Confidence Intervals

PowerPoint slides for Confidence Intervals. Examples are taken from the Medical Literature

Normal distribution and confidence intervals

lecture for COMP6235

Confidence sets for asset correlations in portfolio credit risk

Asset correlations are of critical importance in quantifying portfolio credit risk and economic capitalin financial institutions. Estimation of asset correlation with rating transition data has focusedon the point estimation of the correlation without giving any consideration to the uncertaintyaround these point estimates. In this article we use Bayesian methods to estimate a dynamicfactor model for default risk using rating data (McNeil et al., 2005; McNeil and Wendin, 2007).Bayesian methods allow us to formally incorporate human judgement in the estimation of assetcorrelation, through the prior distribution and fully characterize a confidence set for the correlations.Results indicate: i) a two factor model rather than the one factor model, as proposed bythe Basel II framework, better represents the historical default data. ii) importance of unobservedfactors in this type of models is reinforced and point out that the levels of the implied asset correlationscritically depend on the latent state variable used to capture the dynamics of default,as well as other assumptions on the statistical model. iii) the posterior distributions of the assetcorrelations show that the Basel recommended bounds, for this parameter, undermine the levelof systemic risk.

Point estimates and confidence regions for sequential trials involving selection

A number of authors have proposed clinical trial designs involving the comparison of several experimental treatments with a control treatment in two or more stages. At the end of the first stage, the most promising experimental treatment is selected, and all other experimental treatments are dropped from the trial. Provided it is good enough, the selected experimental treatment is then compared with the control treatment in one or more subsequent stages. The analysis of data from such a trial is problematic because of the treatment selection and the possibility of stopping at interim analyses. These aspects lead to bias in the maximum-likelihood estimate of the advantage of the selected experimental treatment over the control and to inaccurate coverage for the associated confidence interval. In this paper, we evaluate the bias of the maximum-likelihood estimate and propose a bias-adjusted estimate. We also propose an approach to the construction of a confidence region for the vector of advantages of the experimental treatments over the control based on an ordering of the sample space. These regions are shown to have accurate coverage, although they are also shown to be necessarily unbounded. Confidence intervals for the advantage of the selected treatment are obtained from the confidence regions and are shown to have more accurate coverage than the standard confidence interval based upon the maximum-likelihood estimate and its asymptotic standard error.

Hitting emissions targets with (statistical) confidence in multi-instrument Emissions Trading Schemes

A means of assessing, monitoring and controlling aggregate emissions from multi-instrument Emissions Trading Schemes is proposed. The approach allows contributions from different instruments with different forms of emissions targets to be integrated. Where Emissions Trading Schemes are helping meet specific national targets, the approach allows the entry requirements of new participants to be calculated and set at a level that will achieve these targets. The approach is multi-levelled, and may be extended downwards to support pooling of participants within instruments, or upwards to embed Emissions Trading Schemes within a wider suite of policies and measures with hard and soft targets. Aggregate emissions from each instrument are treated stochastically. Emissions from the scheme as a whole are then the joint probability distribution formed by integrating the emissions from its instruments. Because a Bayesian approach is adopted, qualitative and semi-qualitative data from expert opinion can be used where quantitative data is not currently available, or is incomplete. This approach helps government retain sufficient control over emissions trading scheme targets to allow them to meet their emissions reduction obligations, while minimising the need for retrospectively adjusting existing participants’ conditions of entry. This maintains participant confidence, while providing the necessary policy levers for good governance.

Depression, confidence, and decision: evidence against depressive realism

This research examined how retrospective self-assessments of performance are affected by major depression. To test the validity of the depressive realism versus the selective processing hypotheses, aggregate posttest performance estimates (PTPEs) were obtained from clinically depressed patients and an age-matched comparison group across 4 decision tasks (object recognition, general knowledge, social judgment, and line-length judgment). As expected on the basis of previous findings, both groups were underconfident in their PTPEs, consistently underestimating the percentage of questions they had answered correctly. Contrary to depressive realism, and in partial support of the selective processing account, this underconfidence effect was not reduced but modestly exacerbated in the depressed patients. Further, whereas the PTPEs of the comparison group exceeded that expected on the basis of chance alone those of the depressed individuals did not. The results provide no support for the depressive realism account and suggest that negative biases contribute to metacognitive information processing in major depression.

A method for assessing confidence in requirements analysis

Context: During development managers, analysts and designers often need to know whether enough requirements analysis work has been done and whether or not it is safe to proceed to the design stage. Objective: This paper describes a new, simple and practical method for assessing our confidence in a set of requirements. Method: We identified 4 confidence factors and used a goal oriented framework with a simple ordinal scale to develop a method for assessing confidence. We illustrate the method and show how it has been applied to a real systems development project. Results: We show how assessing confidence in the requirements could have revealed problems in this project earlier and so saved both time and money. Conclusion: Our meta-level assessment of requirements provides a practical and pragmatic method that can prove useful to managers, analysts and designers who need to know when sufficient requirements analysis has been performed.