Bootstrap - Example Suppose we have an estimator of a parameter and we want to express its accuracy by its standard error but its sampling distribution is too complicated to derive theoretically. A possible solution for this problem is to use Bootstrap – substitute computation for theory. STA261 week 5

Parametric Bootstrap Suppose data are realization of a random variable with a probability distribution with density fθ(x) with θ unknown. We begin the bootstrap process by first estimating θ from the data to get Next we simulate B “bootstrap samples” from the density fθ(x) with θ being replaced by and for each bootstrap sample we calculate a “bootstrap estimate” of θ denoted by Note that the bootstrap samples are always the same size as the original data set. The bootstrap estimate of the s.e. of is the sample standard deviation of the bootstrap estimates STA261 week 5

Example Consider a data set containing breakdown times of an isolative fluid between electrodes. The theoretical model for this data assumes that this is an i.i.d sample from an exponential distribution… The method of moment estimator of λ is…. We want the s.e of this estimator and for this we use parametric bootstrap. STA261 week 5

Empirical Distribution The empirical distribution is the estimate for the probability distribution that generated the data. The observed data are the possible values and are equally likely. The empirical distribution assign a probability of 1/n to each data value. STA261 week 5

Nonparametric Bootstrap If we could take an infinite number of samples of size n from the probability distribution that generated the data and for each sample find , we would know the sampling distribution of . In the nonparametric bootstrap procedure we get bootstrap samples of size n by re-sampling from the data. Re-sampling is sampling with replacement from this empirical distribution. STA261 week 5

Parametric Versus Nonparametric Bootstrap In the parametric bootstrap we have to make an assumption about the form of the distribution that generated the data Non-parametric – if n is small can behave oddly. STA261 week 5