . n, where x 0 , x n+1 are some bounds of the X n+1 support
(possibly infinite). The corresponding lower and upper probabilities may be derived
by Eqs. (2.12) and (2.13). It was shown that the result of the inference is an ∞monotone capacity and an F-probability [2]. The NPI has also been extended to
include situations with censored observations without the need of including any
restrictive censoring assumptions. An example of NPI lower and upper CDFs is
shown in Fig. 2.9 compared with the empirical distribution.