Abstract

Bootstrap technique has an extensive use in almost all branches of statistics including regression analysis. The classical regression theory heavily depends on normality assumption of errors, which does not often hold in practice. Bootstrap technique can be readily applied to a situation when we do not assume any distribution for the error and that is why bootstrapping in regression is becoming very popular. In reality the true errors are not known, in that case we need a proper substitute of the residuals. The OLS residuals are being used as substitutes of errors from the very beginning but it is now evident that they may break down easily in the presence of unusual observations. In our bootstrap study we consider some other types of residuals such as PRESS and observed that it may produce results even worse than the OLS in the presence of outliers. Then we have considered deletion residuals based on robust fits. The usefulness and/or limitations of bootstrap techniques based on different types of residuals such as PRESS and deletion residuals are studied through some real life examples and a Monte Carlo simulation experiments under a variety of situations.

摘要

Bootstrap 技术在包括回归分析在内的几乎所有统计分支中都有广泛的应用。经典回归理论在很大程度上依赖于错误的正态性假设，这在实践中并不经常成立。Bootstrap 技术可以很容易地应用于我们不假设任何错误分布的情况，这就是回归中的 bootstrapping 变得非常流行的原因。实际上，真正的错误是未知的，在这种情况下，我们需要残差的适当替代。OLS 残差从一开始就被用作误差的替代品，但现在很明显，在出现不寻常的观察结果时，它们很容易失效。在我们的 bootstrap 研究中，我们考虑了一些其他类型的残差，例如 PRESS，并观察到它可能会在存在异常值的情况下产生比 OLS 更差的结果。然后我们考虑了基于稳健拟合的删除残差。通过一些现实生活中的例子和各种情况下的蒙特卡罗模拟实验，研究了基于不同类型残差（例如 PRESS 和删除残差）的自举技术的有用性和/或局限性。