Bootstrapcondenceintervals2014JonathanLearningGoals.pdfVIP

Bootstrap con?dence intervals Class 24, 18.05, Spring 2014 Jeremy Orlo? and Jonathan Bloom 1 Learning Goals 1. Be able to construct and sample from the empirical distribution of data. 2. Be able to explain the bootstrap principle. 3. Be able to design and run an empirical bootstrap to compute con?dence intervals. 4. Be able to design and run a parametric bootstrap to compute con?dence intervals. 2 Introduction The empirical bootstrap is a statistical technique popularized by Bradley Efron in 1979. Though remarkably simple to implement, the bootstrap would not be feasible without modern computing power. The key idea is to perform computations on the data itself to estimate the variation of statistics that are themselves computed from the same data. That is, the data is ‘pulling itself up by its own bootstrap.’ (A google search of ‘by ones own bootstraps’ will give you the etymology of this metaphor.) Such techniques existed before 1979, but Efron widened their applicability and demonstrated how to implement the bootstrap e?ectively using computers. He also coined the term ‘bootstrap’ 1. Our main application of the bootstrap will be to estimate the variation of point estimates; that is, to estimate con?dence intervals. An example will make our goal clear. Example 1. Suppose we have data x1, x2, . . . , xn If we knew the data was drawn from N(μ, σ2) with the unknown mean μ and known variance σ2 then we have seen that x ? 1.96 √σ n , x + 1.96 √σ n is a 95% con?dence interval for μ. Now suppose the data is drawn from some completely unknown distribution. To have a name we’ll call this distribution F and its (unknown) mean μ. We can still use the sample mean x as a point estimate of μ. But how can we ?nd a con?dence interval for μ around x? Our answer will be to use the bootstrap! In fact, we’ll see that the bootstrap handles other statistics as easily as it handles the mean. For example: the median, other percentiles or the trimmed mean. These are statistics where, even