Exploiting Stein's paradox in analysing sparse data from genome-wide association studies

Unbiased estimation appeared to be an accepted golden standard of statistical analysis ever until the Stein's discovery of a surprising phenomenon attributable to multivariate spaces. So called Stein's paradox arises in estimating the mean of a multivariate standard normal random variable. Stein showed that both natural and intuitive estimate of a multivariate mean given by the observed vector itself is not even admissible and may be improved upon under the squared-error loss when the dimension is greater or equal to three. Later Stein and his student James developed so called ‘James–Stein estimator’, a shrunken estimate of the mean, which had uniformly smaller risk for all values in the parameter space. The paradox first appeared both unintuitive and even unacceptable, but later it was recognised as one of the most influential discoveries of all times in statistical science. Today the ‘shrinkage principle’ literally permeates the statistical technology for analysing multivariate data, and in its application is not exclusively confined to estimating the mean, but also the covariance structure of multivariate data. We develop shrinkage versions of both the linear and quadratic discriminant analysis and apply them to sparse multivariate gene expression data obtained at the Centre for Biomedical Informatics (CBI) in Prague.