Bootstrap based confidence regions in multi-parameter problems

The importance of capturing uncertainty cannot be overemphasized when conducting scientific research. The International Committee of Medical Journal Editors in their guidelines for statistical reporting stated that: “When possible, quantify findings and present them with appropriate indicators of measurement error or uncertainty (such as confidence intervals)”, and “Avoid sole reliance on statistical hypothesis testing such as the use of P-values, which fails to convey important quantitative information.” (Bailar and Mosteller, 1988). Confidence intervals (CIs) are used to quantify the uncertainty of a point estimate associated with a single parameter. For multivariate situations, confidence regions (CRs) can be used to quantify uncertainty in the simultaneous estimation of a multiple parameter vector (Krishnamoorthy and Mathew, 2009). Many methods published in the statistical literature for the construction of CRs require distributional assumptions (e.g., multivariate normality) (Fuchs and Sampson, 1987). In practice,however, these assumptions are not always satisfied which led to development of non-parametric methods which relax assumptions required by parametric methods while still providing valid results. In particular, bootstrap methods exist for CI estimation for individual parameters (e.g., for each parameter in a linear or non-linear model). Currently, statistical software (including bootstrap software) only produces CIs for individual parameters.

This research aims to develop a methodology that extends the bootstrap technique by usingParticle Swarm Optimization (PSO) algorithms to construct CRs. The proposed PSO algorithm will generate a CR that is a minimum volume ellipsoid that captures 95% (or any confidence level) of these multi-parameter bootstrap estimates. The primary research outcomes are to: