| The dissertation explores the extension of the James-Stein estimator in a direction that enables it to preserve its superiority when the sample size goes to infinity. The first chapter develops the theoretical foundation for the extension. Instead of shrinking a base estimator towards a fixed point, we shrink towards a data-dependent point, which makes it possible that the prior becomes more accurate as the sample size grows. We prove that the extended James-Stein estimator shrunk towards a data-dependent point has smaller asymptotic risk than the base estimator. It turns out that shrinking an estimator toward a data-dependent point is equivalent to combining two random variables using the James-Stein rule. We propose a general combination scheme which includes random combination and non-random combination as special cases. The result allows us to apply the extended James-Stein estimator to robust regression, especially to the Least Absolute Deviations Estimator. We show analytically and by simulation that if we shrink the LAD Estimator, then we have smaller risk.; The second chapter provides a way to obtain the sampling distributions and confidence intervals for the James-Stein type combination estimator using a bootstrapping method. It is well known that in order to get a better bootstrap confidence interval, one should use pivotal or asymptotically pivotal statistics. We use Ullah (1990)'s results to derive the first moment and the second moment. We use a consistent estimator for this asymptotic variance to obtain the bootstrapping pivotal statistics.; The third applies shrinkage estimation to the construction of optimal portfolios as proposed by Treynor and Black (1973) using alpha and beta forecast data from a financial institution. The Treynor Black model provides a method to exploit security analysis; however, its success depends critically on both the predictive ability of abnormal return forecasts and the conversion of this predictive power into the portfolio construction. We use the OLS estimator, the LAD estimator and shrinkage LAD estimators to extract predictive ability from raw alpha forecasts. The estimated correlation between ex-post abnormal returns and alpha forecasts is as low as 0.04, yet out-of-sample experiments show that the use of robust estimators can yield superior portfolios for the forecast database. |
