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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. - PowerPoint PPT Presentation
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STA261 week 5 1
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 2
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
.
.*
.ˆ...,,ˆ,ˆ **2
*1 B
STA261 week 5 3
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 4
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 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 6
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.