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Unformatted text preview: Bootstrap 1 RESAMPLING AND PORTFOLIO ANALYSIS Introduction Computer simulation is widely used in OR. Applications of simulation to statistics are widespread. Topic of this chapter: simulation technique called the bootstrap or resampling Will study the effects of estimation error on portfolio selection. bootstrap from the phrase pulling oneself up by ones bootstraps bootstrap = resampling Bootstrap 2 Statistics from a random sample are random variables. The sample is only one of many possible samples. Each possible sample gives a possible value of X . We only see one value of X * But it was selected at random from the many possible values. Thus, X is a random variable. Bootstrap 3 Confidence intervals and hypothesis tests are based on the randomness of statistics. Example: confidence coefficient tells us the probability that an interval constructed from a random sample will contain the parameter. Confidence intervals are sometimes derived using probability theory. Often the necessary probability calculations are intractable. In that case we can replace theoretical calculations by Monte Carlo simulation. Bootstrap 4 How do we simulate sampling from an unknown population? We cannot do this exactly. However, a sample is a good representative of the population. We can simulate sampling from the population by sampling from the sample. this is usually called resampling Bootstrap 5 X X X X X X X X X X Population Sample Resample Resample Resample Resample Resample . . . . Samples not taken . . . Bootstrap 6 Each resample has the same sample size, n , as the original sample. We are trying to simulate the original sampling, We want the resampling to be as similar as possible to the original sampling. Bootstrap 7 The resamples are drawn with replacement . Only sampling with replacement give independent observations. We want the resamples to be i.i.d. just like the original sample. If the resamples were drawn without replacement then every resample would be exactly the same as the original sample. So the resamples would show no random variation. This wouldnt be very satisfactory, of course. Bootstrap 8 The number of resamples taken should be large. Just how large depends on context and will be discussed more fully later. Sometimes, tens of thousands of resamples are taken. We will let B denote the number of resamples. Bootstrap 9 There is some good news and some bad news about the bootstrap. The good news is that computer simulation replaces difficult mathematics. Bootstrap 10 The bad news is that resampling is a new and unfamiliar concept....
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 Spring '07
 ANDERSON

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