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Register a new userEpistemic confidence, the Dutch Book and relevant subsets
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Youngjo Lee
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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We use a logical device called the Dutch Book to establish epistemic confidence, defined as the sense of confidence emph{in an observed} confidence interval. This epistemic property is unavailable -- or even denied -- in orthodox frequentist inference. In financial markets, including the betting market, the Dutch Book is also known as arbitrage or risk-free profitable transaction. A numerical confidence is deemed epistemic if its use as a betting price is protected from the Dutch Book by an external agent. Theoretically, to construct the Dutch Book, the agent must exploit unused information available in any relevant subset. Pawitan and Lee (2021) showed that confidence is an extended likelihood, and the likelihood principle states that the likelihood contains all the information in the data, hence leaving no relevant subset. Intuitively, this implies that confidence associated with the full likelihood is protected from the Dutch Book, and hence is epistemic. Our aim is to provide the theoretical support for this intuitive notion.
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The recent paper Simple confidence intervals for MCMC without CLTs by J.S. Rosenthal, showed the derivation of a simple MCMC confidence interval using only Chebyshevs inequality, not CLT. That result required certain assumptions about how the estimator bias and variance grow with the number of iterations $n$. In particular, the bias is $o(1/sqrt{n})$. This assumption seemed mild. It is generally believed that the estimator bias will be $O(1/n)$ and hence $o(1/sqrt{n})$. However, questions were raised by researchers about how to verify this assumption. Indeed, we show that this assumption might not always hold. In this paper, we seek to simplify and weaken the assumptions in the previously mentioned paper, to make MCMC confidence intervals without CLTs more widely applicable.
Consider a linear regression model with independent and identically normally distributed random errors. Suppose that the parameter of interest is a specified linear combination of the regression parameters. We prove that the usual confidence interval for this parameter is admissible within a broad class of confidence intervals.
Consider X_1,X_2,...,X_n that are independent and identically N(mu,sigma^2) distributed. Suppose that we have uncertain prior information that mu = 0. We answer the question: to what extent can a frequentist 1-alpha confidence interval for mu utilize this prior information?
This was a revision of arXiv:1105.2454v1 from 2012. It considers a variation on the STIV estimator where, instead of one conic constraint, there are as many conic constraints as moments (instruments) allowing to use more directly moderate deviations for self-normalized sums. The idea first appeared in formula (6.5) in arXiv:1105.2454v1 when some instruments can be endogenous. For reference and to avoid confusion with the STIV estimator, this estimator should be called C-STIV.
The general problem of constructing confidence regions is unsolved in the sense that there is no algorithm that provides such a region with guaranteed coverage for an arbitrary parameter $psiinPsi.$ Moreover, even when such a region exists, it may be absurd in the sense that either the set $Psi$ or the null set $phi$ is reported with positive probability. An approach to the construction of such regions with guaranteed coverage and which avoids absurdity is applied here to several problems that have been discussed in the recent literature and for which some standard approaches produce absurd regions.
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